CONTINUOUS-TIME ASSET PRICING THEORY a martingale-based approach. [2 ed.] 9783030744106, 3030744108

Citation preview

Springer Finance Textbook

Robert A. Jarrow

Continuous-Time Asset Pricing Theory A Martingale-Based Approach Second Edition

Springer Finance Springer Finance Textbooks

Series Editors Marco Avellaneda, New York University, New York, NY, USA Giovanni Barone-Adesi, Universita della Svizzera Italiana, Lugano, Switzerland Francesca Biagini, Munich University of Applied Sciences, Munich, Germany Bruno Bouchard, Paris Dauphine University, PARIS, France Mark Broadie, Columbia University, New York, NY, USA Emanuel Derman, Columbia University, New York, NY, USA Paolo Guasoni, Dublin City University, Dublin, Ireland Mathieu Rosenbaum, École Polytechnique, Palaiseau, France

This subseries of Springer Finance consists of graduate textbooks.

More information about this subseries at http://www.springer.com/series/11355

Robert A. Jarrow

Continuous-Time Asset Pricing Theory A Martingale-Based Approach Second edition

Robert A. Jarrow Samuel Curtis Johnson Graduate School Cornell University Ithaca, NY, USA

ISSN 1616-0533 ISSN 2195-0687 (electronic) Springer Finance Springer Finance Textbooks ISBN 978-3-030-74409-0 ISBN 978-3-030-74410-6 (eBook) https://doi.org/10.1007/978-3-030-74410-6 Mathematics Subject Classification: 91-XX, 91Bxx, 91Gxx, 62-XX, 62P05, 62P20 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

This book is dedicated to my wife, Gail.

Preface

The fundamental paradox of mathematics is that abstraction leads to both simplicity and generality. It is a paradox because generality is often thought of as requiring complexity, but this is not true. This insight explains both the beauty and power of mathematics.

Philosophy My philosophy in creating models for practice and for understanding is based on two simple principles: 1. Always impose the least restrictive set of assumptions possible to achieve maximum generality. 2. When choosing among assumptions, it is better to impose an assumption that is observable and directly testable versus an assumption that is unobservable and only indirectly testable. This philosophy affects the content of this book.

The Key Topics Finance’s asset pricing theory has three topics that uniquely identify it: 1. Arbitrage pricing theory, including derivative valuation/hedging and multiplefactor beta models 2. Portfolio theory, including equilibrium pricing 3. Market informational efficiency

vii

viii

Preface

These three topics are listed in order of increasing structure (set of assumptions), from the general to the specific. In some sense, topic 3 requires less structure than topic 2 because market efficiency only requires the existence of an equilibrium, not a characterization of the equilibrium. But, because to understand the existence of an equilibrium one needs to first understand what an equilibrium is, topic 2 comes first. The more assumptions imposed, the less likely the structure depicts reality. Of course, this depends crucially on whether the assumptions are true or false. If the assumptions are true, then no additional structure is being imposed when an assumption is added. But in reality, all assumptions are approximations, therefore all assumptions are in some sense “false.” This means, of course, that the less assumptions imposed, the more likely the model is to be “true.”

The Key Insights There are at least nine important insights from asset pricing theory that need to be understood. These insights are obtained from the three fundamental theorems of asset pricing. The insights are enriched by the use of preferences, characterizing an investor’s optimal portfolio decision, and the notion of an equilibrium. These nine insights are listed below: 1. The existence of a state price density or an equivalent local martingale measure (first fundamental theorem) 2. Hedging and exact replication (second fundamental theorem) 3. The risk-neutral valuation of derivatives (third fundamental theorem) 4. Asset price bubbles (third fundamental theorem) 5. Spanning portfolios (mutual fund theorems) (third fundamental theorem) 6. The meaning of systematic versus idiosyncratic risk (third fundamental theorem) 7. The meaning of diversification (third fundamental theorem and the law of large numbers) 8. The importance of the market portfolio (portfolio optimization and equilibrium) 9. Market informational efficiency (third fundamental theorem and equilibrium) Insight 1 requires the first fundamental theorem. Insight 2 requires the second fundamental theorem. Insights 3–7 require the first and third fundamental theorems of asset pricing. Insight 7 also requires the law of large numbers. Insight 8 requires portfolio optimization and the notion of an equilibrium. Finally, insight 9 requires the third fundamental theorem and the notion of an equilibrium. There are three important aspects of insights 1–9 that need to be emphasized. The first is that all of these insights are derived in incomplete markets, including markets with trading constraints. The second is that all of these insights are derived for discontinuous sample path processes, i.e., asset price processes that contain jumps. The third is that all of these insights are derived in models where traders have heterogeneous beliefs,

Preface

ix

and in certain subcases, differential information as well. As such, these insights are very robust and relevant to financial practice. All of these insights are explained in detail in this book.

The Martingale Approach The key topics of asset pricing theory have been studied, refined, and extended for over 40 years, starting in the 1970s with the capital asset pricing model (CAPM), the notion of market efficiency, and option pricing theory. Much knowledge has been accumulated and there are many different approaches that can be used to present this material. Consistent with my philosophy, I choose the most abstract yet the simplest and most general approach for explaining this topic. This is the martingale approach to asset pricing theory—the unifying theme is the notion of an equivalent local martingale probability measure (and all of its extensions). This theme can be used to understand and to present the known results from arbitrage pricing theory up to, and including, portfolio optimization and equilibrium pricing. The more restrictive historical and traditional approach based on dynamic programming and Markov processes is left to the classical literature.

Discrete Versus Continuous-Time There are three model structures that can be used to teach asset pricing: 1. Static (single period) 2. Discrete time and multiple periods 3. Continuous-time Static models are really only useful for pedagogical purposes. The math is simple and the intuition easy to understand. They do not apply in practice/reality. Consistent with my philosophy, this reduces the model structure choice to only two for this book, between discrete time multiple periods and continuous-time models. We focus on continuous-time models in this book because they are the better model structure for matching reality for three reasons. One, a discrete time model implies that one can only trade on the grid represented by the discrete time points. This is not true in practice because one can trade at any time during the day. Second, trading times are best modeled as a finite (albeit very large) sequence of random times on a continuous-time interval. It is a very large finite sequence because with computer trading, the time between two successive trades is very small (milli- and even microseconds). This implies that the limit of a sequence of random times on a continuous-time interval should provide a reasonable approximation. This is, of course, continuous trading. Three, continuous-time has a number of phenomena that are not present in discrete time models—the most

x

Preface

important of which are strict local martingales (see Jarrow and Protter [107]). Strict local martingales will be shown to be important in understanding asset price bubbles in the subsequent theory.

Mean-Variance Efficiency and the Static CAPM As an epilogue to Part III of this book, its last chapter studies the static CAPM. The static CAPM is studied after the dynamic continuous-time model to emphasize the omissions of a static model and the important insights obtained in dynamic models. This is done because the static model is not a good approximation to actual security markets. This book only briefly discusses the mean-variance efficient frontier. Consequently, an in depth study of this material is left to independent reading (see Back [5], Duffie [52], Skiadas [179]). Generalizations of the static CAPM to continuous-time—the intertemporal CAPM due to Merton [144] and the consumption CAPM due to Breeden [22]—are included as special cases of the models presented in this book.

Stochastic Calculus Finance is an application of stochastic process and optimization theory. Stochastic processes because asset prices evolve randomly across time. Optimization because investors trade to maximize their preferences. Hence, this mathematics is essential to developing the theory. This book is not a mathematics book, but an economics book. The math is not emphasized, but used to obtain results. The emphasis of the book is on the economic meaning and implications of assumptions and results. The proofs of most results are included within the text, except those that require a knowledge of functional analysis. Most of the excluded proofs are related to “existence results,” examples include the first fundamental theorem of asset pricing and the existence of a saddle point in convex optimization. For those proofs not included, references are provided. The mathematics assumed in this book is that obtained from a first-level graduate course in real analysis and probability theory, sources of this knowledge include Ash [3], Billingsley [13], Jacod and Protter [76], and Klenke [129]. Excellent references for stochastic calculus include Karatzas and Shreve [123], Medvegyev [143], Protter [158], Roger and Williams [164], and Shreve [177], while those for optimization include Borwein and Lewis [19], Guler [66], Leunberger [140], Ruszczynski [169], and Pham [156].

Preface

xi

ASSET PRICING THEORY: TRADITIONAL VERSUS MARKET MICROSTRUCTURE Asset Pricing Theory 1. Continuous/discrete time 2. Frictionless/frictions 3. Equal/differential beliefs 4. Equal/differential information (Traditional) Competitive markets ⇐⇒Walrasian equilibrium (Market Microstructure) Competitive markets ⇐⇒Nash equilibrium or zero expected profit

Traditional Asset Pricing Theory Versus Market Microstructure Although the distinction between traditional asset pricing theory and market microstructure is not “black and white,” one useful classification of the difference between these two fields is provided in the previous table. In this classification, traditional asset pricing theory and market microstructure have in common the structures (1)–(4). They differ in the meaning of a competitive market, in particular, the notion of an equilibrium. Traditional asset pricing uses the concept of a Walrasian equilibrium (supply equals demand, price takers) whereas market microstructure uses Nash equilibrium or a zero expected profit condition (strategic traders, non price takers). This difference is motivated by the questions that each literature addresses. Asset pricing abstracts from the mechanism under which trades are executed. Consequently, it assumes that investors are price takers whose trades have no quantity impact on the price. This literature focuses on characterizing the price process, optimal trading strategies, and risk premium. In contrast, the market microstructure literature seeks to understand the trade execution mechanism itself, and its impact on market welfare. This alternative perspective requires a different equilibrium notion, one that explicitly incorporates strategic trading. This book presents asset pricing theory using the traditional representation of market clearing. For a book that reviews the market microstructure literature see O’Hara [154].

xii

Preface

Themes The themes in this book differ from those contained in most other asset pricing books in four notable ways. First, the emphasis is on price processes that include jumps, not just continuous diffusions. Second, stochastic optimization is based on martingale methods using convex analysis and duality, and not diffusion processes with stochastic dynamic programming. Third, asset price bubbles are an important consideration in every topic presented herein. Fourth, the existence and characterization of economic equilibrium is based on the use of a representative trader, which significantly simplifies the proof of many results. Other excellent books on asset pricing theory, using the more traditional approach to the topic, include Back [5], Bjork [14], Dana and Jeanblanc [42], Duffie [52], Follmer and Schied [63], Huang and Litzenberger [73], Ingersoll [75], Karatzas and Shreve [124], Merton [147], Pliska [157], and Skiadas [179].

Changes to the Second Edition Besides correcting typos and uniformly improving the exposition of the text and proofs, the second edition significantly modifies the presentation and adds more content to Chaps. 1–5, 9, 14, 16, and 17. Chapter 1 includes new material on the uniqueness of the stochastic integral representation for a stochastic process. Chapter 2 adds two topics: the definition of a non-redundant asset market and suicide trading strategies. Chapter 3 integrates suicide trading strategies into the understanding of asset price bubbles, greatly enriching the presentation. Because Chaps. 1–3 are the basis of the models used in the remainder of the book, the additions to these Chaps. 1–3 result in enhancements in all of the subsequent chapters. New terminology related to basis assets and risk factors is now included in Chap. 4, which clarifies the various topics discussed. The original derivation of the Black-Scholes-Merton formula and Merton’s structural model for credit risk are now included in Chap. 5. Characterizations of strict concavity of a stochastic utility function and state independence of utility functions are added to Chap. 8. These characterizations are new to the literature. Chapter 14 on the representative trader provides two new lemmas which simplify the hypotheses of all the theorems involving a representative trader in that chapter. Chapter 16 is significantly updated to include a more general presentation of market efficiency. Again, the results included therein are new to the literature as well. Finally, the static model in Chap. 17 is expanded to include a discussion of the fundamental theorems of asset pricing. Acknowledgment I am grateful for a lifetime of help and inspiration from family, colleagues, and students.

Ithaca, NY, USA

Robert A. Jarrow

Contents

Part I Arbitrage Pricing Theory 1

Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Stochastic Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Quadratic Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Uniqueness of the Stochastic Integral Representation . . . . . . . . . . . . 1.5 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Ito’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Girsanov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Essential Supremum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Optional Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Martingale Representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Equivalent Probability Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 10 13 16 17 17 18 18 19 19 20 20

2

The Fundamental Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Trading Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Admissibility and Doubling Strategies . . . . . . . . . . . . . . . . . . . 2.1.3 Suicide Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 The Frictionless Market Assumption. . . . . . . . . . . . . . . . . . . . . 2.2 Change of Numeraire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Cash Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Reinvest in the MMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Reinvest in the Risky Asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Non-redundant Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The First Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 No Arbitrage (NA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 No Unbounded Profits with Bounded Risk (NUPBR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 22 24 28 31 31 33 34 34 35 36 37 38 41 xiii

xiv

Contents

2.5.3 Properties of Dl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 No Free Lunch with Vanishing Risk (NFLVR) . . . . . . . . . . 2.5.5 The First Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.6 Equivalent Local Martingale Measures . . . . . . . . . . . . . . . . . . 2.5.7 The State Price Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The Second Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Attainable Securities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Complete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 The Third Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Risk Neutral Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Synthetic Derivative Construction . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Finite Dimension Brownian Motion Market. . . . . . . . . . . . . . . . . . . . . . . 2.8.1 The Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 NFLVR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.3 Complete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.4 ND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 46 47 48 49 50 51 52 58 62 64 65 65 66 70 72 73 73

3

Asset Price Bubbles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Market Price and Fundamental Value . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Asset Price Bubble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Complete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Incomplete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Theorems Under NFLVR and ND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 76 78 78 83 86 90

4

Basis Assets, Multiple-Factor Beta Models, and Systematic Risk . . . . . 4.1 The Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Basis Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Multiple-Factor Beta Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Positive Alphas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 The State Price Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Arrow Debreu Securities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Systematic Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Diversification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 91 93 96 98 99 100 101 103 107

5

The Black Scholes Merton Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 NFLVR, Complete Markets, and ND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The BSM Call Option Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Synthetic Call Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Original Derivation of the BSM Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Merton’s Structural Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109 109 111 114 115 116 118

Contents

xv

6

The Heath Jarrow Morton Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Term Structure Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Arbitrage-Free Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Ho and Lee Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Lognormally Distributed Forward Rates . . . . . . . . . . . . . . . . . 6.4.3 Vasicek Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Cox Ingersoll Ross Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.5 Affine Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Forward and Futures Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Forward Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Futures Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 The Libor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 119 120 124 129 130 130 131 132 133 133 134 137 139 143

7

Reduced Form Credit Risk Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Risky Firm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Existence of an Equivalent Martingale Measure . . . . . . . . . . . . . . . . . . 7.4 Risk Neutral Valuation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Cash Flow 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Cash Flow 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Cash Flow 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.4 Cash Flow 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Coupon Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Credit Default Swaps (CDS). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.3 First-to-Default Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 145 146 148 151 152 152 152 153 155 155 156 157 159

8

Incomplete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 The Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Super-Replication Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Super-Replication Trading Strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 The Sub-replication Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

161 161 161 163 165 166

Part II Portfolio Optimization 9

Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Preference Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 State Dependent EU Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Rationality Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Additional Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Risk Aversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169 169 172 172 174 175

xvi

Contents

9.3

Strict Concavity and Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Independent Gambles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Risk Aversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Characterization Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measures of Risk Aversion for Independent Gambles . . . . . . . . . . . . State Dependent Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conjugate Duality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reasonable Asymptotic Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differential Beliefs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

178 179 179 180 182 185 187 189 191 192

10

Complete Markets (Utility Over Terminal Wealth). . . . . . . . . . . . . . . . . . . . . 10.1 The Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Existence of a Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Characterization of the Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 The Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 The Shadow Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 The Local Martingale Deflator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 The Optimal Trading Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.1 The Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.2 The Utility Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.3 The Optimal Wealth Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.4 The Optimal Trading Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.5 The Value Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193 193 194 199 200 200 203 203 204 206 207 207 208 208 209 209 210 210

11

Incomplete Markets (Utility Over Terminal Wealth) . . . . . . . . . . . . . . . . . . . 11.1 The Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Existence of a Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Characterization of the Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 The Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 The Shadow Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 The Supermartingale Deflator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 The Optimal Trading Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8.1 The Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8.2 The Utility Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8.3 The Optimal Supermartingale Deflator . . . . . . . . . . . . . . . . . . 11.8.4 The Optimal Wealth Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

213 213 214 222 224 224 228 229 229 231 232 232 234 234 235

9.4 9.5 9.6 9.7 9.8 9.9

Contents

12

xvii

11.8.5 The Optimal Trading Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8.6 The Value Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9 Differential Beliefs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.10 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

236 237 237 239 240

Incomplete Markets (Utility Over Intermediate Consumption and Terminal Wealth) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 The Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Existence of a Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Characterization of the Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Utility of Consumption (U2 ≡ 0) . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Utility of Terminal Wealth (U1 ≡ 0) . . . . . . . . . . . . . . . . . . . . . 12.4.3 Utility of Consumption and Terminal Wealth. . . . . . . . . . . . 12.5 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

243 243 246 252 255 255 263 264 267 267

Part III Equilibrium 13

Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 The Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.1 Supply of Shares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.2 Traders in the Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.3 Aggregate Market Wealth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.4 Trading Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.5 An Economy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Intermediate Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.1 Supply of the Consumption Good . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Demand for the Consumption Good . . . . . . . . . . . . . . . . . . . . . 13.4.3 An Economy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

271 271 272 272 274 275 275 276 276 280 280 281 281 282

14

A Representative Trader Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 The Aggregate Utility Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 The Portfolio Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Representative Trader Economy Equilibrium . . . . . . . . . . . . . . . . . . . . . 14.4 Pareto Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Existence of an Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Uniqueness of the Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.1 Uniqueness of the Equilibrium Price Process . . . . . . . . . . . . 14.6.2 Uniqueness of the Supermartingale Deflators . . . . . . . . . . . 14.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7.1 Identical Traders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7.2 Logarithmic Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

283 283 290 294 301 304 309 309 309 311 311 312

xviii

Contents

14.8 14.9

Intermediate Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

15

Characterizing the Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 The Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 The Supermartingale Deflator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Asset Price Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1 Complete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.2 Incomplete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Systematic Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Consumption CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Intertemporal CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 Intermediate Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7.1 Systematic Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7.2 Consumption CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7.3 Intertemporal CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.8 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

317 317 318 320 321 321 322 323 324 326 326 327 327 328

16

Market Informational Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 The Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 The Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 The Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Information Sets and Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 Testing for Market Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.1 Profitable Trading Strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.2 Positive Alphas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.3 Asset Price Evolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6 Random Walks and Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6.1 The Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6.2 Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6.3 Market Efficiency  Random Walk . . . . . . . . . . . . . . . . . . . . . 16.6.4 Random Walk  Market Efficiency . . . . . . . . . . . . . . . . . . . . . 16.7 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

329 329 332 334 336 337 337 337 338 339 339 339 340 342 343

17

Epilogue (The Fundamental Theorems and the CAPM) . . . . . . . . . . . . . . . 17.1 The Fundamental Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.1 The First Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.2 The Second Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . 17.1.3 Risk Neutral Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.4 Finite State Space Market. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Basis Assets, Multi-Factor Beta Models, and Systematic Risk . . . 17.3 Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.1 The Dual Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.2 The Primal Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.3 The Optimal Trading Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . .

345 345 348 349 352 353 355 359 360 365 367 367

Contents

17.5 17.6

Beta Model (Revisited) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Efficient Frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.6.1 The Solution (Revisited) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.6.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.6.3 The Risky Asset Frontier and Efficient Frontier . . . . . . . . . 17.7 Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.8 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xix

370 371 371 372 373 373 378 378

Part IV Trading Constraints 18

The Trading Constrained Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 The Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Trading Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Support Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4 Examples (Trading Constraints and Their Support Functions) . . . 18.4.1 No Trading Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4.2 Prohibited Short Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4.3 No Borrowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4.4 Margin Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5 Wealth Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

389 389 390 392 394 394 395 395 396 397

19

Arbitrage Pricing Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1 No Unbounded Profits with Bounded Risk (NUPBRC ). . . . . . . . . . . 19.2 No Free Lunch with Vanishing Risk (NFLVRC ) . . . . . . . . . . . . . . . . . . 19.3 Asset Price Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4 Systematic Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

401 401 402 403 404

20

The Auxiliary Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 20.1 The Auxiliary Markets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 20.2 The Normalized Auxiliary Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

21

Super- and Sub-Replication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 The Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Auxiliary Market (0, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Auxiliary Markets (ν0 , ν) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Local Martingale Deflators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Wealth Processes Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Super-Replication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Sub-Replication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

411 411 411 412 412 413 415 416

22

Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1 The Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Wealth Processes (Revisited) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 The Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Existence of a Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.5 Characterization of the Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

419 419 421 422 424 425

xx

Contents

22.6 22.7 22.8 22.9 22.10 23

The Shadow Price of the Budget Constraint . . . . . . . . . . . . . . . . . . . . . . . The Supermartingale Deflator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Shadow Prices of the Trading Constraints . . . . . . . . . . . . . . . . . . . . Asset Price Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Systematic Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

425 426 427 427 429

Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.1 The Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2 Representative Trader . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2.1 The Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2.2 Buy and Hold Trading Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 23.3 Existence of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.4 Characterization of Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

433 433 434 435 436 437 438

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

List of Notation

For easy reference, this section contains the notation used consistently throughout the book. Notation that is used only in isolated chapters is omitted from this list, but complete definitions are included within the text. •

x = (x1 , . . . , xn ) ∈ Rn where the prime denotes transpose is a column vector.

n×1

• t ∈ [0, T ] represents time in a finite horizon and continuous-time model. • (Ω, F , F, P) is a filtered probability space on [0, T ] with F = FT where Ω is the state space, F is a σ -algebra, F = (Ft )t∈[0,T ] is a filtration, and P is a probability measure on Ω. • E [·] is expectation under the probability measure P. • E Q [·] is expectation under the probability measure Q given (Ω, F , Q) where Q = P. • Q ∼ P means that the probability measure Q is equivalent to P. • rt is the default-free spot rate of interest. t • Bt = e 0 rs ds , B0 = 1 is the value of a money market account. • St = (S1 (t), . . . , Sn (t)) ≥ 0 represents the prices of n of risky assets (stocks), semimartingales, adapted to Ft . t • Bt := B Bt = 1 for all t ≥ 0 represents the normalized value of the money market account. • St = (S1 (t), . . . , Sn (t)) ≥ 0 represents prices when normalized by the value of i (t) the money market account, i.e. Si (t) = SB(t) . • (S, F, P) is a market. • B(0, ∞) is the Borel σ -algebra on (0, ∞). • L0 := L0 (Ω, F , P) is the space of all FT -measurable random variables. • L0+ := L0+ (Ω, F , P) is the space of all nonnegative FT -measurable random variables. • L1+ (P) := L1+ (Ω, F , P) is the space of all nonnegative FT -measurable random variables X such that E [X] < ∞. • O is the set of optional stochastic processes. • L (S) is the set of predictable processes integrable with respect to S. • L(B) to be the set of optional processes that are integrable with respect to B. xxi

xxii

List of Notation

• L 0 is the set of adapted, right continuous with left limit existing (cadlag) stochastic processes. • L+0 is the set of adapted, right continuous with left limit existing (cadlag) stochastic processes that are nonnegative. A (x) = {(α0 , α) ∈ (O, L (S)) : Xt = α0 (t) + αt ·St , ∃c ≤ 0, t • Xt = x + 0 αu · dSu ≥ c, ∀t ∈ [0, T ] is the set of admissible, self-financing trading strategies. • M = {Q ∼ P : S is a Q martingale} is the set of martingale measures. • Ml = {Q ∼ P : S is a Q local martingale}  = {Q ∼ P : X isaQlocalmartingale, X = 1+ α ·dS, (α0 , α) ∈ A (1)} is the set of local martingale measures. • Ms = {Q ∼ P : S is a Q supermartingale} is the set of supermartingale measures.  Dl = Y ∈ L+0 : Y0 = 1, XY is a P local martingale,  • X = 1 + α · dS, (α0 , α) ∈ A (1) is the set of local martingale deflator processes. • Dl = {YT ∈ L0+ : ∃Z ∈ Dl , YT = ZT } is the set of local martingale deflators. is the set of local martingale deflator • Ml = Y ∈ Dl : ∃Q ∼ P, YT = dQ dP processes  with respect to P.  generated by a probability density • Ml = YT ∈ L0+ : ∃Z ∈ Ml , YT = ZT is the set of local martingale deflators that are probability densities with respect to P. • M = {Y ∈ L+0 : YT = dQ dP , Yt = E [YT |Ft ] , Q ∈ M} = {Y ∈ L+0 : Y ∈ Ml , YT = dQ dP , Q ∈ M} is the set of martingale deflator processes generated by martingale measures. • M = {YT ∈ L0+ : YT = dQ dP , Q ∈ M} 0 = {Y ∈ L+ : ∃Z ∈ M , YT = ZT } is the set of martingale deflators generated by martingale measures. N (x) = {(α0 , α) ∈ (O, L (S)) : Xt = α0 (t) + αt · St , t • Xt = x + 0 αu · dSu ≥ 0, ∀t ∈ [0, T ] is the  set of nonnegative wealth, self-financing trading strategies. Ds = Y ∈ L+0 : Y0 = 1, XY is a P supermartingale,  • X = 1 + α · dS, (α0 , α) ∈ N (1) is the set of supermartingale deflator processes. • Ds = {YT ∈ L0+ : ∃Z ∈ Ds , YT = ZT } = {YT ∈ L0+ : Y0 = 1, ∃(Zn (T ))n≥1 ∈ Ml , YT ≤ lim Zn (T ) a.s.} is n→∞

the set of supermartingale deflators.   t • X e (x) = X ∈ L+0 : ∃(α0 , α) ∈ N (x), Xt = x + 0 αu · dSu , ∀t ∈ [0, T ] is the set of nonnegative wealth processes generated by self-financing trading strategies.  t • X (x) = X ∈ L+0 : ∃(α0 , α) ∈ N (x), x + 0 αu · dSu ≥ Xt , ∀t ∈ [0, T ] is the set of nonnegative wealth processes dominated by the value process of a selffinancing trading strategy.

List of Notation

•

•

• • • • • • • •

xxiii

T C e (x) = {XT ∈ L0+ : ∃(α0 , α) ∈ N (x), x + 0 αt · dSt = XT } = {XT ∈ L0+ : ∃Z ∈ X e (x), XT = ZT } is the set of nonnegative random variables generated by self-financing trading strategies.  T C (x) = XT ∈ L0+ : ∃(α0 , α) ∈ N (x), x + 0 αt · dSt ≥ XT is the set of nonnegative random variables dominated by the value process of a self-financing trading strategy. βt = St − E Q [ST |Ft ] is an asset’s price bubble with respect to the equivalent local martingale measure Q. p(t, T ) is the time t price of a default-free zero-coupon bond paying $1 at time T with t ≤ T . )) is the time t default-free (continuously compounded) f (t, T ) = − ∂log(p(t,T ∂T forward rate for date T with  t ≤ T . p(t,T ) − 1 is the time t default-free discrete forward rate for L(t, T ) = 1δ p(t,T +δ) the time interval [T , T + δ] with t ≤ T . D(t, T ) is the time t price of a risky zero-coupon bond paying $1 at time T with t ≤ T. Ui (x, ω) : (0, ∞) × Ω → R is the state dependent utility function of wealth for investor i = 1, . . . , I . 

I  (F, P) , (N0 , N) , Pi , Ui , e0i , ei i=1 is an economy. U (x, ω) : (0, ∞) × Ω → R is the aggregate utility function of wealth for a representative trader.

Part I

Arbitrage Pricing Theory

Overview The key results of finance that are successfully used in practice are based on the three fundamental theorems of asset pricing. Part 1 presents these three theorems. The applications of these three theorems are also discussed, including equivalent local martingale measures (state price densities), systematic risk, multiple-factor beta models, derivatives pricing, derivatives hedging, and asset price bubbles. All of these implications are based on the existence of an equivalent local martingale measure. The three fundamental theorems of asset pricing relate to the existence of an equivalent local martingale measure, its uniqueness, and its extensions. Roughly speaking, the first fundamental theorem of asset pricing equates no arbitrage with the existence of an equivalent local martingale measure. The second fundamental theorem relates market completeness to the uniqueness of the equivalent local martingale measure. The third fundamental theorem states that there exists an equivalent martingale measure, without the prefix “local,” if and only if there is no arbitrage and no dominated assets in the economy. There are three major models used in derivatives pricing: the Black, Scholes, Merton (BSM) model, the Heath, Jarrow, Morton (HJM) model, and the reduced form credit risk model. These models are discussed in this part. Other extensions and refinements of these models exist in the literature. However, if you understand these three models, then their extensions and refinements are easy to understand. These models are divided into three cases: complete markets, extended complete markets, and incomplete markets. In complete markets, there is unique pricing of derivatives and exact hedging is possible. The two models falling into this category are the BSM and the HJM models. In a complete market model, the local martingale measure never needs to be explicitly identified for pricing, making these models useful in practice. There are also two models for studying credit risk: structural and reduced form models. Structural models assume that markets are complete. Reduced form models,

2

I Arbitrage Pricing Theory

depending upon the structure imposed, usually (implicitly) assume that the markets are incomplete. In reduced form models, market incompleteness is due to the use of inaccessible stopping times to model default (jump processes). Extended complete market models contain the reduced form models studied herein. This class of models is called extended complete because to obtain unique pricing in such a model, one assumes that the market studied is embedded in a larger market that may be complete, and therefore the equivalent local martingale measure is unique. If still incomplete after the embedding, the larger market is assumed to have a sufficient number of traded derivatives (e.g. call and put options with different strikes and maturities) on the primary traded assets (e.g. stocks, zero-coupon bonds) such that the equivalent local martingale measure is uniquely determined. In either case, just as in a complete market, the local martingale measure never needs to be explicitly identified for pricing. It is important to note that in this circumstance, however, exact hedging of derivatives is impossible without the use of the traded derivatives. The primary use of extended complete models is for pricing and static hedging using derivatives, and not dynamic hedging using the primary traded assets. Extended complete markets are used for studying credit risk and asset price bubbles. Last, when such an extended incomplete market does not apply, exact pricing of derivatives is impossible because a unique local martingale measure cannot be identified. This is the case for an incomplete market. In this situation, upper and lower bounds for derivative prices are obtained by super- and sub-replication. Superand sub-replication are studied here as well.

Chapter 1

Stochastic Processes

We need a basic understanding of stochastic processes to study asset pricing theory. Excellent references are Karatzas and Shreve [123], Medvegyev [143], Rogers and Williams [164], and Protter [158]. This chapter introduces some terminology, notation, and key theorems. Few proofs of the theorems are provided, only references for such. The basics concepts from probability theory are used below without any detailed explanation (see Ash [3] or Jacod and Protter [76] for this background material).

1.1 Stochastic Processes We consider a continuous-time setting with time denoted t ∈ [0, ∞). We are given a filtered probability space (Ω, F , F, P) where Ω is the state space with generic element ω ∈ Ω, F is a σ -algebra representing the set of events, F = (Ft )0≤t≤∞ is a filtration, and P is a probability measure defined on F . A filtration is a collection of σ -algebras, which are increasing, i.e. Fs ⊆ Ft for 0 ≤ s ≤ t ≤ ∞. A random variable is a mapping Y : Ω → R such that Y is F -measurable, i.e. Y −1 (A) ∈ F for all A ∈ B(R) where B(R) is the Borel σ -algebra on R, i.e. the smallest σ -algebra containing all open intervals (s, t) with s ≤ t for s, t ∈ R (see Ash [3, p. 8]). A stochastic process is a collection of random variables indexed by time, i.e. a mapping X : [0, ∞) × Ω → R, denoted variously depending on the context, X(t, ω) = X(t) = Xt . It is adapted if Xt is Ft -measurable for all t ∈ [0, ∞). A sample path of a stochastic process is the graph of X(t, ω) across time t keeping ω fixed. We assume that the filtered probability space satisfies the usual hypotheses. The usual hypotheses are that F0 contains the P null sets of F and that the filtration F is right continuous. Right continuous means that Ft = ∩u>t Fu for all 0 ≤ © Springer Nature Switzerland AG 2021 R. A. Jarrow, Continuous-Time Asset Pricing Theory, Springer Finance, https://doi.org/10.1007/978-3-030-74410-6_1

3

4

1 Stochastic Processes

t < ∞. Letting F0 contains the P null sets of F facilitates the measurability of various events, random variables, and stochastic processes. Right continuity implies the important result that given a random variable τ : Ω → [0, ∞], {τ (ω) ≤ t} ∈ Ft for all t if and only if {τ (ω) < t} ∈ Ft for all t, see Protter [158, p. 3]. This fact will be important with respect to the mathematics of stopping times, which are introduced below. One can think of right continuity as implying that the information at time t + is known at time t, see Medvegyev [143, p. 9]. A stochastic process is said to be cadlag if it has sample paths that are right continuous with left limits existing a.s. P. It is said to be caglad if its sample paths are left continuous with right limits existing a.s. P. Both of these stochastic processes allow sample paths that contain jumps, i.e. a sample path which exhibits at most a countable number of discontinuities (jumps) over any compact interval (see Medvegyev [143, p. 5]). An interval in the real line is compact if and only if it is closed and bounded. A stochastic process is said to be predictable if it is measurable with respect to the predictable σ -algebra. The predictable σ -algebra is the smallest σ -algebra generated by the processes that are caglad and adapted, see Protter [158, p. 102]. A stochastic process is said to be optional if it is measurable with respect to the optional σ -algebra. The optional σ -algebra is the smallest σ -algebra generated by the processes that are cadlag and adapted, see Protter [158, p. 102]. It can be shown that the predictable σ -algebra is always contained in the optional σ -algebra (see Medvegyev [143, p. 27]). Hence, we get the following relationship among the two types of stochastic processes, predictable ⊆ optional. A stochastic process is said to be continuous if its sample paths are continuous a.s. P, i.e. it is both cadlag and caglad. Definition 1 (Nondecreasing Process) Let X be a cadlag process. X is a nondecreasing process if the paths Xt (ω) are nondecreasing in t for all ω ∈ Ω a.s. P. Definition 2 (Finite Variation Process) Let X be a cadlag process. X is a finite variation process if the paths Xt (ω) are of finite variation on compact intervals for all ω ∈ Ω a.s. P. A real valued function f : R → R being of finite variation on a compact interval means that the function can be written as the difference of two nondecreasing (monotone) real valued functions, see Royden [167, p. 100]. From Royden [167], Lemma 6, page 101 we get the next lemma. Lemma 1 (Lebesque Integrals) Let Y be a cadlag and adapted process such that  Xt (ω) =

t

Ys (ω)ds 0

exists for all t ∈ [0, T ] and for all ω ∈ Ω a.s. P.

1.1 Stochastic Processes

5

Then, Xt is a continuous and adapted process of finite variation on [0, T ]. Lemma 2 (Increasing Functions of Continuous Finite Variation Processes) Let Xt be a continuous and adapted process of finite variation on [0, T ]. Let f : R →R be differentiable with f  continuous. Then, f (Xt ) is a continuous and adapted process of finite variation on [0, T ]. Proof The continuity of f (Xt ) follows trivially, and the continuity of f implies f (Xt ) is adapted. Consider a partition of the time interval [0, T ] denoted t0 , · · · , tn where max[ti − ti−1 ] → 0 as n →

∞.

Fix ω ∈ Ω. Note that ni=1 f (Xti ) − f (Xti−1 ) = ni=1 f  (ξi ) Xti − Xti−1 for some ξi ∈ (Xti , Xti−1 ) by the mean value theorem. Since Xt is continuous, I = [min{Xt ; t ∈ [0, T ]}, max{Xt ; t ∈ [0, T ]}] is a compact interval on the real line. Since f  is continuous, there exists a ξ ∈ I such that f  (ξi ) ≤ f  (ξ ) for all ξi ∈ I .

Hence, ni=1 f  (ξi ) Xti − Xti−1 ≤ f  (ξ ) ni=1 Xti − Xti−1 . Since Xt is of finite variation the supremum across all such

on [0, T ], taking partitions of time gives sup ni=1 Xti − Xti−1 < ∞, which implies

sup ni=1 f (Xti ) − f (Xti−1 ) < ∞. This completes the proof. Definition 3 (Martingales) A stochastic process X is a martingale with respect to F if (i) X is cadlag and adapted, (ii) E[|Xt |] < ∞ all t, and (iii) E[Xt |Fs ] = Xs a.s. for all 0 ≤ s ≤ t < ∞. It is a submartingale if (iii) is replaced by E[Xt |Fs ] ≥ Xs a.s. It is said to be a strict submartingale if it is a submartingale but not a martingale, i.e. the inequality is strict with positive probability for some 0 ≤ s ≤ t < ∞. It is a supermartingale if (iii) is replaced by E[Xt |Fs ] ≤ Xs a.s. It is said to be a strict supermartingale if it is a supermartingale but not a martingale, i.e. the inequality is strict with positive probability for some 0 ≤ s ≤ t < ∞. For the definition of an expectation and a conditional expectation, see Ash [3, Chapter 6]. Within the class of martingales, uniformly integrable martingales play an important role (see Protter [158, Theorem 13, p. 9]). Definition 4 (Uniformly Integrable Martingales) A stochastic process X is a uniformly integrable martingale with respect to F if (i) X is a martingale, (ii) Y = lim Xt a.s. P exists, E [|Y |] < ∞, and t→∞

(iii) E[Y |Ft ] = Xt a.s. for all 0 ≤ t < ∞. Remark 1 (Uniformly Integrable Martingales) Suppose we are given a filtered probability space (Ω, F , (Ft )t∈[0,T ] , P) for a finite time horizon T < ∞ with X : [0, T ] × Ω → R where F = FT . Then, if X is a martingale, we have

6

1 Stochastic Processes

E[XT |Fs ] = Xs a.s. P for all s ∈ [0, T ]. This implies that all martingales on a finite horizon are uniformly integrable. This completes the remark. The following lemma will prove useful when studying supermartingales on finite horizon filtered probability spaces. Lemma 3 (Sufficient Condition for a Supermartingale to be a Martingale) Let Xt be a supermartingale on the finite horizon filtered probability space (Ω, F , (Ft )t∈[0,T ] , P) for T < ∞. Then, Xt is a martingale if and only if E [XT ] = X0 . Proof If Xt is a martingale, then E[XT |F0 ] = X0 a.s., i.e. E [XT ] = X0 . For the converse, Xt a supermartingale implies E[Xt |Fs ] ≤ Xs a.s. for all 0 ≤ s ≤ t ≤ T . Suppose for some s and t that E[Xt |Fs ] < Xs on a set of positive probability. Then, taking expectations gives E [Xt ] < E [Xs ]. But, being a supermartingale implies E [XT ] ≤ E [Xt ] < E [Xs ] ≤ X0 . Now, if E [XT ] = X0 , we get a contradiction. Hence, E[Xt |Fs ] = Xs a.s. for all 0 ≤ s ≤ t ≤ T . This completes the proof. Definition 5 (Stopping Time) A random variable τ : Ω → [0, ∞] is a stopping time if {ω ∈ Ω : τ (ω) ≤ t} ∈ Ft for all t ∈ [0, ∞]. Note that +∞ is included in the range of the stopping time. And, as previously mentioned, right continuity of the filtration implies that τ is stopping time if and only if {ω ∈ Ω : τ (ω) < t} ∈ Ft for all t ∈ [0, ∞] as well. Definition 6 (Stopping Time σ -Algebra) Let τ be a stopping time. The stopping time σ -algebra is Fτ = {A ∈ F : A ∩ {τ ≤ t} ∈ Ft f or all t} . Let τ be a stopping time. Then, the stopped process is defined as  Xt∧τ =

Xt if t < τ Xτ if t ≥ τ.

Definition 7 (Local Martingales) A stochastic process X is a local martingale with respect to F if (i) X is cadlag and adapted, (ii) there exists a sequence of stopping times (τn ) such that lim τn = ∞ a.s. P n→∞

where Xt∧τn is a martingale for each n, i.e. Xs∧τn = E[Xt∧τn |Fs ] a.s. P for all 0 ≤ s ≤ t < ∞. A local martingale that is not a martingale is called a strict local martingale.

1.1 Stochastic Processes

7

Remark 2 (Finite Horizon Local Martingales) Suppose we are given a filtered probability space (Ω, F , (Ft )t∈[0,T ] , P) for a finite time horizon T < ∞ with X : [0, T ] × Ω → R where F = FT . In the definition of a local martingale, condition (ii) is modified to the existence of a sequence of stopping times (τn ) such that lim τn = T a.s. P where Xt∧τn is a martingale for each n. This completes the n→∞

remark. Remark 3 (Local Submartingales and Supermartingales) The notion of a local process extends to both submartingales and supermartingales. Indeed, in the definition of a local martingale replace the word “martingale” with either “submartingale” or “supermartingale.” This completes the remark. Lemma 4 (Sufficient Condition for a Local Martingale to be a Supermartingale) Let X be a local martingale that is bounded below, i.e. there exists a constant a > −∞ such that Xt ≥ a for all t a.s. P. Then, X is a supermartingale. Proof Xt ≥ a all t, implies Zt = Xt − a ≥ 0 a.s. P. Note that Z is a local martingale. Hence, without loss of generality we can consider only nonnegative processes. By definition of a local martingale, let the sequence of stopping time (τn ) ↑ ∞ be such that E[Xt∧τn |Fs ] = Xs∧τn . Keeping s, t fixed, taking limits of both the left and right sides gives lim E[Xt∧τn |Fs ] = lim Xs∧τn = Xs . Now, by Fatou’s n→∞

n→∞

lemma lim E[Xt∧τn |Fs ] ≥ E[ lim Xt∧τn |Fs ] = E[Xt |Fs ] . n→∞

n→∞

Combined these give Xs ≥ E[Xt |Fs ] . This completes the proof. Lemma 5 (Sufficient Condition for a Local Martingale to be a Martingale) Let X be a local martingale. Let Y be a martingale such that |Xt | ≤ |Yt | for all t ≥ 0 a.s. P. Then, X is a martingale. Proof For a fixed T , by Remark 1, Yt is a uniformly integrable martingale on [0, T ]. By Medvegyev [143, Proposition 1.144, p. 107], the set {Yτ : τ is a finite valued stopping time} is of class D. Hence {Xτ : τ is a finite valued stopping time} is of class D.   This follows because lim sup {|Yτ |≥n} |Yτ | dP = 0 implies lim sup {|Xτ |≥n} n→∞ τ

n→∞ τ

|Xτ | dP = 0 (by the definition of uniform integrability Protter [158, p. 8]). By Medvegyev [143, Proposition 1.144, p. 107] again, X is a uniformly integrable martingale on [0, T ].

8

1 Stochastic Processes

Since this is true for all T , X is a martingale. This completes the proof. Remark 4 (Bounded Local Martingales Are Martingales) Let X be a local martingale that is bounded, i.e. there exists a constant k > 0 such that |Xt | ≤ k for all t ≥ 0 a.s. P. Then, Yt = k for all t ≥ 0 is a (uniformly integrable) martingale. Applying Lemma 5 shows that X is a martingale. This completes the remark. Definition 8 (Semimartingales) A stochastic process X is a semimartingale with respect to F if it has a decomposition Xt = X0 + Mt + At where (i) M0 = A0 = 0, (ii) A is adapted, cadlag, and of finite variation on compact intervals of the real line, and (iii) M is a local martingale (hence cadlag). Semimartingales are important because they are the class of processes for which one can construct stochastic integrals (see Protter [158, Chapter 2]). Definition 9 (Independent Increments) A stochastic process X has independent increments with respect to F if (i) X0 = 0, (ii) X is cadlag and adapted, and (iii) whenever 0 ≤ s < t < ∞, Xt − Xs is independent of Fs . By independence of Fs we mean that P(Xt − Xs ∈ A |Fs ) = P(Xt − Xs ∈ A) for all A ∈ B(R). Definition 10 (Poisson Process) Let Xt be an adapted process with respect to F taking values in the set {0, 1, 2, . . .} with X0 = 0. It is a Poisson process if (i) for any s, t with 0 ≤ s < t < ∞, Xt − Xs is independent of Fs and (ii) for any s, t, u, v with 0 ≤ s < t < ∞, 0 ≤ u < v < ∞, t − s = v − u, the distribution of Xt − Xs is the same as that of Xv − Xu . Remark 5 (Poisson Process Distribution) It can be shown (see Protter [158, p. 13]) that for all t > 0, P(Xt = n) =

e−λt (λt)n , n!

n = 0, 1, 2, · · ·

for some constant λ ≥ 0 where E(Xt ) = λt and V ar(Xt ) = E([Xt − E(Xt )]2 ) = λt. A Poisson process is a discontinuous sample path process, i.e. its sample paths have at most a countable number of jumps. And, Xt −λt is a martingale. This follows

1.1 Stochastic Processes

9

because E (Xt − Xs |Fs ) = E (Xt − Xs ) = λ(t − s) for 0 ≤ s < t < ∞. The first equality is due to property (i). Rearranging terms yields E (Xt − λt |Fs ) = Xs −λs, which proves the result. Definition 11 (Brownian Motion) Let Xt be an adapted process with respect to F taking values in R with X0 = 0. It is a one-dimensional Brownian motion if (i) for any s, t with 0 ≤ s < t < ∞, Xt − Xs is independent of Fs and (ii) for 0 < s < t, Xt − Xs is normally distributed with E(Xt − Xs ) = 0 and V ar(Xt − Xs ) = (t − s). Xt − Xs being normally distributed with mean zero and variance (t − s) means that  P (Xt − Xs ≤ x) =

x

−∞

√

1 2 1 e− 2(t−s) z dz 2π(t − s)

for all x ∈ R. Remark 6 (Continuous Sample Path Brownian Motions) It can be shown that conditions (i) and (ii) imply that a Brownian motion process Xt always has a modification that has continuous sample paths a.s. P (see Protter [158, p. 17]). A modification of a stochastic process X is another stochastic process Y that is equal to X a.s. P for each t (see Protter [158, p. 3]). When discussing Brownian motions, without loss of generality, we will always assume that the Brownian motion process has continuous sample paths. This completes the remark. A Brownian motion Xt is a martingale. This follows because E (Xt − Xs |Fs ) = E (Xt − Xs ) = 0 for 0 ≤ s < t < ∞. The first equality is due to property (i). Rearranging terms yields E (Xt |Fs ) = Xs , which proves the result. Definition 12 (Levy Process) A stochastic process X is a Levy process if (i) for any s, t with 0 ≤ s < t < ∞, Xt − Xs is independent of Fs (called independent increments), and (ii) the distributions of Xt+s −Xt and Xs −X0 are the same for all 0 ≤ s < t < ∞. Remark 7 (Examples of Levy Processes) Both Brownian motions and Poisson processes are examples of Levy Processes (see Medvegyev [143, Chapter 7]). This completes the remark. Definition 13 (Cox Process) Let Xt be an adapted process with respect to F taking values in the set {0, 1, 2, . . .} with X0 = 0. Let Yt be an adapted process with respect to F taking values in Rd . Denote FtY = σ (Ys : 0 ≤ s ≤ t) the σ -algebra generated by Y up to and Y = ∨∞ F Y the smallest σ -algebra containing F Y for all including time t and F∞ t t t=0 t ≥ 0. d Let  tλ : [0, ∞)×R −→ [0, ∞), denoted λt (y) ≥ 0, be jointly Borel measurable with 0 λu (Yu )du < ∞ for all t ≥ 0 a.s. P.

10

1 Stochastic Processes

X is a Cox process if for all 0 ≤ s < t and n = 0, 1, 2, · · · 



Y ∨ Fs = P Xt − Xs = n| F∞

e−

t s

λu (Yu )du

 t s

n λu (Yu )du

n!

Y ∨ F is the smallest σ -algebra containing both F Y and F . We note where F∞ s s ∞ that for all t > 0: Y t Y t (i) E Xt F∞ = λu (Yu )du, and  V ar Xt F∞ 0 λu (Yu )du,  = t Y

0  t

(ii) E (Xt ) = E 0 λu (Yu )du , E V ar Xt F∞ = E 0 λu (Yu )du .

This is sometimes called a doubly stochastic process or a conditional Poisson process (see Bremaud [23, p. 21], Bielecki and Rutkowski [12, p. 193]). Intuitively, conditioned on the entire history of Y over [0, ∞), X is a Poisson process with a continuous and deterministic compensator (see Klebaner [128, p. 256]).

1.2 Stochastic Integration This section introduces the notion of a stochastic integral based on Protter [158]. We define two integrals in this section. The first, the Ito-Stieltjes integral, is with respect to a finite variation process. The second, the (Ito) stochastic integral, is with respect to a semimartingale. Definition 14 (Ito-Stieltjes Integrals) Let X be an adapted process of finite variation. Let Y be an adapted cadlag process. Then, 

t

Ys (ω)dXs (ω), 0

the pathwise Lebesgue-Stieltjes integral exists for all t ≥ 0 and ω ∈ Ω a.s. P (Medvegyev [143, Proposition 2.9, p. 115]). This pathwise integral is called the Ito-Stieltjes integral. For a definition of the Lebesgue-Stieltjes integral see Royden [167, p. 263]. For future use, let L(X) denote the set of optional processes that are Ito-Stieltjes integrable with respect to X. We next start the process of defining a stochastic integral for semimartingales. Definition 15 (Simple Predictable Processes) A stochastic process αt (ω) is a simple predictable process if it has a representation αt = α0 +

n  i=1

αi 1(Ti ,Ti+1 ] (t)

1.2 Stochastic Integration

11

for all t ≥ 0 where 0 = T1 ≤ · · · ≤ Tn+1 < ∞ is a finite sequence of stopping times and αi is FTi -measurable for all i = 1, . . . , n. Let the set of simple predictable processes be denoted S. Note that this process is adapted and left continuous with right limits existing. Definition 16 (Stochastic Integrals) Let X be a semimartingale. For α ∈ S, the stochastic integral is defined by 

t

αs dXs = α0 X0 +

0

n 

αi (XTi+1 − XTi )1[Ti+1 ,∞) (t)

i=1

for all t ≥ 0. We need to extend these integrands to a larger class of stochastic processes. First, we consider the set of all adapted, left continuous with right limits existing processes, denoted α ∈ L. We endow this set of processes with the uniformly on compacts in probability (ucp) topology (see Protter [158, p. 57]). We note that the space S is dense in L under the ucp topology. Hence, given any α ∈ L there exists a sequence α n ∈ S such that α n → α. Also, endow the set of cadlag and adapted processes with the ucp topology. The stochastic integral defined above is in this set. We can now define the stochastic integral for a semimartingale X and α ∈ L. Definition 17 (Stochastic Integrals) Let X be a semimartingale. For α ∈ L, choose a sequence α n ∈ S such that it converges to α in the ucp topology, then the stochastic integral is defined by  0

t

 αs dXs = ucp − lim n→∞

0

t

αsn dXs .

t In this notation, we interpret 0 αs dXs as a stochastic process defined on t ∈ [0, ∞). We need to extend this stochastic integral to even a larger class of integrands, the set of predictable processes. Let the set of predictable processes be denoted α ∈ P. This stochastic integral is extended from L to the class P, again, by taking limits. We sketch this construction. The construction is rather complicated. It proceeds by first restricting the set of semimartingales for which the stochastic integral is defined. To obtain this restriction, we introduce the H 2 norm on the set of semimartingales and consider the set of semimartingales with finite norm (see Protter [158, p. 154]). This space of semimartingales is a Banach space. This norm induces a topology on the set of semimartingales. This gives the appropriate notion of limits in the space of semimartingales. The set of semimartingales with finite H 2 norm becomes the set of integrators. Next, we consider the predictable processes that are bounded, i.e. α ∈ P such that |α(w, t)| ≤ K for all t a.s. P where K is a positive constant. We denote this set α ∈ bP. We define a distance function dX (α 1 , α 2 ) on this set of bounded predictable

12

1 Stochastic Processes

processes α 1 , α 2 ∈ bP (see Protter [158, p. 155]). This distance function induces a topology, the dX -topology, on the space of bounded predictable processes. This gives the appropriate notion of limits in the space of bounded predictable processes. Denote the set of bounded, adapted, and left continuous processes by α ∈ bL. We note that the space of bounded, adapted, and left continuous processes is dense in bP using the dX -topology (see Protter [158, p. 156]). Hence, given any α ∈ bP there exists a sequence α n ∈ bL such that α n → α in the dX -topology. The set bP gives the set of integrands. a semimartingale X ∈ H 2 . We note that for α n ∈ bL, stochastic integral  t Fix  t the n n dX ∈ H 2 . Given α dX is well-defined by the previous definition and α s s 0 s 0 s these preliminaries, we can now define the stochastic integral for X ∈ H 2 and α ∈ bP. Definition 18 (Stochastic Integrals) Let X ∈ H 2 be a semimartingale. For α ∈ bP, choose a sequence α n ∈ bL such that it converges to α in the dX -topology, then 

t 0

 αs dXs =

H 2 − lim n→∞ 0

t

αsn dXs .

t As before, here we interpret 0 αs dXs as a stochastic process defined on t ∈ [0, ∞). Finally, this integral can be extended to a semimartingale X and a predictable process α ∈ P using a localization argument via a sequence of stopping times τn approaching infinity where the stopped processes Xt∧τn , αt∧τn are in H 2 and bP, respectively. We leave a description of this localization argument to Protter [158, p. 163]. This completes the construction. We define α ∈ L (X) ⊂ P to be the set of predictable processes where t the stochastic integral 0 αs dXs with respect to X exists (after the localization argument). By construction, the stochastic integral is a semimartingale (see Protter [158, Theorem 13, p. 162]). But, not all stochastic integrals are local martingales. The following lemma gives sufficient conditions for a stochastic integral to be a local martingale. Lemma 6 (Sufficient Condition for a Stochastic Integral to be a Local Martingale) Let H ∈ L (X) where X is a local martingale. Consider  t Y t = Y0 + Hs dXs 0

for all t ≥ 0. Let Y be bounded below, i.e. there exists a constant c > −∞ such that Yt ≥ c for all t ≥ 0 a.s. P. Then, Y is a local martingale.

1.3 Quadratic Variation

13

Proof By Protter [158, Theorem 89, p. 234], Y is a σ -martingale. Since Y is bounded below, by Ansel and Stricker [2], Y is a local martingale. This completes the proof.

1.3 Quadratic Variation Definition 19 (Quadratic Variation and Quadratic Covariation) Let X, Y be semimartingales. The quadratic variation of X denoted [X, X]t is defined by  [X, X]t = Xt2 − 2

t

Xs− dXs 0

for t ≥ 0. The quadratic covariation of X, Y denoted [X, Y ]t is defined by 

t

[X, Y ]t = Xt Yt −



0

for t ≥ 0 where Xs− =

lim Xt and Ys− =

t→s, t 0 be a probability dP measure. Then, 

t

Xt =

Hs ds + Wt

0

is a standard Brownian motion under Q for 0 ≤ t ≤ T . Remark 12 (Equivalent Probability Measures) The probability measure Q defined in Girsanov’s theorem is equivalent to P, written Q ∼ P. This means that Q agrees with P on zero probability events, i.e. P(A) = 0 ⇔ Q(A) = 0 for all A ∈ F . This completes the remark.

1.8 Essential Supremum The following theorem (see Pham [156, p. 174]; note the proof here does not depend on S being a continuous process) will be important in super- and sub-replication. Theorem 6 (Essential Supremum) Let XT ≥ 0 be FT -measurable. Let S be a Rn -valued semimartingale. Then, the cadlag modification of the process Xt = ess sup E Q [XT |Ft ] Q∈Ml

1.10 Martingale Representation

19

for t ∈ [0, T ] is a supermartingale for all Q {Q ∼ P : S is a Q local martingale}.

∈

Ml where Ml

=

1.9 Optional Decomposition The next theorem (see Follmer and Kabanov [62]) will also prove useful be in superand sub-replication. Theorem 7 (Optional Decomposition) Let S be a Rn -valued semimartingale. Let Ml = ∅ where Ml = {Q ∼ P : S is a Q local martingale}. Let X be a local supermartingale with respect to all Q ∈ Ml . Then, for any Q ∈ Ml there exists a nondecreasing cadlag adapted process C with C0 = 0 and a predictable integrand α ∈ L (S) such that 

t

Xt = X0 +

αs dSs − Ct

0

for t ≥ 0. If S ≥ 0, then

t 0

αs dSs is a local martingale with respect to Q.

1.10 Martingale Representation The following theorem (see Protter [158, p. 187]) will prove useful in hedging derivatives. Theorem 8 (Martingale Representation) Let W = (W 1 , . . . , W n ) be an n-dimensional Brownian motion on (Ω, F , (Ft )t∈[0,T ] , P) with the filtration (Ft )t∈[0,T ] its completed natural filtration where F = FT . Let ZT be FT -measurable with E [|ZT |] < ∞. T Then, there exist predictable processes α i in L (W i ) with 0 (αsi )2 ds < ∞ for i = 1, . . . , n such that ZT = E[ZT ] +

n   i=1

T

0

αsi dWsi

and where Zt = E[ZT ] +

n   i=1

is a martingale.

0

t

αsi dWsi

20

1 Stochastic Processes

1.11 Equivalent Probability Measures The next theorem characterizes equivalent probability measures on a Brownian filtration. It will subsequently prove useful to understand arbitrage-free markets in a Brownian motion market. Theorem 9 (Equivalent Probability Measures on a Brownian Filtration) Let W = (W 1 , . . . , W n ) be an n-dimensional Brownian motion on (Ω, F , (Ft )t∈[0,T ] , P) with the filtration (Ft )t∈[0,T ] its completed natural filtration where F = FT . Q∼P if and only if T (1) there exist predictable processes α i in L (W i ) with 0 (αsi )2 ds < ∞ for i = 1, . . . , n such that  t i 2 n  t i i 1 n (2) Zt = e− i=1 0 αs dWs − 2 i=1 0 αs ds , (3) ZT = dQ dP > 0, and (4) E [ZT ] = 1 where E [·] is expectation under P. Proof (⇐) This direction is trivial. Assuming the hypotheses (1)–(4), the measure Q defined by condition (2) is a probability measure equivalent to P. (⇒) Assume Q ∼ P. Then, define ZT = dQ dP > 0. This gives E [ZT ] = 1. Protter [158, Corollary 4, p. 188], gives conditions (1) and (2). This completes the proof.

1.12 Notes Excellent references for stochastic calculus include Karatzas and Shreve [123], Medvegyev [143], Protter [158], and Roger and Williams [164]. For books that present both the basics of stochastic calculus and its application to finance, see Baxter and Rennie [10], Jeanblanc et al. [117], Klebaner [128], Korn and Korn [130], Lamberton and Lapeyre [135], Mikosch [148], Musiela and Rutkowski [152], Shreve [177], and Sondermann [180].

Chapter 2

The Fundamental Theorems

This chapter presents the three fundamental theorems of asset pricing. These theorems are the basis for pricing and hedging derivatives, characterizing price bubbles, and understanding the risk return relations among assets including the notion of systematic risk, idiosyncratic risk, portfolio optimization, and equilibrium pricing.

2.1 The Set-Up We consider a continuous-time model with a finite horizon [0, T ] where trading takes place at times t ∈ [0, T ) and the outcome of all trades are realized at time T . No trades take place at time t = T because all traded assets are liquidated and their proceeds distributed at that time. We are given a complete filtered probability space (Ω, F , F, P) where the filtration F = (Ft )t∈[0,T ] satisfies the usual hypotheses and F = FT . Here P is the statistical probability measure. By the statistical probability measure P we mean that from which historical time series data are generated (drawn by nature). Hence, standard statistical methods can be used to estimate this probability measure P from historical time series data. For simplicity of notation, we adopt the convention that all of the subsequent equalities and inequalities given are assumed to hold almost surely (a.s.) with respect to P, unless otherwise noted. Traded in this market are a money market account and n risky assets. The markets are assumed to be frictionless and competitive. By frictionless we mean that there are no transaction costs, no differential taxes, shares are infinitely divisible, and there are no trading constraints, e.g. short sales restrictions, borrowing limits, or margin requirements. By competitive we mean that traders act as price takers, i.e. they can trade any quantity of shares desired without affecting the market price. Alternatively stated, there is no liquidity risk. Liquidity risk is when there is a © Springer Nature Switzerland AG 2021 R. A. Jarrow, Continuous-Time Asset Pricing Theory, Springer Finance, https://doi.org/10.1007/978-3-030-74410-6_2

21

22

2 The Fundamental Theorems

quantity impact from trading on the price. In Part IV of this book we will relax the no trading constraints assumption. However, the competitive market assumption is maintained throughout this book. t Let Bt = e 0 rs ds denote the time t value of a money market account (mma), initialized with a dollar investment, B0 = 1. Here  t rt is the default-free spot rate of interest, adapted to F, and integrable with 0 |rs | ds < ∞ for all t ≥ 0. By construction, the value of the mma is continuous in time t and of finite variation on compact intervals (use Lemmas 1 and 2 in Chap. 1). Note that Bt > 0 for all t ≥ 0. Let St = (S1 (t), . . . , Sn (t)) > 0 denote the time t prices of the n risky assets, which are strictly positive semimartingales with respect to F. The prime denotes transpose. With a slight loss of generality, we assume that all risky asset prices are strictly positive, so that returns are well defined. We also assume that the risky assets have no cash flows (dividends) over [0, T ). The time T value of the risky assets can be viewed as a liquidating cash flow (dividend). Recall that there is no trading at time T . Surprisingly perhaps, it is shown in Sect. 2.3 below that the assumption of no cash flows over [0, T ) is without loss of generality. It is important to note that the risky asset price processes can be discontinuous (with jumps). All of the subsequent results apply to these general price processes unless otherwise noted. Remark 13 (Risky Assets May Include Derivatives) In the collection of risky assets, some could be derivatives on a subset of the remaining risky assets. For example, (S1 (t), . . . , Sk (t)) for k < n could be stocks, and (Sk+1 (t), . . . , Sn (t)) could be European call options on these stocks (e.g. Sk+1 (t) could be a European call option on S1 (t)). This completes the remark.

2.1.1 Trading Strategies A trading strategy is defined to be a specification of the number of units of the mma and risky assets held for all state-time pairs (ω, t) ∈ Ω × [0, T ], i.e. the specification of the stochastic processes (α0 (t), αt ) ∈ Rn+1 where αt = α(t) = (α1 (t), . . . , αn (t)) ∈ Rn . The stochastic processes (α0 (t), αt ) are assumed to be adapted to F. Being adapted means that when the trading strategy (α0 (t), αt ) is formed at time t only information known at time t can be used in the construction. Alternatively stated, these trading strategies cannot be constructed using information from the future. Let x be the initial value or wealth invested in the trading strategy. Then, an accounting identity states that x = X0 = α0 (0)B0 + α0 · S0

(2.1)

2.1 The Set-Up

23

where z · y = nj=1 zj yj for z, y ∈ Rn . This equals the number of shares in the trading strategy at time 0 multiplied by the price per share and then summed across all traded assets. The trading strategy’s value at any time t ∈ [0, T ] is Xt = α0 (t)Bt + αt · St .

(2.2)

Define L(B) to be the set of optional processes that are integrable with respect to B, and L (S) to be the set of predictable processes that are integrable with respect to S, see Sect. 1.2 for the formal definition of a stochastic integral. We assume that the holdings in the mma α0 ∈ L(B) and the holdings in the risky asset α ∈ L (S). Note the difference in measurability requirements on the position in the mma versus the risky assets. From a mathematical perspective, this difference is due to the restrictions needed to guarantee the existence of the relevant integrals. This distinction also has an economic interpretation. Recall that a predictable process is a stochastic process generated by the set of adapted and caglad (left continuous with right limits existing) processes. This means that when forming the position in the risky assets, only past information and current information generated by the flow of time from the left can be utilized. In contrast (this will be more clearly seen after normalizing prices by the value of the mma below), the position in the mma is a “residual” decision. The mma position is completely determined by the dynamic holdings in the risky assets (expression (2.4)) and the value of the portfolio at any time (expression (2.2)). This implies that its measurability requirements are determined by the position in the risky assets α ∈ L (S) and the risky asset price process St via expression (2.2). Since St is an adapted and cadlag (right continuous with left limits existing) process (not necessarily predictable), and α ∈ L (S) can be a constant (a buy and hold), this implies the more general measurability of the position in the mma (an optional process) is needed. As defined, a trading strategy may require cash infusions or generate cash outflows after time 0 and before time T . To understand why, consider the trading strategy consisting of buying 1 unit of the mma at time 0 and purchasing another unit of the mma at time T2 , i.e.  (α0 (t), αt ) =

(1, (0, 0, . . . , 0)) if 0 ≤ t < T2 (2, (0, 0, . . . , 0)) if T2 ≤ t ≤ T .

At time T2 , the change in the value of this trading strategy is ΔX T = 2B T − B T = 2 2 2 B T > 0. To finance this change in value, a cash infusion of B T dollars is needed. 2 2 Trading strategies that require no cash outflows or inflows between (0, T ) are called self-financing.

24

2 The Fundamental Theorems

Definition 20 (Self-financing Trading Strategy (s.f.t.s.)) A trading strategy (α0 , α) ∈ (L(B), L (S)) is self-financing if t t Xt = x + 0 α0 (u)dBu + 0 αu · dSu t t = x + 0 α0 (u)ru Bu du + 0 αu · dSu or

(2.3)

dXt = α0 (t)rt Bt dt + αt · dSt

(2.4)

for all t ∈ [0, T ]. As seen by expression (2.4), the change in the trading strategy’s value is completely determined by the change in the value of the underlying assets’ prices. There are no cash flows in or out of the trading strategy, hence the terminology.

2.1.2 Admissibility and Doubling Strategies Not all s.f.t.s. are reasonable. With continuous trading, without further restrictions on the trading strategy, it is possible to construct a s.f.t.s. that generates positive values for sure from zero initial wealth. These s.f.t.s.’s are called doubling strategies. Definition 21 (Doubling Strategies) A doubling strategy is a s.f.t.s. (α0 , α) ∈ (L(B), L (S)) such that (i) X0 = 0, (ii) Xt ≤ x for all t ∈ [0, T ], and (iii) P(XT = x) = 1 for x > 0. A doubling strategy is an “arbitrage opportunity” (to be defined later) because it starts with zero investment and it is terminated with a strictly positive value equal to x dollars with probability one. Doubling strategies have the following properties. Si (t) Bt

t are local martingales under P for i = 1, . . . , n, then the value process X Bt   T is a strict submartingale under P with X0 = 0 < E X BT = x. In this case, the value process of the doubling strategy is not bounded below.

1. If

t Proof By Lemma 6 in Chap. 1, − X Bt is a local martingale under P because it is bounded below by −x < 0, and it is a supermartingale under P by Lemma 4 T = −x, in Chap. 1. Finally, by taking expectations, −X0 = 0 > E − X BT

t making it a strict supermartingale under P (Lemma 3 in Chap. 1). Thus, X Bt is a strict submartingale under P. Last, if the value process of the doubling strategy is bounded below, then it a local martingale bounded above and below, and hence a martingale (see Remark 4 in Chap. 1), which contradicts the value process being a strict submartingale. This completes the proof.

2.1 The Set-Up

25

2. A doubling strategy being a strict submartingale is invariant to a change in equivalent probability measures Q ∼ P as long as SBi (t) are still local martingales t under Q for i = 1, . . . , n. This follows due to the hypothesis of property 1 above, and because the definition of a doubling strategy only depends on probability one events. Example 2 (A Doubling Strategy) This example illustrates a s.f.t.s. that is a doubling strategy. Fix a sequence of dates {ti }i=0,1,2,... → T with t0 = 0. For the doubling strategy to be successful, we need an infinite number of trading times represented by this sequence. Consider a risky asset whose market price at time t0 = 0 is S0 = 1, and at each subsequent time ti its price changes discretely by  S ti =

2Sti−1 with probability 12 1 1 2 Sti−1 with probability 2 .

Let the stock price process be right continuous with left limits existing. This asset price process is strictly positive and discontinuous. Let the mma be identically one, i.e. Bt = 1 for all t ≥ 0. The trading strategy starts with zero investment, holding one share of the risky asset and shorting the money market account to finance the position. It liquidates the first time the risky asset price increases. Otherwise, it retrades and “doubles-up” as follows. The s.f.t.s. is constructed such that the first time the risky asset price increases, the value process equals 1 dollar. We define the trading strategy recursively. At time t0 = 0, α(0) = 1 and α0 (0) = −1 < 0 is chosen so that X0 = α0 (0) + α(0)S0 = −1 + 1 · 1 = 0. At time t1 , if the stock price jumps up to St1 = 2, the value process is: up

Xt1 = α0 (0) + α(0)St1 = −1 + 1 · 2 = 1, and liquidate. If the stock price jumps down to St1 = 12 , the value process is Xdown = α0 (0) + α(0)St1 = −1 + 1 · t1

1 1 =− . 2 2

We need to double up by choosing α0 (t1 ), α(t1 ) such that the trading strategy is self-financing, i.e. −

  1 1 = α0 (t1 ) + α(t1 ) 2 2

26

2 The Fundamental Theorems

and if the stock price jumps up to St2 = 2

  1 2

= 1, then

up

Xt2 = α0 (t1 ) + α(t1 ) · 1 = 1. Solving this system of two linear equations in two unknowns gives α0 (t1 ) = −2 and α(t1 ) = 3. Continuing, at time ti for i ≥ 1, the stock price Sti is known.  i−1 If Sti = 2Sti−1 = 2 12 , then liquidate the position. The shares α0 (ti−1 ), α(ti−1 ) were chosen such that the liquidated value is up Xti

If Sti = ( 12 )Sti−1 =

 i 1 2

 i−1 1 = α0 (ti−1 ) + α(ti−1 )2 = 1. 2 , then retrade. The value at time ti before retrading is

Xdown ti

 i 1 = α0 (ti−1 ) + α(ti−1 ) < 0. 2

We want to buy more shares in the risky asset α(ti ) and short additional units of the mma α0 (ti ) keeping the strategy self-financing and such that if the price increases, the value of the trading strategy is one dollar. Hence, (α0 (ti ), α(ti )) are chosen so that Xdown ti

 i 1 = α0 (ti ) + α(ti ) 2

and up

Xti+1 = α0 (ti ) + α(ti )2

 i 1 = 1. 2

and α(ti ) = (1 − Xdown )2i . The solution is α0 (ti ) = −1 + 2Xdown ti ti This value process equals one dollar at time T with probability one because an up jump will occur before time T . Indeed, a jump up occurs with probability one because the realizations of the stock price across time are independent and identically distributed, and there are an infinite number of realizations possible before time T . Hence, the probability that the stock price never jumps up is lim ( 12 )i = 0.

i→∞

We now prove that this strategy’s value has a strictly positive probability of falling below any negative constant c < 0.

2.1 The Set-Up

27

Suppose we are at time t1 with an asset price equal to St1 = 12 . The value is = α0 (0) + α(0) Xdown t1

1 1 = − < 0. 2 2

Continuing, at time ti suppose we enter with Xti < 0 where the asset price is Sti+1 = ( 12 )i ( 12 ). Such a consecutive sequence of down jumps occurs with positive probability for all i ≥ 1. The value of the trading strategy is Xdown ti+1 = α0 (ti ) + α(ti )

 i   1 1 < 0. 2 2

Substitution for the trading strategy’s holdings gives 1 3 down 3 Xdown < Xdown < 0. ti+1 = − + Xti 2 2 2 ti Thus, Xdown ti+1

3 < Xdown < 2 ti

 2  i  i 3 3 3 1 down down . Xti−1 < · · · < Xt1 =− 2 2 2 2

 i Given any c < 0, there exists a large enough i such that 32 21 > c. Since the  i probability that i down jumps occur is 12 > 0, there is a positive probability that the value process evolves below any fixed lower bound. This completes the example. To exclude doubling strategies, we add the following restriction on a s.f.t.s. Definition 22 (Admissible s.f.t.s.) A s.f.t.s. (α0 , α) ∈ (L(B), L (S)) with initial value x and value process Xt is called admissible if there exists a constant c ≤ 0 such that  t  t α0 (u)dBu + αu · dSu ≥ c Xt = x + 0

0

for all t ∈ [0, T ]. The additional restriction imposed by admissibility is that the value of the trading strategy cannot fall below some fixed negative level c ≤ 0. The set of admissible s.f.t.s. for a given initial wealth x is denoted by A (x) = {(α0 , α) ∈ (L(B), L (S)) : Xt = α0 (t)Bt + αt · St , ∃c ≤ 0, 

t

Xt = x + 0



t

α0 (u)dBu + 0

 αu · dSu ≥ c, ∀t ∈ [0, T ] .

28

2 The Fundamental Theorems

Lemma 10 (Admissibility Excludes Doubling Strategies) Let Q ∼ P be such that BS are local martingales under Q, then the set of admissible s.f.t.s. A (x) contain no doubling strategies. Proof Suppose there exists an (α0 , α) ∈ A (x) that is a doubling strategy. Then, it is a strict submartingale under Q by property 2 of doubling strategies. But, the t value process X Bt is a local martingale under Q because it is bounded below by c < 0, and it is a supermartingale under Q by Lemma 4 in Chap. 1. This yields the contradiction, which completes the proof. The admissibility condition, the fixed lower bound c < 0, imposes an implicit borrowing constraint on the trading strategy’s value. This condition has some implications for the set of s.f.t.s. A (x). Remark 14 (Admissibility and (Naked) Short Selling) If a risky asset’s price process Si (t) is unbounded above, i.e. for any c > 0, P( sup Si (t) > c) > 0, t∈[0,T ]

the admissibility condition excludes (naked) short selling of the risky asset from the set of admissible s.f.t.s.’s A (x) where the initial wealth is x = −Si (0) < 0. Indeed, (naked) short selling of the ith risky asset is the s.f.t.s. (α0 (t), α1 (t), . . . , αi−1 (t), αi (t), αi+1 (t), . . . αn (t)) = (0, 0, . . . , 0, −1, 0, . . . , 0) for all t ∈ [0, T ]. Because the value process of this trading strategy is X(t) = α0 (t)Bt + αt · St = −Si (t) for all t ≥ 0, we see that for any c > 0, P( inf X(t) < −c) = P( sup Si (t) > c) > 0, violating the admissibility t∈[0,T ]

t∈[0,T ]

condition. This completes the remark.

2.1.3 Suicide Strategies The set of admissible s.f.t.s. A (x) may contain a collection of “bad” trading strategies called suicide strategies. Definition 23 (Suicide Strategies) A suicide strategy is an admissible s.f.t.s. (α0 , α) ∈ A (0) such that (i) X0 = 0, (ii) Xt ≥ −x for all t ∈ [0, T ], and (iii) P(XT = −x) = 1 for x > 0. A suicide strategy starts with a zero investment and loses x dollars with probability one. It is easily seen that a suicide strategy is equivalent to being short a doubling strategy. The only difference is that because the suicide’s value process is bounded below, it is admissible, while a doubling strategy’s value process is not. Consequently, the following properties follow directly from those for a doubling strategy.

2.1 The Set-Up

29

Si (t) Bt

t are local martingales under P for i = 1, . . . , n, then the value process X Bt   T is a strict supermartingale under P with X0 = 0 > E X BT = −x. In this case, suicide strategies are not bounded above. 2. A suicide strategy is invariant to a change in equivalent probability measures Q as long as SBi (t) are local martingales under Q for i = 1, . . . , n. t

1. If

Remark 15 (Strictly Positive Initial Value Suicide Strategy) It is easy to see that an admissible s.f.t.s. (α˜ 0 , α) ˜ ∈ A (x) such that (i) X0 = x > 0, (ii) Xt ≥ 0 for all t ∈ [0, T ], and (iii) P(XT = 0) = 1 can be obtained from the suicide strategy with zero initial investment by adding x additional dollars in the mma account at all times, i.e. (α˜ 0 , α) ˜ = (α0 +x, α) ∈ A (x) where (α0 , α) ∈ A (0) is the s.f.t.s. in Definition 23 above. We will call such admissible s.f.t.s. suicide strategies as well. This completes the remark. Example 3 (A Suicide Strategy) Shorting the doubling strategy in Example 2 is an example of a suicide strategy. The following is from Harrison and Pliska [67]. Consider a market consisting of a money market account (mma) and a risky asset with evolutions 1

Bt = 1, St = eμt− 2 σ

2 t+σ W t

for all t ∈ [0, T ] where S0 = 1, μ, σ are strictly positive constants and Wt is a standard Brownian motion with W0 = 0 that generates the filtration F. This evolution for the risky asset is called geometric Brownian motion. Let X0 = 0 be the initial value of the trading strategy. We want to construct an (α0 , α) ∈ A (0) that loses x > 0 with probability one by time T . Fix a sequence of trading dates {ti }i=0,1,2,... → T with t0 = 0. Choose the trading strategy (α0 (t), α(t)) for t ∈ [0, T ] with initial value X0 = α0 (0) + α(0)S0 = 0 recursively as follows. Time t0 = 0. Short α(t0 ) = −k1 shares of the risky asset, and buy α0 (t0 ) = k1 units of the mma. Define τ (k1 ) = inf {t ≥ t0 : St = (−x + α0 (t0 )) /k1 } = inf {t ≥ t0 : Xt = −x} where Xt = α0 (t0 ) − k1 St . Choose k1 such that P(τ (k1 ) ≤ t1 ) = p > 0. This is always possible under the assumed evolution. Do no additional trading until time t1 . Time t1 . If τ (k1 ) ≤ t1 happened, then Xt1 = −x. Stop.

30

2 The Fundamental Theorems

If τ (k1 ) > t1 , then entered time t1 with the value process Xt1 = α0 (t0 ) + α(t0 )St1 = α0 (t0 ) − k1 St1 > −x. Short k2 − k1 more shares of the risky asset so that α(t1 ) = −k2 . Buy (k2 − k1 ) St1 additional units of the mma, so that α0 (t1 ) = α0 (0) + (k2 − k1 ) St1 . This makes the trading strategy self-financing. Define τ (k2 ) = inf {t ≥ t1 : St = (−x + α0 (t1 )) /k2 } = inf {t ≥ t1 : Xt = −x} where Xt = α0 (t1 ) − k2 St . Choose k2 such that P(τ (k2 ) ≤ t2 ) = p. This is always possible under the assumed evolution. The trading strategy is rebalanced to: Xt1 = α0 (t1 ) − k2 St1 . Do no additional trading until time ti for i ≥ 2. If τ (ki ) ≤ ti happened, then Xti = −x. Stop. If τ (ki ) > ti , then entered time ti with the value process Xti = α0 (ti−1 ) + α(ti−1 )Sti = α0 (ti−1 ) − ki Sti > −x. Short ki+1 − ki more shares of the risky asset so that so α(ti ) = −ki+1 . Buy (ki+1 − ki ) Sti additional units of the mma, so that α0 (ti ) = α0 (ti−1 ) + (ki+1 − ki ) Sti . This makes the trading strategy self-financing. Define τ (ki+1 ) = inf {t ≥ ti : St = (−x + α0 (ti )) /ki+1 } = inf {t ≥ ti : Xt = −x} where Xt = α0 (ti ) − ki+1 St . Choose ki+1 such that P(τ (ki+1 ) ≤ ti+1 ) = p. This is always possible under the assumed evolution. The trading strategy is rebalanced to Xti = α0 (ti ) − ki+1 Sti . Continue for i + 1, . . . . Following this strategy, the probability that

Xti never hits −x over [0, T ] is equal to lim (1 − p)i = 0, i.e. P ∃i : Xti = −x = 1. i→∞

Last, the trading strategy is easily seen to be admissible because, by construction, Xt ≥ −x for all t ∈ [0, T ]. This completes the example. Suicide strategies are important for understanding the second fundamental theorem of asset pricing.

2.2 Change of Numeraire

31

2.1.4 The Frictionless Market Assumption For subsequent usage, a market is defined to be collection ((B, S), F, P) representing the stochastic processes for the traded assets, the market’s information set, and the statistical probability measure. The underlying state space and σ -algebra, (Ω, F ), are always implicit in this collection and not included. A market is always assumed to be frictionless, with the exception of the admissibility condition (unless otherwise indicated), and competitive. The assumption of frictionless markets implicitly appears in the definition of the set of admissible s.f.t.s. A (x) in that, except for the admissibility constraint, the accumulated value of the trading strategy has no additional adjustments for other frictions, e.g. transaction costs, taxes, indivisible shares, explicit short sale constraints, or margin requirements. As noted previously, the admissibility condition is, in fact, a trading constraint imposed on the aggregate value of all shorts in the s.f.t.s. The assumption of competitive markets also implicitly appears in the definition of A (x) because the price processes (Bu , Su ) do not depend on the trading strategy (α0 , α). The trader is a price-taker because there is no quantity impact on the price processes from trading the shares (α0 , α). Although a misnomer, the convention in the literature is to still call a market frictionless if the only restriction imposed is the admissibility condition. Because admissibility is needed to exclude doubling strategies, its imposition is thought to be very mild. It is also standard in portfolio optimization problems, in the context of a frictionless market, to impose an analogous constraint that a trader’s wealth is always nonnegative (see Part II of this book). For the remainder of the book, a market is always assumed to be frictionless in the sense just discussed. It is important to keep this misnomer in mind when using the phrase “frictionless markets” in the subsequent models. Doing so one can more easily understand why asset price bubbles often exist as an implication of the model. For example, in the asset price bubbles Chap. 3 this clarifies why asset price bubbles exist in a frictionless and competitive market where there are no arbitrage opportunities (to be defined). Second, in the portfolio optimization Chaps. 10–12, this clarifies how an optimal wealth and consumption path can exist in the presence of asset price bubbles in a frictionless and competitive market. And finally, in Chaps. 13–16 that study economic equilibrium, this also clarifies how asset price bubbles can exist in a frictionless and competitive market rational equilibrium.

2.2 Change of Numeraire Normalization by the money market account, which is a change of numeraire, simplifies the notation and is almost without loss of generality. The lost of generality is that the set of trading strategies A (x) after the change of numeraire may differ

32

2 The Fundamental Theorems

from the set of trading strategies before due to the modified integrability conditions needed to guarantee that the relevant integrals exist. This section presents the new notation and the evolutions for the mma and the risky assets under this change of numeraire. t Let Bt = B Bt = 1 for all t ≥ 0, this represents the normalized value of the money market account (mma). Let St = (S1 (t), . . . , Sn (t)) ≥ 0 represent the risky asset prices when normalized by the value of the mma, i.e. Si (t) = SBi (t) . Then, t dBt =0 Bt

and

dSt dSt = − rt dt. St St

(2.5)

(2.6)

Proof Using the integration by parts formula Theorem 3 in Chap. 1, one obtains (dropping     the t’s) S S dB d BS = B1 dS + Sd B1 = dS S B − B B .   The first equality uses d S, B1 = 0, since B is continuous and of finite variation (use Lemmas 2 and 7 in Chap. 1). Substitution yields dB dS = dS S S − S B . Algebra completes the proof. Recall that L (S) is the set of predictable processes integrable with respect to S and O is the set of optional processes. We have that the new set of admissible self-financing trading strategies under the change of numeraire is A (x) = {(α0 , α) ∈ (O, L (S)) : Xt = α0 (t) + αt · St , ∃c ≤ 0, t Xt = x + 0 αu · dSu ≥ c, ∀t ∈ [0, T ] . The normalized value of a trading strategy is Xt St = α0 (t) + αt · Bt Bt or Xt = α0 (t) + αt · St . The value process evolution dXt = α0 (t)rt Bt dt + αt · dSt

(2.7)

2.3 Cash Flows

33

is transformed to T XT = X0 + 0 αt · dSt or dXt = αt · dSt .

(2.8)

As easily seen, these expressions are obtainable from the non-normalized evolutions by setting Bt = 1 and rt = 0 for all t ≥ 0. Proof All of the next steps are reversible. dX  = α · dS   S d X B =α·d B . The integration by parts formula Theorem 3 in Chap. 1, using the fact that d X, B1 = 0 since B is continuous and of finite variation (use Lemmas 2 and 7 in Chap.  1), yields     1 1 dX X dB dS S dB d X B = B dX − Xd B = B − B B = α · B − B B . This equality uses X = α0 B + α · S. dX − Xrdt = α · (dS − Srdt). Again using X = α0 B + α · S we get dX = (α0 B + α · S)rdt + α · (dS − Srdt). Simplifying yields dX = α0 rBdt + α · dS. This completes the proof. A normalized frictionless (unless otherwise noted) and competitive market, is defined to be the collection (S, F, P).

2.3 Cash Flows This section shows that the assumption of no cash flows to the risky assets over [0, T ) is without loss of generality. Of course, the time T value of the risky assets can be viewed as a liquidating cash flow since there is no trading at that date. Indeed, it is shown that if there are cash flows, a simple transformation of the risky asset price processes generates an “equivalent” market with no cash flows. All the theorems can be generated for this equivalent market with no cash flows and then the transformation reversed to obtain the results in the original market. We are given a normalized market (S, F, P). For simplicity, assume that the market consists of just one risky asset with price process denoted St . Let this risky asset have a cumulative cash flow process which is a nonnegative, nondecreasing, cadlag, adapted stochastic process that equals 0 at time 0, i.e. Gt ≥ 0 with G0 = 0. Since we are using the normalized market representation, the cumulative cash flow process is in units of the mma. These conditions imply that the cash flow process is of finite variation.

34

2 The Fundamental Theorems

To transform the market with cash flows into an equivalent market without cash flows, we need to reinvest all cash flows into either the mma or the risky asset. We consider both possibilities.

2.3.1 Reinvest in the MMA Denote the new risky asset’s price with reinvestment in the mma by St∗ . The transformation is St∗ = St + Gt . Note that this transformation is additive. A useful special case of this transformation is where for each ω ∈ Ω, Gt is absolutely continuous, i.e. there exists a nonnegative, cadlag, adapted stochastic process Gt ≥ 0 such that  Gt =

t

Gu du. 0

In this case, dSt∗ = dSt + Gt dt. Here, the price change in the risky asset with reinvestment in the mma over [t, t +dt] equals the price change in the risky asset plus the cash flows received.

2.3.2 Reinvest in the Risky Asset Denote the new risky asset’s price with reinvestment in the risky asset by St∗ . The transformation is  t 1 dGu . St∗ = St + St S u 0 To understand this expression, note that at time u a cash flow of dGu occurs. This enables the purchase of S1u dGu shares of the risky asset. The accumulated value of this position at time t is the right side of this expression. A useful special case of this transformation is where there exists a nonnegative, nondecreasing, cadlag, adapted stochastic process Gt ≥ 0 with G0 = 1 such that dGt = St dGt .

2.3 Cash Flows

35

Substitution yields St∗ = St + St



t

dGu = St + St (Gt − 1) = St Gt .

0

Note that this transformation is multiplicative. Define gt by the following expression Gt = egt . Then, St∗ = St egt Assume, in addition, that gt is a continuous process. Then, dSt∗ dSt = + dgt . ∗ St St In this case, the return on the risky asset with cash flows reinvested over [t, t + dt] equals the return on the risky asset plus the cash flows received. Proof Using the integration by parts Theorem 3 in Chap. 1, we obtain dSt∗ = dSt · egt + St · egt dgt =

dSt · St egt + St · egt · dgt . St

This uses the fact that [S, g] = 0 since g is continuous and of finite variation (Lemma 7 in Chap. 1). Dividing by St∗ completes the proof.

2.3.3 Summary As just proved, a market (St , Gt ) with cash flows over [0, T ) can be transformed into an equivalent market St∗ , 0 with no cash flows over [0, T ) using either of the transformations given above. All the analysis can be done in the equivalent market without cash flows, then the transformation reversed to get the results in the original market with cash flows. For the remainder of this book, without loss of generality, we will only consider the equivalent market with no cash flows.

36

2 The Fundamental Theorems

2.4 Non-redundant Assets For many applications, it is important to know when the admissible s.f.t.s. that generates a value process is unique. For this section, we consider a normalized market (S, F, P). We start with a definition. Definition 24 (Non-redundant Assets) The risky assets S are non-redundant if given any admissible s.f.t.s. (α0 , α) ∈ A (x) with value process  t αu · dSu , Xt = α0 (t) + αt · St = x + 0

(α0 , α) is unique for almost all (t, ω), d[S1 , S1 ]t (ω) × · · · × d[Sn , Sn ]t (ω) × dP. If the market does not satisfy this condition, the risky assets are said to be redundant. If the risky assets in a market are non-redundant, then any admissible s.f.t.s. value process X can only be generated by a unique trading strategy. Note that uniqueness of the admissible s.f.t.s. is invariant to a change in equivalent probability measures Q ∼ P because the uniqueness holds with probability one under P. Not all markets have the risky assets satisfying this condition. Indeed, a simple example of a market where the risky assets are redundant is where two assets have different indices, but they are otherwise identical, e.g. if S1 (t) = S2 (t) for all t ≥ 0. For another example, the traded bonds in the Heath-Jarrow-Morton model in Chap. 6 are redundant. The following theorem provides a set of sufficient conditions for a market to be non-redundant. Theorem 10 (Sufficient Conditions for a Non-redundant Market) Let the risky assets’ price processes be Si (t) = Si (0) +

K  

t

σij (s)dMj (s)

j =1 0

for all t ∈ [0, T ] and i = 1, . . . , n, where Mj are local martingales under some probability Q ∼ P with Mj (0) = 0 and σij (t) ∈ L (Mj ) for j = 1, . . . , K. If [Mi , Mj ] = 0 for all i = j , K ≥ n, and there is a non-singular n × n matrix embedded in ⎡ ⎤ σ11 σ12 · · · σ1n ⎢ σ21 σ22 · · · σ2n ⎥ ⎢ ⎥, ⎣ ⎦ σK1 σK2 · · · σKn then the risky assets S are non-redundant.

2.5 The First Fundamental Theorem

37

Proof Let Y (t) = y +

n   i=1

t

αi (s)dSi (s).

0

This is a local martingale under Q because of the admissibility condition on Y . By associativity of stochastic integrals (Protter [158, Theorem 22, p.165]) and changing the order of summation, Y (t) = y +

K   j =1 0

t

! n 

" αi (s)σij (s) dMj (s).

i=1

Theorem n 2 in Chap. 1 implies that i=1 αi (s)σij (s) = hj for j = 1, . . . , K are unique almost all (t, ω), d[S1 , S1 ]t (ω) × · · · × d[Sn , Sn ]t (ω) × dQ. Fix (t, ω). Viewing this as the system of equations: ⎤⎡α ⎤ ⎡ h ⎤ 1 1 σ11 σ12 · · · σ1n ⎥ ⎢ h2 ⎥ α ⎢ σ21 σ22 · · · σ2n ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎥⎢ ⎢ . ⎥=⎢ . ⎥ ⎦⎢ ⎣ ⎣ .. ⎦ ⎣ .. ⎦ σK1 σK2 · · · σKn αn hK ⎡

the solutions α exist by construction. If K ≥ n, the solutions are unique because there is a n × n matrix embedded in the matrix on the left side which is non-singular, which uniquely determines α. If K < n, the solutions are not unique. This completes the proof. Two special cases are worth mentioning at this time. First, for markets consisting of only one risky asset, the risky asset in the market is trivially non-redundant if its price process is not a constant. An example of such a market is the Black-ScholesMerton model in Chap. 5. Second, for a market where the local martingales Mi (t) are independent Brownian motions under some Q ∼ P, the condition [Mi , Mj ] = 0 for all i = j is automatically satisfied, see Lemma 8 in Chap. 1. We will revisit this market in Sect. 2.8 below.

2.5 The First Fundamental Theorem This section presents the first fundamental theorem of asset pricing. Before presenting the first fundamental theorem, however, there are a number of concepts that need to be introduced, so that the theorem is well understood. We consider a normalized market (S, F, P).

38

2 The Fundamental Theorems

2.5.1 No Arbitrage (NA) We start with a definition of an arbitrage opportunity. Definition 25 (No Arbitrage (NA)) An admissible s.f.t.s. (α0 , α) ∈ A (x) with value process X is a (simple) arbitrage opportunity if (i) X0 = x = 0, (zero investment) T (ii) XT = 0 αt · dSt ≥ 0 with P probability one, and (iii) P (XT > 0) > 0. A market satisfies no arbitrage (NA) if there are no trading strategies that are arbitrage opportunities. As defined, a simple arbitrage opportunity requires zero wealth at time 0. It is called a zero investment trading strategy. In addition, an arbitrage opportunity never loses any wealth and with strictly positive probability it generates positive wealth at time T . This is equivalent to getting a free lottery ticket, where the lottery is yet to occur. Remark 16 (Other Simple Arbitrage Opportunities) Two other types of simple arbitrage opportunities, that are also excluded by NA, need to be understood. The first is where the admissible s.f.t.s. (α0 , α) ∈ A (x) for X0 = x is not zero initial investment, but it generates an immediate positive cash flow. In particular, it satisfies (i) X0 = x < 0,  Tand (ii) XT = x + 0 αt · dSt ≥ 0 with P probability one. Using the mma, one can easily transform this type of arbitrage opportunity into that used in the definition. This is done by immediately purchasing x units of the mma with the positive cash flow generated by condition (i), so that the time 0 value of the transformed trading strategy is zero. Note that a negative value at time 0, as in condition (i), implies a positive cash flow. With this reinvestment, condition (ii) then implies that the trading strategy’s terminal value will be greater than or equal T to 0 αt · dSt , which is strictly greater than zero with P probability one. Hence, this transformed trading strategy is now an arbitrage opportunity as in Definition 25. The second type of alternative arbitrage trading strategy is where the admissible s.f.t.s. (α0 , α) ∈ A (x) for X0 = x has strictly positive initial investment, and it generates payoffs in excess of its initial investment. In particular, it satisfies (i) X0 = x > 0, T (ii) XT = x + 0 αt · dSt ≥ x with P probability one, and (iii) P (XT > x) > 0. Using the mma, one can easily transform this type of arbitrage opportunity into that used in the definition. This is done by immediately shorting x units of the mma, and adding this to the initial s.f.t.s. Hence, because of these considerations, restricting our attention to zero initial investment trading strategies as in Definition 25 is without-loss-of-generality. This completes the remark.

2.5 The First Fundamental Theorem

39

Remark 17 (Numeraire Invariance) In order for normalization by the mma to be without-loss-of generality when studying no arbitrage, it is important to note that the definition of an arbitrage opportunity is invariant to normalization by the mma. Indeed, since Bt > 0 all t, X0 > 0 XT ≥ 0

⇔ ⇔

X0 = XT =

X0 B0 XT B T

P (XT > 0) > 0 ⇔ P (XT > 0) = P

>0 ≥0 XT BT

 >0 >0

This completes the verification and the remark. Remark 18 (Invariance with Respect to a Change in Equivalent Probability Measures) The admissible s.f.t.s.’s in the definitions of NA are also invariant with respect to a change in equivalent probability measures. This follows because the definitions only depend on events of probability zero or strictly positive probability, which are preserved by an equivalent probability measure. This completes the remark. Example 4 (Asset Price Processes Violating NA) This example illustrates asset price processes that violate NA. 1. (Reflected Brownian Motion) Let the market consist of a single risky asset and a mma with the risky asset price process given by S1 (t) = S0 |Wt | for all t ∈ [0, T ] where S0 > 0 and Wt is a standard Brownian motion with W0 = 1. This process is known as a reflected Brownian motion. This price process admits a simple arbitrage opportunity. Indeed, consider the following trading strategy. Let τ > 0 correspond to this first time St hits 0 after time 0. Note that P (0 < τ < T ) > 0. Buy the risky asset at τ . Let τ ∗ be the first time after τ that Sτ ∗ ≥ ε where ε > 0 is a constant. Note that P ( τ ∗ ≤ T | 0 < τ < T ) = P( sup St ≥ ε |Sτ = 0 ) > 0. Then, sell the risky t∈(τ,T ]

asset at τ ∗ . After selling, invest the proceeds in the mma until time T . Let Xt denote the value process of this trading strategy. Note that at time 0, X0 = 0 because no position is taken in the risky asset or the mma. At time τ , Xτ = 1 · Sτ = 0 because the asset is purchased when it is of zero value, and ⎧ ⎨ ε if 0 < τ < T and τ ∗ ≤ T XT = 0 if 0 < τ < T and τ ∗ > T ⎩ 0 if τ ≥T

40

2 The Fundamental Theorems

This trading strategy is self-financing and it is admissible since Xt ≥ 0 for all t ∈ [0, T ]. Finally, since P (XT = ε > 0) = P (0 < τ < T and τ ∗ ≤ T ) = P (τ ∗ ≤ T |0 < τ < T ) P (0 < τ < T ) > 0, and P (XT ≥ 0) = 1, it is a simple arbitrage opportunity. This example can be generalized to any risky asset price process that hits zero at some stopping time τ ∈ (0, T ] with strictly positive probability, and after hitting zero the value process has a strictly positive probability of becoming strictly positive again before time T. The same admissible s.f.t.s. works to create an arbitrage opportunity. Hence, we get the following fact. Assume NA. Given a risky asset price process St , if P (Sτ = 0) > 0 for any τ ∈ (0, T ) and P( sup St ≥ ε |Sτ = 0 ) > 0 for t∈(τ,T ]

some ε > 0 then St = 0 for all t ∈ [τ, T ]. 2. (Redundant Assets with Different Drifts) Let the market consist of two risky assets and a mma with the risky asset price processes given by S1 (t) = S0 eμ1 t+σ Wt , and S2 (t) = S0 eμ2 t+σ Wt for all t ∈ [0, T ] where S0 > 0, μ1 > μ2 ≥ 0 are constants, and Wt is a standard Brownian motion with W0 = 0. Both of these risky asset prices follow geometric Brownian motions with the same volatility σ > 0. These price processes admit a simple arbitrage opportunity. Indeed, consider the following trading strategy. At time 0, buy risky asset 1 and short risky asset 2. Hold these positions until time T . Next, let Xt denote the value process of this trading strategy. Note that at time 0, X0 = S0 − S0 = 0, and XT = S0 (eμ1 T − eμ2 T )eσ WT > 0 a.s. P. This strategy is admissible since Xt ≥ 0 for all t ∈ [0, T ], it is self-financing and, therefore, it is a simple arbitrage opportunity. This completes the example. Unfortunately, this intuitive definition of no arbitrage is not strong enough to obtain the first fundamental theorem of asset pricing. The reason is because an admissible s.f.t.s.’s value is characterized as a stochastic integral, and stochastic integrals are themselves defined as limits of sums. Hence, approximate arbitrage opportunities that become arbitrage opportunities in the limit must also be excluded. We need to study additional types of admissible s.f.t.s. that characterize these “approximate arbitrage opportunities.”

2.5 The First Fundamental Theorem

41

2.5.2 No Unbounded Profits with Bounded Risk (NUPBR) Another type of admissible s.f.t.s. relates to a limiting type of arbitrage opportunity called no unbounded profits with bounded risk (NUPBR). This definition is from Karatzas and Kardaras [121, p. 465]. Definition 26 (No Unbounded Profits with Bounded Risk (NUPBR)) A sequence of admissible s.f.t.s.’s (α0 , α)n ∈ A (x) with value processes Xn generate unbounded profits with bounded risk (UPBR) if (i) X0n = x > 0, (positive investment) (ii) Xtn ≥ 0 for all t ∈ (0, T ] a.s. P (always nonnegative), and (iii) (XTn )n ≥ 0 are unbounded in probability, i.e. it is not the case that lim (sup P(XTn > m)) = 0. m→∞

n

If no such sequence of trading strategies exists, then the market satisfies NUPBR. A sequence of admissible s.f.t.s. whose value processes are always nonnegative with an initial value of x > 0 (condition (i)) generates UPBR if it never loses more than the initial investment of x units (condition (ii)), and it generates unboundedly large payoffs at time T with strictly positive probability (condition (iii)). Because condition (ii) implies that this admissible s.f.t.s. can incur losses, the trading strategy has risk. But, the risk is bounded because the loss is limited to the initial investment of x > 0. And, condition (iii) that the sequence Xn is unbounded in probability means that there exists an ε > 0 such that for all m > 0, sup P(XTn > m) ≥ ε. Alternatively stated, there is no largest payoff to the n

sequence of admissible s.f.t.s. Given this explanation, it is understandable why such a sequence of admissible s.f.t.s. that generates an UPBR is an attractive trading strategy. Without-loss-of-generality, one can replace x > 0 with x = 1 in Definition 26. Indeed, if the admissible s.f.t.s. (α0 , α)n ∈ A (1) satisfies all of the conditions of Definition 26, then the admissible s.f.t.s. (α˜ 0 , α) ˜ n = (xα0 , xα)n ∈ A (x) satisfies Definition 26 for an arbitrary x > 0. An equivalent definition of NUPBR is given in the appendix to this chapter. Remark 19 (No Arbitrage of the First Kind) The definition of an UPBR can be clarified by relating it to the notion of an arbitrage opportunity of the first kind. This definition is from Kardaras [126, p. 653]. Definition (No Arbitrage of the First Kind) A FT -measurable random variable χ generates an arbitrage opportunity of the first kind if (i) P(χ ≥ 0) = 1, (ii) P(χ > 0) > 0, and (iii) for all x > 0 there exists an admissible s.f.t.s. (α0 , α) ∈ A (x) with value process Xx,α ≥ 0 such that XTx,α ≥ χ .

42

2 The Fundamental Theorems

The market satisfies no arbitrage of the first kind if there exits no arbitrage opportunities of the first kind. As defined, no arbitrage of the first kind excludes a related, but different type of admissible s.f.t.s. Indeed, to be an arbitrage opportunity, there needs to exist a random variable χ satisfying conditions (i) and (ii), and a zero investment (x = 0) admissible s.f.t.s. (α0 , α) ∈ A (0) such that XT0,α ≥ χ . Here, however, the admissible s.f.t.s must have a strictly positive initial investment x > 0 with a value process that is nonnegative at time T . Key here is that the initial investment can be made arbitrarily small since condition (ii) must be true for all x > 0. The fact that the initial investment must be strictly positive is also crucial. It implies that a “small” loss on the trading strategy is possible, although the loss is bounded by the initial value of x. Note too that the magnitude of this random variable χ can be made unboundedly large by multiplying both it and the initial investment x by an arbitrary constant c > 0 and letting c → ∞. An amazing result, shown by Kardaras [126] (see also Kabanov et al. [119, Lemma A.1, p. 1105]), is that NUPBR is equivalent to no arbitrage of the first kind. This completes the remark. Remark 20 (Invariance with Respect to a Change in Equivalent Probability Measures) It is important to note that the admissible s.f.t.s.’s in the definition of NUPBR are invariant with respect to a change of equivalent probability measures. This follows because the definitions only depend on events of probability zero or strictly positive probability. This completes the remark. Let us denote by L+0 the set of adapted and cadlag (right continuous with left limits existing) stochastic processes Y (t, ω) on [0, T ] × Ω such that Yt ≥ 0. Next, we define a subset of L+0 that are needed to give an alternative characterization of NUPBR.  Dl = Y ∈ L+0 : Y0 = 1, XY is a P local martingale,  (2.9) X = 1 + α · dS, (α0 , α) ∈ A (1) . A stochastic process in this set is called a local martingale deflator process. It is called this because when the value process of an admissible s.f.t.s. with unit initial wealth is multiplied by a Y ∈ Dl (a deflator or change of numeraire) the resulting product XY is a P local martingale. This set of local martingale deflator processes depends on the probability measure P. We can now present the characterization of NUPBR, which is stated without proof (see Takaoka and Schweizer [181] or Kardaras [126]). Theorem 11 (Probability Characterization of NUPBR) The market satisfies NUPBR if and only if Dl = ∅, i.e. there exists a local martingale deflator process. This result will prove useful in the portfolio optimization Chap. 11 below in an incomplete market (this concept is defined below). The notion of NUPBR in conjunction with NA is sufficient to obtain the first fundamental theorem of asset

2.5 The First Fundamental Theorem

43

pricing. Before stating and proving this theorem, however, it is necessary to note some useful properties of the set of local martingale deflator processes. Associated with the set of local martingale deflator (stochastic) processes is the set of local martingale deflators (random variables), Dl = {YT ∈ L0+ : ∃Z ∈ Dl , YT = ZT } where L0+ is the set of nonnegative FT -measurable random variables.

2.5.3 Properties of Dl A number of observations are important with respect to the set of local martingale deflator processes in Dl . 1. Note that buying and holding only the mma generates an admissible s.f.t.s. with X0 = 1 and Xt = 1 for all t. This implies that any Y ∈ Dl is a P local martingale. And, since it is bounded below by zero, Y ∈ Dl is a P supermartingale (see Lemma 4 in Chap. 1). Finally, since Y0 = 1, the supermartingale property implies E [YT ] ≤ 1. 2. Because the admissible s.f.t.s. X is bounded below and Y ≥ 0, Yt Xt is bounded below. Yt Xt being a P local martingale implies that for any Y ∈ Dl , Yt Xt is a P supermartingale, i.e. Yt Xt ≥ E [YT XT |Ft ] for all t ≥ 0 (see Lemma 4 in Chap. 1). 3. A special subset of the set of local martingale deflator processes are those Y ∈ Dl such that YT = dQ dP > 0 for some probability measure Q equivalent to P, i.e.  E [YT ] = E

 dQ = 1 = Y0 . dP

By Lemma 3 in Chap. 1, because Y is a P supermartingale, this implies that Y is a P martingale, i.e. Yt = E [YT |Ft ] . We denote this subset of local martingale deflator processes by   dQ Ml = Y ∈ Dl : ∃Q ∼ P, YT = . dP This set of stochastic processes generates the associated set of local martingale deflators (random variables) Ml = {YT ∈ L0+ : ∃Z ∈ Ml , YT = ZT }. Given these properties, we can prove the next lemma.

44

2 The Fundamental Theorems

Lemma 11 (X Is a Q Local Martingale) Let Q be such that for some YT ∈ Ml , YT = dQ dP . Then, t X is a Q local martingale for Xt = 1 + 0 αu · dSu , (α0 , α) ∈ A (1). Proof Since YT ∈ Ml , Yt Xt is a local martingale under P where Yt = E [YT |Ft ]. Hence, there exists a sequence of stopping times (τn ) such that lim τn = T n→∞

a.s. P and Yt∧τn Xt∧τn is a martingale for each n. To prove this lemma  it is  sufficient to show that Xt∧τn is a martingale under Q, i.e. E YYTt XT ∧τn |Ft = Xt dQ dP

|t = YYTt (see Karatzas and Shreve [123], Lemma,     = = Xt . But, E YYTt XT ∧τn |Ft p. 193) implies that E Q XT ∧τn |Ft       

 E E YYTt XT ∧τn FT ∧τn |Ft = E E YT FT ∧τn Y1t XT ∧τn |Ft . Since Y is a uniformly integrable martingale P, by Doob’s Optional Sampling

 under

F Theorem (Protter [158, p. 9]), E Y = YT ∧τn . Substitution yields T T ∧τn    Yt Xt  YT ∧τn 1 E Yt XT ∧τn |Ft = Yt E YT ∧τn XT ∧τn |Ft = Yt = Xt which completes the proof.

for τn ≥ t since

This lemma enables us to prove that the set of equivalent probability measures Q generated by the local martingale deflators YT ∈ Ml is equivalent to the set of equivalent local martingale measures, denoted Ml = {Q ∼ P : S is a Q local martingale} , as the next lemma states. Lemma 12 (Characterization of Ml )   dQ Ml = Q : ∃YT ∈ Ml , YT = dP    = Q ∼ P : X is a Q local martingale, X = 1 + α · dS, (α0 , α) ∈ A (1) . Proof Define   N = Q ∼ P : X is a Q local martingale, X = 1 + α · dS, (α0 , α) ∈ A (1) . (Step 1) Show N ⊂ Ml . Choose Q ∈ N. Consider a buy and hold trading strategy in a single asset with a unit holding in that asset. This is an admissible s.f.t.s. For this buy and hold trading strategy Xt = SS0t is a Q local martingale by the definition of N. Hence, S is a Q local martingale, implying Q ∈ Ml . (Step 2) Show Ml ⊂ N.  Choose Q ∈ Ml . Since S is a Q local martingale, and given that X = 1+ α ·dS is bounded below by admissibility of the s.f.t.s., by Lemma 6 in Chap. 1, X is a Q local martingale. This implies Q ∈ N.

2.5 The First Fundamental Theorem

45

(Step 3) The first equality follows from the definitions of the various sets and Lemma 11. This completes the proof. The subset of equivalent probability measures making S a Q martingale, and not just a local martingale, will also prove to be an important set. This is called the set of equivalent martingale measures and denoted by M = {Q ∼ P : S is a Q martingale} . Associated with the set of equivalent martingale measures is the set of martingale deflator processes M = {Y ∈ L+0 : YT =

dQ , Yt = E [YT |Ft ] , Q ∈ M} dP

= {Y ∈ L+0 : Y ∈ Ml , YT =

dQ , Q ∈ M} dP

and the set of martingale deflators M = {YT ∈ L0+ : YT =

dQ , Q ∈ M} dP

= {YT ∈ L0+ : ∃Z ∈ M , YT = ZT }. For easy reference, we summarize the relationships among the various sets. probability measures : stochastic processes : random variables :

M ⊂ Ml ⊂ {Q : Q ∼ P} M ⊂ Ml ⊂ Dl ⊂ L+0 M ⊂ Ml ⊂ Dl ⊂ L0+

Remark 21 (Properties of M) We note that unlike the result that holds for local martingale measures as given in Lemma 12,    M = Q ∼ P : X is a Q martingale, X = 1 + α · dS, (α0 , α) ∈ A (1) .  This is true because given a Q ∼ P, X = 1 + α · dS for an arbitrary (α0 , α) ∈ A (1) is in general only a local martingale and not a martingale. Hence for a given martingale measure Q, to ensure that X is also a martingale under Q, additional restrictions on the trading strategies (α0 , α) ∈ A (1) are needed. For a set of sufficient conditions on both the martingale measure Q (therefore S) and the

46

2 The Fundamental Theorems

stochastic process X (therefore the trading strategies (α0 , α) ∈ A (1)) that guarantee that X is a martingale under Q, see Theorem 17 below. This completes the remark.

2.5.4 No Free Lunch with Vanishing Risk (NFLVR) Finally, we get to the strengthening of NA needed to prove the first fundamental theorem of asset pricing. Because the following definition is new to the literature, the appendix to this chapter shows that it is equivalent to the standard definition contained in Delbaen and Schachermayer [44]. Definition 27 (No Free Lunch with Vanishing Risk (NFLVR)) A free lunch with vanishing risk (FLVR) is a sequence of admissible s.f.t.s.’s (α0 , α)n ∈ A (x) with initial value x ≥ 0, value processes Xtn , lower admissibility bounds 0 ≥ cn where ∃c ≤ 0 with cn ≥ c for all n, and a FT -measurable random variable χ ≥ x with P(χ > x) > 0 such that XTn → χ in probability. The market satisfies NFLVR if there exist no FLVR trading strategies. As defined, a market satisfies FLVR if there is a sequence of admissible s.f.t.s. (α0 , α)n ∈ A (x) that approaches a simple arbitrage opportunity in the limit. Note that the sequence of admissible s.f.t.s.’s have a uniform lower admissibility bound c ≤ 0. It is easy to see that a simple arbitrage opportunity is a FLVR. Indeed, consider the simple arbitrage opportunity (α0 , α) ∈ A (0) with value process Xt and admissibility bound c that satisfies X0 = 0, XT ≥ 0, and P(XT > 0) > 0. Define the (constant) sequence of admissible s.f.t.s.’s using the simple arbitrage opportunity, i.e. (α0 , α)n = (α0 , α) whose time T value process XTn = XT approaches (equals) χ = XT where χ ≥ 0 with P(χ > 0) > 0. This sequence (α0 , α)n is a FLVR. Hence, NFLVR excludes both simple and approximate arbitrage opportunities. Remark 22 (Equivalent Definitions of NFLVR) Equivalent, but different formulations of NFLVR appear in the literature (see Karatzas and Kardaras [121, p. 466]). Two of these are worth mentioning. In the first, one can replace x > 0 with x = 1. To see this, suppose (α0 , α)n ∈ A (1) has an initial value 1, value processes Xtn , lower admissibility bounds 0 ≥ cn where ∃c ≤ 0 with cn ≥ c for all n, and a FT -measurable random variable χ ≥ 1 with P(χ > 1) > 0 such that XTn → χ in probability. This sequence of admissible s.f.t.s. can be transformed into a FLVR. Indeed, for an arbitrary x > 0, define the admissible s.f.t.s. (α˜ 0 , α) ˜ n = (xα0 , xα)n ∈ A (x). A straightforward verification shows that this is a FLVR with x > 0. In the second, one can replace c ≤ 0 with c = 0. However, in this case one must restrict the initial wealth x > 0. To see this, suppose (α0 , α)n ∈ A (x) has an initial value x > 0, value processes Xtn , lower admissibility bounds cn ≥ 0 for all n, and a FT -measurable random variable χ ≥ x with P(χ > x) > 0 such that XTn → χ in probability. This sequence of admissible s.f.t.s. can be transformed into a FLVR.

2.5 The First Fundamental Theorem

47

Indeed, let c < 0 be an arbitrary uniform lower bound. If x + c ≥ 0, then a straight forward verification shows that (α˜ 0 , α) ˜ n = (α0 + c, α)n ∈ A (x + c) is a FLVR with the uniform lower admissibility bound equal to c < 0. If x + c < 0, then first transform (α0 , α)n ∈ A (x) to (α˜ 0 , α) ˜ n = ( xx˜ α0 , xx˜ α)n ∈ A (x) ˜ where x˜ + c ≥ 0, which is a FLVR, and then apply the previous modification with x˜ + c ≥ 0 to complete the argument. This completes the remark. An alternative characterization of NFLVR is given in the next theorem, which is stated without proof (see Delbaen and Schachermayer [44, Corollary 3.8] or Kabanov et al. [119, Proposition A.4, p. 1106]). Theorem 12 (Characterization of NFLVR) The market satisfies NA and NUPBR if and only if NFLVR. This theorem characterizes NFLVR as equivalent to both NA and NUPBR. As shown, NFLVR is a stronger assumption than either of NUPBR or NA alone. Remark 23 (Invariance with Respect to a Change in Equivalent Probability Measures) It is important to note that the admissible s.f.t.s.’s in the definitions of NFLVR are invariant with respect to a change in equivalent probability measures. This follows because the definitions only depend on events of probability zero or strictly positive probability. This completes the remark. Example 5 (Asset Price Processes Violating NFLVR) A collection of asset price processes that violate NFLVR are contained in the Brownian motion market of Sect. 2.8 below. In this section, Theorem 19 gives necessary and sufficient conditions on the asset price processes such that NFLVR is satisfied. This completes the example. NFLVR is the basis of the first fundamental theorem of asset pricing to which we now turn.

2.5.5 The First Fundamental Theorem Given the economic characterization of NFLVR, to prove theorems, we need a probabilistic characterization. This is given by the first fundamental theorem of asset pricing. Theorem 13 (The First Fundamental Theorem) NFLVR if and only if Ml = ∅, i.e. there exists a Q ∼ P such that S is a Q local martingale. Q ∈ Ml is called an equivalent local martingale measure. Proof (Ml = ∅ ⇒ NFLVR) This is a proof by contradiction. Assume Ml = ∅. Let Q be such an equivalent local martingale measure. Let there exist a FLVR with initial value x = 0. Then, there exists a c ≤ 0 and a sequence of admissible s.f.t.s.’s (α0 , α)n ∈ A (0) with lower admissibility bounds

48

2 The Fundamental Theorems



0 ≥ cn ≥ c for all n, values Xtn , and a FT -measurable random variable f ≥ 0 with P(f > 0) > 0 such that XTn → f in probability. For the admissible s.f.t.s., note that there exists a subsequence such that XTn → f a.s. P (see Jacod and Protter [76, p. 141]). By Fatou’s lemma, lim E Q [XTn ] ≥ n→∞

E Q [ lim XTn ] = E Q [f ] > 0. Since Q is a local martingale measure, by Lemma 12, n→∞

Xtn is a local martingale under Q. Xtn is also a supermartingale under Q by Lemma 4 in Chap. 1 because Xtn is bounded below by the admissibility condition c ≤ 0. Thus, 0 = X0 ≥ E Q [XTn ] for all n, which implies 0 ≥ lim E Q [XTn ] > 0. This is the n→∞

contradiction. (NFLVR ⇒ Ml = ∅) is given in Delbaen and Schachermayer [46]. However, they prove NFLVR implies there exists an equivalent Q such that S is a σ -martingale under Q. But, since S ≥ 0, every nonnegative σ -martingale is a local martingale, see the proof of Lemma 6 in Chap. 1, which implies Q ∈ Ml . This completes the proof. Remark 24 (An Infinite Number of Traded Risky Assets) The statement NFLVR ⇒ Ml = ∅ depends on the fact that there is only a finite number of risky assets trading. The statement Ml = ∅ ⇒ NFLVR holds more generally for an arbitrary number of risky assets (see the proof of the first fundamental theorem). This collection could be a continuum (an uncountably infinite set). However, in this extension, any trading strategy must still consist of only a finite number of traded assets, even though an infinite number of risky assets are available to choose from. This completes the remark. Using Lemma 4 in Chap. 1, this theorem has an important corollary. Corollary 1 If NFLVR, then St is a supermartingale for any Q ∈ Ml , i.e. St ≥ E Q [ST |Ft ] f or all t.

(2.10)

Note that the asset price process is not necessarily a martingale. The difference between the price of the asset and the conditional expectation on the right side of expression (2.10), if strictly positive, will be seen (in Chap. 3) to represent an asset price bubble.

2.5.6 Equivalent Local Martingale Measures This section generates some observations about equivalent local martingale measures useful in subsequent chapters. Fix a Q ∈ Ml . Because Q is mutually absolutely continuous (equivalent) with respect to P, there exists a unique strictly positive Radon Nikodym derivative (see Ash [3, p. 63]) YT =

dQ > 0 ∈ Ml dP

(2.11)

2.5 The First Fundamental Theorem

49

such that E[1A YT ] = E Q [1A ] = Q(A)

(2.12)

for all A ∈ FT . YT is a probability density function with respect to P. This local martingale deflator YT uniquely determines a local martingale deflator process Y ∈ Ml defined by Yt = E [YT |Ft ] .

(2.13)

As seen, Y is a uniformly integrable martingale. Using this stochastic process, we can define conditional expectations of indicator functions with respect to Q, i.e.  YT E Q [1A |Ft ] = E 1A Yt



Ft

(2.14)

for all A ∈ FT . Dividing by Yt on the right side of this equality is necessary to make E Q [1Ω |Ft ] = 1. Last, given an FT -measurable random variable XT that is integrable with respect to Q, we have E Q [XT ] = E [YT XT ]

(2.15)

  YT

Ft . E Q [XT |Ft ] = E XT Yt

(2.16)

and

2.5.7 The State Price Density In this section, we consider the non-normalize market ((B, S), F, P) to define the state price density, sometimes called the stochastic discount factor. Definition 28 (State Price Density) Let Y ∈ Ml . The state price density H ∈ L+0 is Ht =

Yt E [YT |Ft ] = . Bt Bt

Lemma 13 Assume NFLVR in ((B, S), F, P). (St Ht ) is a local martingale under P for any state price density H ∈ L+0 .

(2.17)

50

2 The Fundamental Theorems

Proof NFLVR in the non-normalized market ((B, S), F, P) implies NFLVR in the normalized market (S, F, P), which implies there exists a Y ∈ Ml . Hence, there exists a state price density H ∈ L+0 . Note that St Ht = St Yt . We have that St is a local martingale under Q. This is equivalent to St Yt being a local martingale under P. Indeed, consider a sequence of stopping times (τn ) ↑ T . Using expression (2.15), because the stopped process St∧τn under Q is a martingale, the stopped process St∧τn Yt∧τn is a martingale under P. Hence, St Ht is a local martingale under P. This completes the proof. By Lemma 4 in Chap. 1, this implies that the nonnegative process (St Ht ) is a supermartingale under P, i.e. 

HT St ≥ E ST Ht



Ft .

(2.18)

For subsequence use, the following lemma will be important. Lemma 14 Assume NFLVR in ((B, S), F, P). Consider an admissible s.f.t.s. (α0 , α) ∈ A (x) with initial value x > 0 and value process dXt = α0 (t)rt Bt dt + α(t) · dSt . Then, (Xt Ht ) is a local martingale under P. Proof Note that Xt Ht = Xt Yt where dXt = αt · dSt is a local martingale under Q. Consider a sequence of stopping times (τn ) ↑ T . We have that expression (2.15) holds for the stopped process Xt∧τn Yt∧τn under P. This completes the proof. Since Xt is bounded below by the admissibility condition, again by Lemma 4 in Chap. 1, the nonnegative process (Xt Ht ) is a supermartingale under P, i.e. 

HT Xt ≥ E XT Ht



Ft .

(2.19)

2.6 The Second Fundamental Theorem This section presents the second fundamental theorem of asset pricing which relates the notion of a complete market to local martingale measures. We are given a normalized market (S, F, P), and as before, we start with some definitions. Let L0+ := L0+ (Ω, F , P) be the space of all nonnegative FT -measurable random variables, and L1+ (Q) := L1+ (Ω, F , Q) to be the space of all nonnegative FT -measurable random variables X such that E Q [X] < ∞. The set of random variables L0+ should be interpreted as the set of “derivative securities” and L1+ (Q) should be interpreted as the set of “derivative securities whose payoffs are integrable with respect to the probability measure Q.”

2.6 The Second Fundamental Theorem

51

2.6.1 Attainable Securities Define   C (x) = XT ∈ L0+ : ∃(α0 , α) ∈ A (x), x +

T

 αt · dSt = XT .

0

This is the set of nonnegative random variables generated by all admissible s.f.t.s. starting with an initial value of x ≥ 0. The collection ∪ C (x) is the set of securities that can be obtained—synthetically x≥0

constructed—via the use of admissible s.f.t.s. from nonnegative initial values x ≥ 0. These securities are called attainable. Definition 29 (Attainable Securities) A random variable ZT ∈ L0+ is attainable if ZT ∈ ∪ C (x), i.e. there exists an x ≥ 0 and an (α0 , α) ∈ A (x) such that x≥0



T

x+

αt · dSt = ZT .

(2.20)

0

We say that (α0 , α) ∈ A (x) attains ZT . The attainable securities, although important, are not sufficient for the valuation and hedging of derivatives. To understand why, let us first assume that the market satisfies NFLVR, so that the risky asset price processes S are local martingales under a Q ∈ Ml . There may be many such probability measures Q ∈ Ml , choose one. Second, let us assume that the risky assets S are non-redundant. This means  t that the admissible s.f.t.s. (α0 , α) ∈ A (x) generating the value process Xt = x + 0 αs · dSs is unique. Now suppose all derivatives ZT ∈ L0+ are attainable. Then, there exists an T (α0 , α) ∈ A (x) such that x + 0 αt · dSt = ZT . First, we note that the value t process generated by this trading strategy Xt = x + 0 αs · dSs is a local martingale under Q, bounded below, and hence a supermartingale under Q. We claim that there may exist another admissible s.f.t.s. (α˜ 0 , α) ˜ ∈ A (x + y) with y > 0 such T that x + y + 0 α˜ t · dSt = ZT . Indeed, if they exist, one can choose a suicide strategy (ϕ0 , ϕ) ∈ A (0) that loses y dollars for sure by time T . Then, add this to the original s.f.t.s., i.e. (α˜ 0 , α) ˜ = (α0 , α)+(ϕ0 , ϕ). This admissible s.f.t.s. starts with x + y dollars and attains ZT too, but the value processes differ. Because the suicide strategy’s value process is a strict supermartingale under Q, the value process X˜ generated by (α˜ 0 , α) ˜ ∈ A (x + y) will be a strict supermartingale under Q too. So, there is non-uniqueness of the trading strategy attaining ZT . But, more problematic is that this non-uniqueness affects how one determines the “arbitrage-free” price of the derivative. The first problem is that the value processes X and X˜ generated by both (α0 , α) ∈ A (x) and (α˜ 0 , α) ˜ ∈ A (x + y) are different, with X a supermartingale and X˜ a strict supermartingale under Q. This implies that

52

2 The Fundamental Theorems

x ≥ E Q [ZT ] and x +y > E Q [ZT ]. Since both of these trading strategies attain ZT , logic dictates that one should never pay more than x for the derivative. Generalizing, the most one should pay is inf x≥E Q [ZT ]

{x : ∃ (α0 , α) ∈ A (x) that attains ZT }

where the lower bound on x exists because all of the value processes are supermartingales under Q. Three problems arise. First, we do not know if the infimum equals E Q [ZT ] or not. Second, even if the infimum equals E Q [ZT ], we do not know if there exists a s.f.t.s. (α0 , α) ∈ A (E Q [ZT ]) with initial value E Q [ZT ] that attains ZT . If one does exist, however, the value process is a martingale under Q because X0 = x = E Q [ZT ] (Lemma 3 in Chap. 1), and the admissible s.f.t.s attaining this value process is unique, thereby resolving all the previous issues, except one. Recall that there may be more than one Q ∈ Ml . So, which one do we choose and why? It turns out, surprisingly perhaps, that if such an admissible trading strategy exists whose value process is a martingale under Q for all derivatives, then there can be at most one Q ∈ Ml , and the remaining problem disappears as well. This discussion motivates the need to have a stronger notion for the synthetic construction of a derivative’s payoff than attainability, which is given in the next section.

2.6.2 Complete Markets As seen in the previous section, an important market is one satisfying NFLVR in which given any derivative whose payoff is integrable with respect to a local martingale measure Q ∈ Ml , there exists an admissible s.f.t.s. that attains the derivative’s payoff, and this trading strategy’s value process is a Q martingale. If this is true for all derivatives, a market is called complete. Definition 30 (Complete Market with Respect to Q ∈ Ml ) Given Ml = ∅. (NFLVR) Choose a Q ∈ Ml . The market is complete with respect to Q if given any ZT ∈ L1+ (Q), there exists a x ≥ 0 and (α0 , α) ∈ A (x) such that  T αu · dSu = ZT , x+ 0

and the value process  Xt = α0 (t) + αt · St = x +

t

αu · dSu

0

is a Q martingale, i.e. Xt = E Q [ZT |Ft ] for all t ∈ [0, T ].

2.6 The Second Fundamental Theorem

53

The definition of a complete market depends on the market satisfying NFLVR and the choice of a probability measure Q ∈ Ml . Unlike the definitions of NA, NUPBR, and NFLVR, the definition of a complete market is not invariant with respect to a change in equivalent probability measures. This follows because the set of integrable random variables L1+ (Q) depends on Q. Similar, but different definitions for a complete market can be found in the literature. For a definition of a complete market that is independent of both NFLVR and a particular Q ∈ Ml see Battig and Jarrow [9]. Remark 25 (Attainable Securities and Q Martingales) The definition of a complete market with respect to Q ∈ Ml requires that there exists an admissible s.f.t.s. that T satisfies x + 0 αu · dSu = ZT at time T , i.e. the security ZT is attainable. But, the definition of a complete market is stronger. In addition, the definition also requires t that this admissible s.f.t.s makes the value process Xt = α0 (t) + αt · St = x + 0 αu · dSu a Q martingale.  t This is a stronger condition than attainability because the stochastic integral 0 αu · dSu , being bounded below (here by zero), is only a Q local martingale (see Lemma 6 in Chap. 1). It need not, however, be a Q martingale. The next corollary gives a sufficient condition on ZT such that the value process X is a Q martingale for any admissible s.f.t.s that attains ZT . An alternative set of sufficient conditions for X to be a Q martingale are provided in Theorem 17 later in this chapter. This completes the remark. Corollary 2 (Attainability and Boundedness) Given Ml = ∅. Let ZT ≥ 0 be attainable, i.e. there exists a x ≥ 0 and (α0 , α) ∈ A (x) with value process  Xt = α0 (t) + αt · St = x +

t

αu · dSu

0

such that 

T

XT = x +

αu · dSu = ZT .

0

Let Xt be bounded above, i.e. Xt ≤ k for some k > 0 and all t ∈ [0, T ]. Then, Xt is a Q martingale, i.e. Xt = E Q [ZT |Ft ]. Proof By admissibility Xt is bounded below, hence by Lemma 6 in Chap. 1, Xt is a Q local martingale. By Remark 4 in Chap. 1, Xt is a Q martingale. This complete the proof. Remark 26 (Uniqueness of the s.f.t.s. (α0 , α) ∈ A (x) Attaining ZT ) If the risky assets are non-redundant, then there is only one admissible s.f.t.s. (α0 , α) ∈ A (x) that attains ZT and has a value process Xt = E Q [ZT |Ft ] and initial value X0 = E Q [ZT ]. There may be other admissible s.f.t.s. (α0 , α) ∈ A (z) that attain ZT too, but they will all have z > x and be strict supermartingales under Q. In this case,

54

2 The Fundamental Theorems

the admissible s.f.t.s. that attains ZT will be (α0 , α) ∈ A (x) modified by adding a suicide strategy that loses z − x > 0 for sure by time T . This completes the remark. Remark 27 (Indicator Functions) Note that if the market is complete with respect to Q, then 1A ∈ L1+ (Q) is attainable for all A ∈ FT and the value process of the admissible s.f.t.s. that generates it is a martingale. These indicator random variables will be identified as abstract Arrow Debreu securities in Chap. 4 below. They are used in the proof of the second fundamental theorem of asset pricing. This completes the remark. We now state and prove the second fundamental theorem of asset pricing. Theorem 14 (Second Fundamental Theorem) Assume Ml = ∅, i.e. NFLVR. If the market is complete with respect to Q ∈ Ml , then Ml is a singleton, i.e., the equivalent local martingale measure is unique. Conversely, if M = ∅, i.e. there exists an equivalent martingale measure and M is a singleton, then the market is complete with respect to Q ∈ M. Proof (Ml = ∅. Completeness ⇒ Ml a singleton) Consider two measures Q1 , Q2 ∈ Ml . Choose an arbitrary A ∈ FT . Consider ZT = 1A ≥ 0. Since the market is complete with respect to both Q1 , Q2 ∈ Ml , there exists T a x i ≥ 0 and (α0i , α i ) ∈ A (x i ) such that x i + 0 αui · dSu = 1A and Xti = t i x i + 0 αui · dSu = E Q [1A |Ft ] for all t ≥ 0 and for i = 1, 2. We note that for each i, the value process Xti is bounded below by zero and above by 1 for all t ≥ 0. We claim that x 1 = x 2 . We prove this by contradiction. Suppose, without loss of generality, that x 1 > x 2 at time 0. Then, NFLVR is violated (in fact NA is violated), which contradicts the hypothesis of the theorem. To see that NFLVR is violated, consider the following trading strategy: short {x 1 ≥ 0 and (α01 , α 1 ) ∈ A (x 1 )}, go long {x 2 ≥ 0 and (α02 , α 2 ) ∈ A (x 2 )}, and invest {x 1 − x 2 > 0 in a buy and hold position in the m.m.a}. Since −Xt1 is bounded below by −1 for all t ≥ 0, the trading strategy is admissible. It is trivially self-financing. The initial value of this admissible s.f.t.s. is 0 and its time T value is x 1 − x 2 > 0 with probability one. This is a (simple) arbitrage opportunity, which proves the claim. Hence, we have 1 1 x 1 = E Q [XT ] = E Q [1A ] = Q1 (A), 2 2 x 2 = E Q [XT ] = E Q [1A ] = Q2 (A), and x 1 = x 2 , which yields Q1 (A) = Q2 (A). Since this is true for all A ∈ FT , we have Q1 = Q2 . (M = ∅. M a singleton ⇒ Completeness) This proof is contained in Harrison and Pliska [67]. We prove the theorem under the additional hypothesis that S1 , . . . , Sn are H01 (Q) martingales. (Step 1) Prove a martingale representation theorem. First, we need various definitions and a version of the Jacod and Yor Theorem. Let {X1 , . . . , Xn } be H01 (Q) martingales. We note that these are uniformly integrable martingales because they are martingales on [0, T ].

2.6 The Second Fundamental Theorem

55

Define χ = span{X1 , . . . , Xn } and stable(χ ) to be the smallest closed linear subspace containing χ such that if Xt ∈ χ , then Xt∧τ ∈ χ for all stopping times τ (see Protter [158, p. 178]). By Medvegyev [143, Proposition 5.44, p. 341], this is equivalent to closure under stochastic integration, i.e. if X ∈ stable (χ ) and there exists a α ∈ L (X) such that t α • X ∈ H01 (Q), then α • X ∈ stable (χ ) where (α • X)t := 0 α · dX. We say that χ has martingale representation if stable (χ ) = H01 (Q), see Medvegyev [143, p. 329]. Define MH (χ ) = {P ∼ Q : X ∈ χ implies X ∈ H01 (P)}. These are the equivalent probability measures such that X is a H01 (P) martingale. Note that MH (χ ) = ∅ since Q ∈ MH (χ ). Also note that MH (χ ) ⊂ M, the equivalent probability measures P such that X ∈ χ are P martingales. Finally, MH (χ ) is a convex set. A version of the Jacod and Yor Theorem (Medvegyev [143, p. 341]) states χ ⊂ H01 (Q) has martingale representation if and only if Q is an extremal point of MH (χ ). We can now prove the following lemma. Lemma If Q ∈ M is unique, then stable(χ ) = H01 (Q). Proof Q ∈ M is unique implies that Q ∈ MH (χ ) is unique. This trivially implies Q is an extremal point of MH (χ ). By the Jacod and Yor Theorem given above, χ ⊂ H01 (Q) has martingale representation. Thus, by definition, stable (χ ) = H01 (Q). This completes the proof of the Lemma. (Step 2) Definition of a complete market (revisited). The market is complete with respect to Q if L1+ (Q) = ZT ∈ L1+ (Q) : ∃x ≥ 0, ∃(α0 , α) ∈ A (x), X = x + α • S is a Q martingale with XT = ZT } . Rewritten in terms of stochastic processes, define   G 1 = Z ∈ L+0 : ∃x ≥ 0, ∃(α0 , α) ∈ A (x), Z = x + α • S is a Q martingale   and G 2 = Z ∈ L+0 : Z is a Q martingale . The market is complete with respect to Q if G 1 = G 2 . Without lossof generality we can to the sets  restrict consideration   G01 = G 1 ∩ Z ∈ L+0 : Z0 = 0 and G02 = G 2 ∩ Z ∈ L+0 : Z0 = 0 . Indeed, given Z ∈ G 1 , this Q martingale can be generated by the X ∈ G01 given by X = Z−Z0 . The admissible s.f.t.s.’s determined by G01 that attains X can be used to attain Z by adding Z0 additional units in the mma, which retains admissibility and nonnegativity. Hence, the market is complete with respect to Q if G01 = G02 . (Step 3) Show the market is complete if Q ∈ M is unique. First, by the definitions, we have G01 ⊂ G02 . We need to show that G02 ⊂ G01 . By hypothesis, Xi (t) = Si (t) − Si (0) are uniformly integrable H01 (Q) martingales.

56

2 The Fundamental Theorems

Hence, {X1 , . . . , Xn } ∈ G01 where each Xi represents a buy and hold trading strategy in the mma and the risky asset.

This is an admissible s.f.t.s. Next, span{X1 , . . . , Xn } ∩ L+0 ⊂ G01 . These are buy and hold trading strategies in the mma and all the risky assets whose value processes are nonnegative, hence admissible s.f.t.s.

Finally, stable(χ ) ∩ L+0 ⊂ G01 , because stable(χ ) is closed under stochastic integration. Hence, this set corresponds to all s.f.t.s. that generate nonnegative valued processes, which are admissible.

Then, by the Lemma in (Step 1), H01 (Q) ∩ L+0 ⊂ G01 , i.e. all H01 (Q) martingales are in G01 . We note that given any admissible (nonnegative) s.f.t.s. in H01 (Q) ∩ L+0 = stable(χ ) ∩ L+0 , stopping the trading strategy at a stopping time τ and investing the proceeds in the mma until time T is included within the set. Stopping an admissible s.f.t.s. is an admissible s.f.t.s. Thus, by considering a sequence of stopping times τn with τn → T , the set of admissible s.f.t.s using {X1 , . . . , Xn } generates the set of all Q local martingales with nonnegative values. Hence, since Q martingales are Q local martingales, the set of admissible s.f.t.s using {X1 , . . . , Xn } generates G02 , thus G02 ⊂ G01 , which completes the proof of (Step 3) and the theorem. In a market satisfying NFLVR, the second fundamental theorem relates the notion of a complete market to the uniqueness of a local martingale measure. The first statement in the theorem is that if the market is complete with respect to any Q ∈ Ml , then the local martingale measure is unique. For the converse, the hypothesis needs to be strengthened to the existence of a martingale measure Q ∈ M. Then, if the martingale measure is unique, the market is complete with respect to Q ∈ M. Harrison and Pliska’s [67, 69] original formulation of the second fundamental theorem is an easy corollary. Corollary 3 (Characterization of Complete Markets Under M = ∅) Assume M = ∅, i.e. there exists an equivalent martingale measure. The market is complete with respect to Q ∈ M if and only if M is a singleton, i.e. the equivalent martingale measure is unique. This additional hypothesis for the converse makes it important to understand, in economic terms, the relation between the two sets of equivalent probability measures Ml and M. This economics behind these two sets of measures is studied in the next section. Remark 28 (Implications of an Incomplete Market) We can rephrase the second statement in the second fundamental theorem to the following. Assume Ml = ∅. Choose a Q ∈ Ml . The market is incomplete with respect to Q implies either that there does not exist a Q ∈ M or the local martingale measure Q ∈ Ml is not unique. At this level of generality, it is possible that the market is incomplete with respect to Q ∈ Ml , the local martingale measure is unique, and Q ∈ / M. However, for most

2.6 The Second Fundamental Theorem

57

models used in the literature, given additional structure on the risky asset evolution S, it can be shown that if the market is incomplete, then the local martingale measure is not unique. This is the case for the finite dimension Brownian motion market in Sect. 2.8 below. In an incomplete market, if the set of equivalent local martingale measures is not unique, then the cardinality |Ml | = ∞, i.e. the set contains an infinite number of elements. This follows because if not a singleton, the set contains at least two distinct elements and all convex combinations of these two elements. This completes the remark. Related to this remark, another partial converse of the second fundamental theorem for bounded derivatives ZT ∈ L0+ follows. Theorem 15 Let S have continuous sample paths. If Ml = {Q} is a singleton, then for any bounded ZT ∈ L0+ , the market is complete with respect to Q. Proof Suppose Ml is a singleton, let Ml = {Q}. Consider a bounded ZT ∈ L0+ . Then ZT ∈ L1+ (Q). By Theorem 44 in Chap. 8, there exists an admissible s.f.t.s. (α 0 , α) ∈ A (c0 ) with cash flows, an adapted right continuous nondecreasing process At with A0 = 0, and a value process Xt = α 0 (t) + α t · St such that  X t = c0 +

t

0

α s dSs + At = ct

for all t ∈ [0, T ] where ct = ess inf E Q [ZT |Ft ] ≥ 0. Since Q is unique, ct = Q∈Ml

E Q [ZT |Ft ] . Similarly, by Theorem 43 in Chap. 8, there exists an admissible s.f.t.s. (α¯ 0 , α) ¯ ∈ A (c¯0 ) with cash flows, an adapted right continuous nondecreasing process A¯ t with A¯ 0 = 0, and a value process X¯ t = α¯ 0 (t) + α¯ t · St such that X¯ t = c¯0 +



t

α¯ s · dSs − A¯ t = c¯t

0

for all t ∈ [0, T ] where c¯t = ess sup E Q [ZT |Ft ] ≥ 0. Since Q is unique, c¯t = Q∈Ml

E Q [ZT |Ft ] . t t This implies c¯t = ct . Equivalently 0 α s dSs + At = 0 α¯ s · dSs − A¯ t . t t Rearranging, At +A¯ t = 0 α¯ s ·dSs − 0 α s dSs . The right side is a local martingale with respect to Q and the left side is a nondecreasing process. Given S is continuous, by Protter [158, Theorem 30, p. 173], the right side is continuous. Any continuous local martingale with paths of finite variation on compacts is a.s. constant (see Protter [158, Corollary 1, p 72]). Hence At + A¯ t = 0 which implies At = 0 and A¯ t = 0 because both are non-decreasing. Combined,

58

2 The Fundamental Theorems

this implies there exists an admissible s.f.t.s. (α¯ 0 , α) ¯ ∈ A (c¯0 ) with value process X¯ t = α¯ 0 (t) + α¯ t · St such that X¯ t = c¯0 +



t

α¯ s · dSs

0

is a Q-martingale for all t ∈ [0, T ]. This completes the proof. We note that in this theorem, the market could be incomplete because with respect to ZT ∈ L1+ (Q) that are not bounded above, there may not be an admissible s.f.t.s. that attains ZT and whose value process is a Q-martingale. Equivalently, using the second fundamental theorem, the unique Q ∈ Ml may not be a martingale measure. Remark 29 (Discontinuous Price Processes and Market Completeness) For continuous sample path price processes, the market can be complete or incomplete, depending on properties of the filtration and the evolutions of the risky asset price processes. When the filtrations and the risky asset price processes are generated by a finite dimensional Brownian motion, then the market is complete if and only if the volatility matrix of the risky asset return processes satisfies a nonsingularity condition for all times with probability one (see Theorem 22 in Sect. 2.8 below). This condition fails, for example, when the risky asset price processes have stochastic volatility (see Eisenberg and Jarrow [56]). In contrast, however, for discontinuous sample path price processes the market is almost always incomplete. This is because when a jumps occurs, the distribution for the change in the price process is usually not just discrete with a finite number of jump amplitudes where the number of jump amplitudes is less then the number of traded assets (see Cont and Tankov [34, Chapter 9.2]). In this case the number of traded assets will be insufficient to hedge the different jump magnitudes possible (in mathematical terms, the martingale representation property fails for the value processes of admissible s.f.t.s.). This completes the remark.

2.7 The Third Fundamental Theorem This section presents the third fundamental theorem of asset pricing. The third fundamental theorem is the “work horse” of the three, yielding as a corollary the key tool used to value and hedge derivatives, called risk neutral valuation. To obtain this theorem, as in the previous sections, we are given a normalized market (S, F, P). We start with some definitions. First, we introduce the notion of no dominance (ND) initially employed by Merton [145] to study the properties of option prices.

2.7 The Third Fundamental Theorem

59

Definition 31 (No Dominance) The ith asset Si (t) is undominated if there exists no admissible s.f.t.s. (α0 , α) ∈ A (x) with an initial value x = Si (0) such that  P{x + 0

T

 αt · dSt ≥ Si (T )} = 1

and

T

P{x +

αt · dSt > Si (T )} > 0.

0

A market (S, F, P) satisfies no dominance (ND) if each Si , i = 0, . . . , n are undominated. ND states that it is not possible to find an admissible s.f.t.s. with initial investment equal to an asset’s initial price such that the trading strategy’s payoffs at time T dominate the payoffs to the traded asset. The definition of ND is invariant with respect to a change in equivalent probability measures. A dominated asset need not imply the existence of an arbitrage opportunity. This is due to the admissibility condition in the definition of a trading strategy, which precludes shorting an asset whose price is unbounded above (see Remark 14 above). NFLVR and ND are distinct conditions, although both imply the simpler no arbitrage NA condition. Indeed, that NFLVR implies NA follows directly from Theorem 12 above. Second, since ND implies that the mma (Bt = 1 for all t ≥ 0) is undominated, making the following identifications in the definition of T ND, (Si (0) = B0 = 1, Si (T ) = BT = 1, and XT = 1 + 0 αt · dSt ), yields the definition of NA. ND is a condition related to supply equalling demand. To understand why, note that a NFLVR is a mispricing opportunity that any single trader can exploit in unlimited quantities. As a NFLVR is exploited, it is believed that trading the arbitrage opportunity will cause prices to adjust and eventually eliminate the mispricing. Hence, NFLVR is independent of the aggregate market supply or demand. In contrast, ND compares two different investment alternatives for obtaining the same time T payoff. Of these two investment alternatives, the first, a s.f.t.s. dominates the second, holding the asset directly. A trader, if they desire the time T payoff from holding this position in their optimal portfolio, will never hold the asset. Since all traders see the same dominance, no one in the market will hold the asset. Hence, the supply for the asset (if positive) will exceed market demand. Positive supply can only equal demand if ND holds. Example 6 (NFLVR but not ND) The finite dimension Brownian motion market in Sect. 2.8 below can be used to generate examples of risky asset price evolutions that satisfy NFLVR but not ND. To obtain these examples, use Theorem 19 to guarantee that the market satisfies NFLVR. Then, use the insights from Theorem 23 to obtain a price process that does not satisfy Novikov’s condition and is a strict local martingale. An illustration of such a risky asset price process is contained in the asset price bubbles Chap. 3, Example 7. Another example is given in the asset price bubbles Chap. 3, Example 8, for a CEV price process with α > 1. This completes the example.

60

2 The Fundamental Theorems

We now state and prove the third fundamental theorem of asset pricing. Recall the notation for the set of equivalent martingale measures M = {Q ∼ P : S is a Q martingale} and the set of equivalent local martingale measures Ml = {Q ∼ P : S is a Q local martingale} where M ⊂ Ml , often a strict subset. Theorem 16 (Third Fundamental Theorem) NFLVR and ND if and only if M = ∅, i.e. there exists an equivalent probability Q such that S is a Q martingale. Proof (NFLVR and ND ⇒ M = ∅) By the first fundamental theorem of asset pricing, NFLVR ⇒ Ml = ∅. Hence, we only need to show that ND ⇒ M = ∅. The proof of this is based on Jarrow and Larsson [100]. This proof requires the definition of a maximal trading strategy and two lemmas. First the definition. An admissible s.f.t.s. (α0 (t), αt ) ∈ A (0) with zero initial wealth is maximal  T if for T every other admissible s.f.t.s. (β0 (t), βt ) ∈ A (0) such that 0 βt ·dSt ≥ 0 αt ·dSt , T T 0 βt · dSt = 0 αt · dSt holds. Next, the two lemmas. Lemma 1 (Delbaen and Schachermayer [46], Theorem 5.12) (α0 (t), αt ) ∈ t A (0) is maximal if and only if there exists a Q ∈ Ml such that 0 αs · dSs is a Q martingale. This theorem in Delbaen and Schachermayer [46] applies to our market because local martingales and σ -martingales coincide when the risky asset prices are nonnegative. Lemma 2 (Delbaen and Schachermayer [45], Theorem 2.14) Finite sums of maximal strategies are again maximal. This theorem in Delbaen and Schachermayer [45] applies to our market because the proof in Delbaen and Schachermayer [45] never uses the local boundedness assumption.

For all t ≥ 0, define the trading strategy γ0i (t), γti = (−Si (0), (0, . . . , 1, . . . , 0)) with a 1 in the ith place of the risky assets. This is buying and holding the ith risky asset and shorting and holding −Si (0) units of the mma. This s.f.t.s. (because it is a buy and hold trading strategy) has zero initial value because γ0i (0) + γti · S0 = −Si (0) + Si (0) = 0. It is admissible because the time t t value of this trading strategy is 0 γsi · dSs = −Si (0) + Si (t) ≥ −Si (0) for all t ≥ 0.

First, note that ND implies that for all i, γ0i (t), γti ∈ A (0) is maximal.

2.7 The Third Fundamental Theorem

61



Define (γ0 (t), γt ) = ni=1 γ0i (t), γti = − ni=1 Si (0), (1, . . . , 1) . This is an admissible s.f.t.s. with zero initial value. The time t value of this trading strategy is t γ · dS . s 0 s By Lemma 2, we get (γ0 (t), γt ) ∈ A (0) is maximal. t By Lemma 1, there is a Q making 0 γs · dSs a martingale. t n But, for all i, 0 γs · dSs = i=1 (Si (t) − S0 (t)) ≥ Si (t) − Si (0) for all t ≥ 0. Using Lemma 5 in Chap. 1, we have that a local martingale bounded by a martingale is itself a martingale. This completes the proof. (M = ∅ ⇒ NFLVR and ND). Let Q ∈ M. First, M = ∅ implies Ml = ∅, hence NFLVR by the first fundamental theorem of asset pricing. Second, we prove ND holds by contradiction. Suppose ND is violated. Then there exists an (α0 , α) ∈ A (x) with an initial value x = S0 such that for some risky asset i, 

T

Si (0) +

 αt · dSt ≥ Si (T )

and

T

P{Si (0) +

0

αt · dSt > Si (T )} > 0.

0

  T Taking expectations, we obtain Si (0) + E Q 0 αt · dSt > E Q [Si (T )] = Si (0). The last equality  follows by  the definition of Q being a martingale measure.  T T Q Thus, E 0 αt · dSt > 0. But, using Lemma 14 we have that XT = 0 αt ·   T dSt is a supermartingale, thus, E 0 αt · dSt ≤ 0. This is the contradiction, which completes the proof. The third fundamental theorem gives economic meaning to the difference between markets for which Ml = ∅ versus M = ∅. The first is equivalent to the market satisfying NFLVR. The second is equivalent to the market satisfying both NFLVR and ND. As emphasized previously ND is a much stronger condition, related to supply equalling demand. Under NFLVR and ND, let Q ∈ M and YT = dQ dP > 0 be its Radon Nikodym derivative. Then, the risky asset price processes are Q martingales, i.e. 

YT St = E [ST |Ft ] = E ST Yt Q



Ft

(2.21)

for all t ∈ [0, T ]. Alternatively stated, the asset’s price equals its expected (discounted, due to the normalization by the mma) value using the probability Q. Notice that multiplying the risky asset’s liquidating cash flow by the Radon Nikodym derivative generates the “certainty equivalent” of the cash flow. Indeed, the asset’s present value equals the expected (discounted) value of ST YYTt under the

probability P. This shows that we can interpret YYTt as an adjustment for risk. This is a key economic insight that will be used repeated below.

62

2 The Fundamental Theorems

Remark 30 (An Infinite Number of Risky Assets) The statement NFLVR and ND ⇒ M = ∅ depends on the fact that there is only a finite number of risky assets trading. The statement M = ∅ ⇒ NFLVR and ND holds more generally for an arbitrary number of risky assets (see the proof of the third fundamental theorem). This collection could be a continuum (an uncountably infinite set). However, in this extension, the trading strategy must still consist of only a finite number of the traded assets selected from the infinite collection that trade. This extension will be subsequently used in both Ross’ Arbitrage Pricing Theory (APT) in Chap. 4 and the Heath-Jarrow-Morton model in Chap. 6. This completes the remark.

2.7.1 Risk Neutral Valuation Given the third fundamental theorem, we can now discuss risk neutral valuation. Risk neutral valuation is a method for computing the present value of a derivative’s payoff using an equivalent martingale measure. Theorem 17 (Risk Neutral Valuation) Assume there exists a Q ∈ M (i.e. NFLVR and ND holds).   1

Fix the Q ∈ M. Let E Q [Si , Si ]T2

< ∞ for i = 1, . . . , n.

Given any attainable claim ZT ∈ ∪ C (x) ⊂ x≥0

L0+

with

EQ

  1 2 [X, X]T < ∞

where Xt is the value process of the admissible s.f.t.s. (α0 , α) ∈ A (x) that generates ZT , then Xt = E Q [ZT |Ft ]

(2.22)

for all t ∈ [0, T ] with x = X0 . Proof The integrability condition with respect to Q ∈ M makes Si a H 1 martingale (see Protter [158, p. 193 and p. 238, Ex 17]).  T Since ZT is attainable, there exists an x ≥ 0 and (α0 , α) ∈ A (x)  t such that x + 0 αt · dSt = ZT . Define the value process of this s.f.t.s. X&t = x + 0 α's · dSs . Note   1 XT = ZT . Then, E Q [X, X]T2 < ∞, implies that E Q sup X(t) < ∞ (see t∈[0,T ]

Protter [158, Theorem 48, p. 193]). This imposes an implicit restriction on (α0 , α) ∈ t A (x), which implies that 0 αu · dSu is an H 1 martingale (see Medvegyev [143, p. 341]). Thus, Xt is a H 1 martingale with X0 = x, which completes the proof. Note that in the statement of this theorem, expression (2.22) gives the time t value for a derivative with payoff ZT at time T . The time t value is equal to the payoff’s conditional expectation using the equivalent martingale probability measure Q ∈ M. This theorem is called risk neutral valuation because it provides the valuation

2.7 The Third Fundamental Theorem

63

formula that would exist in equilibrium (see Part III of this book) in an economy populated by risk neutral investors whose beliefs all equal Q. It is important to note that the above theorem does not require the market to be complete with respect to Q ∈ M. Consequently, there could exist many equivalent martingale probability measures. Nonetheless, all attainable claims (if properly integrable) that satisfy the hypotheses of this Theorem have the same price under any of these equivalent martingale probability measures because they generate martingales with the same terminal and initial values. Remark 31 (An Alternative Sufficient Condition for Risk Neutral Valuation) An alternative sufficient condition that yields risk neutral valuation is first to assume there exists a Q ∈ M. Let ZT ∈ ∪ C (x) ⊂ L0+ be attainable with (α0 , α) ∈ A (x) x≥0

and value process Xt . If Xt is bounded above, i.e. Xt ≤ k for some k > 0 for all t, then expression (2.22) applies. This statement is true because the value process X is a Q local martingale by Lemma 11. And, any bounded local martingale is a martingale by Remark 4 in Chap. 1. This completes the remark. In a complete market, we get the following useful theorem. This theorem is the basis for the valuation methodologies used in the BlackScholes-Merton (Chap. 5) and Heath-Jarrow-Morton (Chap. 6) models for pricing derivatives. Theorem 18 (Risk Neutral Valuation in a Complete Market) Assume M = ∅ (i.e. NFLVR and ND holds). Let the market be complete with respect to Q ∈ M. Then, risk neutral valuation works for any ZT ∈ L1+ (Q), i.e. let Xt be the value process of the admissible s.f.t.s. (α0 , α) ∈ A (x) that generates ZT , then Xt = E Q [ZT |Ft ] for all t ∈ [0, T ] with x = X0 . This theorem follows because in a complete market, by Corollary 3, Q ∈ M is unique. And, by the definition of a complete market, all derivatives integrable with respect to Q are attainable, i.e. ZT ∈ L1+ (Q) implies that ZT ∈ ∪ C (x), and the x≥0

value process is a Q martingale. Thus, there exists an admissible s.f.t.s. (α0 , α) ∈ A (x) such that Xt = E Q [ZT |Ft ] with X0 = x. Note that we do not need to assume the additional integrability conditions as in Theorem 17 on either S or ZT with respect to Q.

64

2 The Fundamental Theorems

2.7.2 Synthetic Derivative Construction Given the risk neutral valuation formula for pricing any derivative, we now discuss synthetic construction of the derivative’s payoffs. Assuming NFLVR and ND, i.e. M = ∅, plus assuming that the market is complete with respect to Q ∈ M, we know for a given “traded derivative” ZT ∈ L1+ (Q), there exists an initial investment x > 0 and an admissible s.f.t.s. (α0 , α) ∈ A (x) such that 

t

Xt = x +

αu · dSu = E Q [ZT |Ft ]

where

(2.23)

0

x = X0

and

Xt = α0 (t) + αt · St

(2.24)

for all t ∈ [0, T ]. This trading strategy with value process Xt is called the “synthetic derivative” because its time T payoffs ZT are identical to those of the “traded derivative.” For practical applications, we need to identify this admissible s.f.t.s., i.e. we need to be able to determine both the initial investment x and the trading strategy itself, (α0 (t), αt ). We now explain how this is done. First, taking expectations under Q, we get the initial investment x = X0 = E Q [ZT ] .

(2.25)

In words, the initial cost of constructing the synthetic derivative is equal to the expected value of the derivative’s time T payoffs under the equivalent martingale measure. Second, once we know the holdings in the risky assets αt , then we can compute the position in the mma α0 (t) by solving expression (2.24). This leaves only the determination of αt . For a large class of processes one can determine αt using Malliavin calculus and the generalized Clark Ocone formula (see Nunno et al. [51]). Here, however, we add some additional structure and use more elementary methods. This additional structure is often available in practical applications. In this regard, we assume that S is a continuous price process and that we can write the value process Xt as a C 1,2 function (continuously differentiable in the first argument and twice continuously differentiable in the second argument) f of time t and the risky asset prices St , i.e. Xt = E Q [ZT |Ft ] = f (t, St ).

(2.26)

2.8 Finite Dimension Brownian Motion Market

65

Applying Ito’s formula (see Theorem 4 in Chap. 1) to the function in expression (2.26) determines αt . Indeed, applying Ito’s formula yields the expression ( ) ∂f 2 dt + 12 ni=1 nj=1 ∂ f2 dSi (t), dSj (t) + ∂S · dSt or t ∂Sij  )  T ∂f ∂f ∂2f ( 1 T n n 2 dSi (t), dSj (t) + 0 ∂St · dSt = XT i=1 j =1 ∂t dt + 2 0

dXt =

X0 +

T 0

∂f ∂t

∂Sij

(2.27)





∂f ∂f ∂f where ∂S = ∂S , . . . , ∂S ∈ Rn . Identifying the integrands of dSt across t n 1 Eqs. (2.23) and (2.27) gives the self-financing trading strategy

αt =

∂f . ∂St

(2.28)

These holdings are called the derivative’s deltas. If the normalized market is non-redundant (see Sect. 2.4 in Chap. 2), then the admissible s.f.t.s. obtained above is unique. Otherwise, this approach will generate one of the perhaps numerous trading strategies that attain ZT and whose value processes are Q martingales.

2.8 Finite Dimension Brownian Motion Market This section illustrates an application of the fundamental theorems of asset pricing in a market where the randomness is generated by a finite dimension Brownian motion process. This market is the basis for the Black-Scholes-Merton model in Chap. 5 and the Heath-Jarrow-Morton model in Chap. 6. Given is a normalized market (S, F, P) where the money market account Bt = 1 for all t ≥ 0.

2.8.1 The Set-Up We assume that   dSi (t) = Si (t) (bi (t) − rt )dt + D d=1 σid (t)dWd (t) Si (t) = Si (0)e

t

1 0 (bi (u)−ru )du− 2

  t D 2 d=1 σid (u) du+ 0 d=1 σid (u)dWd (u)

 t  D 0

or (2.29)

for all t ∈ [0, T ] and i = 1, . . . , n where Wt = W (t) = (W1 (t), . . . , WD (t)) ∈ RD are independent Brownian motions with Wd (0) = 0 for all d = 1, . . . , D that generate the filtration F = (Ft )t∈[0,T ] , S0 = S(0) = (S1 (0), . . . , Sn (0)) ∈ Rn is a T vector of strictly positive constants, rt is Ft -measurable with 0 |rt | dt < ∞, bt =

66

2 The Fundamental Theorems

b(t) = (b1 (t), . . . , bn (t)) ∈ Rn is Ft -measurable (adapted) with where x2 = x · x = ni=1 xi2 for x ∈ Rn , and ⎡ ⎢ ⎢ σt = σ (t) = ⎢ ⎣

σ11 (t) · · · σ1D (t)

T 0

bt  dt < ∞

⎤

⎥ ⎥ .. .. ⎥ . . ⎦ σn1 (t) · · · σnD (t)

(2.30)

n×D

T 2 is an n×D matrix which is Ft -measurable (adapted) with nj=1 D d=1 0 σj d dt < ∞. In vector notation, we can write the evolution of the stock price process as dSt = (bt − rt 1)dt + σt dWt St

(2.31)

  dS1 (t) dSn (t) t where dS = , . . . , ∈ Rn and 1 = (1, . . . , 1) ∈ Rn . St S1 (t) Sn (t) This assumption implies that S has continuous sample paths. The quadratic variation is *D + D   ( ) dSi (t), dSj (t) = Si (t)σid (t)dWd (t), Sj (t)σj k (t)dWk (t) d=1

=

k=1

D D  D  ( )  Si (t)σid (t)dWd (t), Sj (t)σj k (t)dWk (t) = Si (t)Sj (t)σid (t)σj d (t)dt. d=1 k=1

d=1

(2.32) In vector notation ,

dSt dSt = σt σt dt. , St St n×n

(2.33)

2.8.2 NFLVR For pricing derivatives or searching for arbitrage opportunities we need to know when the risky asset price evolution in expression (2.29) satisfies NFLVR. By the First Fundamental Theorem 13 of asset pricing we have NFLVR if and only if there exists a Q ∼ P such that S is a Q local martingale. Using Theorem 9

2.8 Finite Dimension Brownian Motion Market

67

in Chap. 1, we have NFLVR if and only if there exists a predictable process θt = θ (t) = (θ1 (t), . . . , θD (t)) ∈ RD with T 2 and    T0 θt  1dt T< ∞ − 0 θt ·dWt − 2 0 θt 2 dt =1 E e

(2.34)

such that  T 1 T dQ 2 = e− 0 θt ·dWt − 2 0 θt  dt > 0, dP

(2.35)

and S is a Q local martingale. Next, we determine necessary and sufficient conditions under which S is a Q local martingale using Girsanov’s theorem (see Theorem 5 in Chap. 1). By Girsanov’s theorem, the stochastic process Wtθ = W θ (t) = (W1θ (t), . . . , WDθ (t)) ∈ RD defined by t Wiθ (t) = Wi (t) + 0 θi (u)du or (2.36) dWiθ (t) = dWi (t) + θi (t)dt is an independent Brownian motion process under the equivalent probability measure Q with W0θ = (0, . . . , 0) . Substitution of expression (2.36) into expression (2.29) yields ! " D D   θ dSi (t) = Si (t) (bi (t) − rt )dt − σid (t)θd (t)dt + σid (t)dWd (t) . d=1

d=1

(2.37) The risky asset price process S is a local martingale under Q if and only if the drift terms in these evolutions are identically zero, i.e. (bi (t) − rt ) =

D 

σid (t)θd (t) for i = 1, . . . .n

(2.38)

d=1

for almost all t ∈ [0, T ] (up to Lebesgue measure zero set) a.s. P (or Q since equivalent). In vector notation, this is bt − rt 1 = σt θt .

(2.39)

We have now proven the following theorem. Theorem 19 (NFLVR Evolution) The risky asset price evolution S in expression (2.29) satisfies NFLVR if and only if there exists a predictable process θt satisfying expression (2.34) and bt − rt 1 = σt θt

(2.40)

68

2 The Fundamental Theorems

for almost all t ∈ [0, T ] (up to Lebesgue measure zero set) a.s. P. The vector process θt = θ (t) = (θ1 (t), . . . , θD (t)) ∈ RD are the “risk premiums” associated with the risky Brownian motions. The next theorem gives a sufficient condition on the evolution of the risky asset price process for expression (2.40) to have a solution. First, we need to define the set of predictable stochastic processes    T 2 ν  K (σ ) = ν ∈ L (W ) : dt < ∞, σt νt = 0 for all t . (2.41) t 0

This is the set of D-dimension processes that are orthogonal to the column space of the matrix σt for all t ∈ [0, T ] (up to Lebesgue measure zero set) a.s. P. Theorem 20 (Sufficient Condition on S for NFLVR) Assume that (1) rank (σt ) = min {n, D} for all t a.s. P, (2) D ≥. n, and .2 T .  −1 . (3) 0 .σt σt σt (bt − rt 1). dy < ∞. Then, the set of solutions to expression (2.40) is nonempty and equals {θ + ν : ν ∈ K(σ )} where  −1 θt = σt σt σt (bt − rt 1) .   T Proof Let A = ψ ∈ L (W ) : 0 ψt 2 dt < ∞, bt − rt 1 = σt ψt and   T B = θ + ν ∈ L (W ) : 0 νt 2 dt < ∞, σt νt = 0 . (Step 1) Fix a (t, ω) ∈ [0, T ] × Ω. Hypothesis

(1) and (2) imply that rank (σt ) = n. Hence, by Theil [183, p. 11], rank σt σt = n. Hence, σt σt is invertible and θt is well defined. Consider the set of solutions νt to the equation σt νt = 0. Since by hypothesis (1) σt is of full rank, this set of equations has a solution (see Perlis [155, p. 47]). Hence, {ν : σt νt = 0} = ∅. The solution set is a singleton, consisting of only {0} if and only if D = n. Note that νt = 0 ∈ B. Hence, B = ∅. (Step 2) Show B ⊂ A . Let θ + ν ∈ B. Then,  −1 σt (θt + νt ) = σt θt = σt σt σt σt (bt − rt 1) = bt − rt 1. Hence, θ + ν ∈ A .  −1 (Step 3) Show A ⊂ B. Let ψ ∈ A . Consider θt = σt σt σt (bt − rt 1). As in (Step 1), σt θt = bt − rt 1. Define νt = ψt − θt . Then, σt νt = ψt θt − σt θt = (bt − rt 1) − (bt − rt 1) = 0. Thus, ψt = θt + νt ∈ B. This completes the proof.

2.8 Finite Dimension Brownian Motion Market

69

Theorem 20 gives sufficient conditions for the market to satisfy NFLVR. Condition (1) is that the volatility matrix must be of full rank for all t. This omits risky assets that randomly change from risky to locally riskless (finite variation) across time. Condition (2) removes redundant assets from the market (see Theorem 10). Finally, condition (3) is a necessary integrability condition for θt . For subsequent use we note that conditions (1) and (2) are true if and only if rank (σt ) = n for all t a.s. P. Given Theorem 20, we can now characterize the set of local martingale measures Ml . Theorem 21 (Characterization of Ml ) Assume that (1) rank (σt ) = n for all t a.s. P and .2 T . −1 . .  (2) 0 .σt σt σt (bt − rt 1). dy < ∞. Then, Ml = {Qν :

T T dQν − 0 (θt +νt )·dWt − 12 0 θt +νt 2 dt = e dP  ν E dQ dP = 1, ν ∈ K(σ )} = ∅

> 0,

where  −1 θt = σt σt σt (bt − rt 1) . Proof By Theorems 19 and 20, the market satisfies NFLVR and by the First Fundamental Theorem 13 of asset pricing Ml = ∅. Define  ν T T dQν − 0 (θt +νt )·dWt − 12 0 θt +νt 2 dt ν Al = {Q : dP = e = 1, ν ∈ > 0, E dQ dP K(σ )}. (Step 1) Show Ml ⊂ Al . Take Q ∈ Ml . Then, by Theorem 9 in Chap. 1, there exists a predictable process ψ satisfying the appropriate integrability conditions such that dQ dP =   T T dQ − 0 ψt ·dWt − 12 0 ψt 2 dt e > 0 and E dP = 1. By Girsanov’s Theorem 5 in Chap. 1, ψ dWi = dWi (t) + ψi (t)dt is a Brownian motion under  Q. Substituting into D dSi (t) = Si (t) (bi (t) − rt )dt + d=1 σid (t)dWd (t) gives   D ψ dSi (t) = Si (t) (bi (t) − rt )dt + D σ (t)ψ (t)dt + σ (t)dW (t) . id id id d=1 d=1 d D But, since Q ∈ Ml , this implies (bi (t) − rt ) + d=1 σid (t)ψid (t) = 0. Or, in vector notation bt − rt 1 = σt ψt . By the proof of Theorem 20 above, this implies ψt = θt + νt for some νt satisfying σt νt = 0. Hence, Q ∈ Al . This completes the proof of (Step 1). (Step 2) Show Al ⊂ Ml . Take Qν ∈ Al . Then by Girsanov’s Theorem 5 in Chap. 1, dWiν = dWi (t) + (θi (t) + νi (t)) dt is a Brownian motion under Qν .

70

2 The Fundamental Theorems

Substituting into   dSi (t) = Si (t) (bi (t) − rt )dt + D d=1 σid (t)dWd (t) gives  D dSi (t) = Si (t) (bi (t) − rt )dt + D d=1 σid (t) (θi (t) + νi (t)) dt + d=1 σid (t)

ν dWd (t) . By the definition of θ and ν this implies  D ν (t) , which is a Qν local martingale. Hence, σ (t)dW dSi (t) = Si (t) id d=1 d Qν ∈ Ml . This completes the proof of (Step 2) and the theorem.

2.8.3 Complete Markets This subsection studies when the Brownian motion market is complete. We assume that the evolution S in expression (2.29) satisfies (1) rank (σt ) = n for all t a.s. P and .2 T . −1 .  . (2) 0 .σt σt σt (bt − rt 1). dy < ∞. Then, by Theorem 20 above, NFLVR is satisfied. Choose an equivalent local martingale measure Qν ∈ Ml . We have, using vector notation, that under this probability measure the risky assets evolve as dSt = σt dWtν . St

(2.42)

We also know from the Second Fundamental Theorem 14 that a necessary condition for the market to be complete with respect to Qν is that the equivalent local martingale measure must be unique. Using the previous Theorem 21, Qν is unique if and only if the solution θt to the NFLVR expression bt − rt 1 = σt θt is unique. And, this is true if and only if K(σ ) = {0}. But, K(σ ) = {0} if and only if rank(σt ) = D for all t a.s. P, see Perlis [155]. We now show that rank(σt ) = D for almost all t ∈ [0, T ] is also sufficient for the market to be complete. Because rank(σt ) = D, without loss of generality we can remove redundant assets from the market (remove rows from the matrix) so that σt is a D × D matrix and σt−1 exists. Then, we can rewrite expression (2.42) as dWtν = σt−1

dSt . St

(2.43)

2.8 Finite Dimension Brownian Motion Market

71

Now, consider an arbitrary ZT ∈ L1+ (Qν ). By the Martingale Representation Theorem 8 in Chap. 1, there exists predictable processes Hd in L (Wdν ) with T 2 0 (Hd (t)) dt < ∞ for d = 1, . . . , D such that ZT = E

Qν

[ZT ] +

D  

T

d=1 0

Hd (t)dWdν (t),

and where the process Zt = E Q [ZT ] + ν

D   d=1 0

t

Hd (s)dWdν (s)

is a Qν martingale. In vector notation, we can rewrite this as dZt = Ht · dWtν

(2.44)

with Z0 = E Q [ZT ] and Ht = (H1 (t), · · · , HD (t)) . Substituting expression (2.43) into this expression gives ν

dZt = Ht · σt−1

Zt = E

Qν

[ZT ] +

D   i=1

0

t

dSt St

or

!D "  Hd (t)σ −1 (t) di

d=1

Si (t)

dSi (t).

(2.45)

(2.46)

−1 ν Hsd σdi (t) Consider the trading strategy defined by x = E Q [ZT ], αi (t) = D for d=1 S i (t) i = 1, · · · , D, and α0 (t) = Zt − αt · St for all t. This trading strategy generates ZT , it is self-financing by expression (2.46), and it is admissible because Zt is a Qν ν martingale, which implies that Zt = E Q [ZT |Ft ] ≥ 0 for all t. Hence, the market is complete. We have now proven the following theorem.

Theorem 22 (Complete Market Evolution) Assume NFLVR. Suppose that the risky asset price evolution S in expression (2.29) satisfies (1) rank (σt ) = n for all t a.s. P and .2 T . −1 .  . (2) 0 .σt σt σt (bt − rt 1). dy < ∞. Choose a Q ∈ Ml . The market is complete with respect to Q if and only if the volatility matrix σ has rank D for almost all t ∈ [0, T ] (up to Lebesgue measure zero set) a.s. P.

72

2 The Fundamental Theorems

When the market is complete, then without loss of generality we can always remove redundant assets from the market (remove rows from the matrix) so that σ is an D × D matrix and the NFLVR expression for the risk premium can be rewritten as θt = σt−1 (bt − rt 1).

(2.47)

2.8.4 ND Up to this point, Theorem 19 provides conditions characterizing when the market satisfies NFLVR. These conditions only ensure the existence of a local martingale measure Q. As shown when discussing the first fundamental theorem, NFLVR only implies that the price process S is a supermartingale under Q. To guarantee the existence of a martingale measure, by the third fundamental theorem, ND must also hold. We now study conditions under which ND holds for the risky asset price evolution S given in expression (2.29). First, we assume that the market satisfies NFLVR. Hence, there exists a local martingale measure Qν such that under Qν the evolution for a risky asset can be written as dSi (t) = Si (t) Si (t) = Si (0)e

 D



ν d=1 σid (t)dWd (t)    t D  t D ν 2 − 12 0 d=1 σid (u) du+ 0 d=1 σid (u)dWd (u)

or (2.48) .

Whether or not this risky asset’s price process is a martingale under Qν is completely determined by the integrability properties of the volatility matrix σ . A necessary and sufficient condition for Si (t) to be a Qν martingale is that E

Qν

  1  t  D   t D ν 2 −2 0 d=1 σid (u) du+ 0 d=1 σid (u)dWd (u) e =1

(2.49)

(see Protter [158, p. 138]). This is often a difficult condition to verify. A stronger but easier sufficient (an almost necessary) condition to check that guarantees Si (t) is a Qν martingale is called Novikov’s condition (see Protter [158, p. 140]), i.e. E

Qν

   1  T  D 2 d=1 σid (t) dt 2 0 e < ∞.

We have now proven the following theorem. Theorem 23 (ND Evolution) Assume NFLVR. The market satisfies ND if there exists a Ft -measurable θt satisfying expression (2.34) where bt − rt 1 = σt θt

(2.50)

Appendix

73

for almost all t ∈ [0, T ] (up to Lebesgue measure zero set) a.s. P and EQ

ν

   1  T  D 2 d=1 σid (t) dt e2 0 < ∞ for all i = 1, . . . , n.

2.9 Notes This chapter assumes that markets are competitive and frictionless. When the competitive market assumption is relaxed, then strategic trading and market manipulation are possible. For models relaxing the competitive market assumption see Jarrow [83, 84, 96], Bank and Baum [6], and Cetin et al. [30]. In Part IV of this book, the frictionless market assumption is relaxed with the inclusion of trading constraints. Excellent references on the theory of no arbitrage for the general semimartingale market is Delbaen and Schachermayer [47], and Karatzas and Shreve [124] for the Brownian motion market.

Appendix This appendix proves that the original definition of NFLVR in Delbaen and Schachermayer [44, Proposition 3.6], is equivalent to Definition 27 of NFLVR given in the text. And, it uses Theorem 12 to give another characterization of NUPBR. Definition 32 (No Free Lunch with Vanishing Risk (D&S)) A free lunch with vanishing risk (D&S) is: (i) a simple arbitrage opportunity (NA), or (ii) a sequence of zero initial investment admissible s.f.t.s.’s (α0 , α)n ∈ A (0) with value processes Xtn , lower admissibility bounds cn ≤ 0, and a FT -measurable random variable χ ≥ 0 with P(χ > 0) > 0 such that cn → 0 and XTn → χ in probability. We denote condition (ii) as D&S(ii). The market satisfies D&S if there exist no D&S trading strategies. As seen, a market satisfies D&S if there are no simple arbitrage opportunities and no approximate arbitrage opportunities (α0 , α)n ∈ A (0), that in the limit, become simple arbitrage opportunities. Both conditions (i) and (ii) are needed in this definition. Indeed, condition (ii) does not imply condition (i). To see why, consider a (simple) arbitrage opportunity (α0 , α) ∈ A (0) with admissibility bound c and value process Xt that satisfies X0 = 0, XT ≥ 0, and P(XT > 0) > 0. Define the (constant) sequence of admissible s.f.t.s.’s using the simple arbitrage opportunity, i.e. (α0 , α)n = (α0 , α) whose time T value process XTn = XT approaches (equals) χ = XT where χ ≥ 0 with P(χ > 0) > 0. This constant sequence of admissible s.f.t.s.’s satisfies all of the properties of condition (ii) except that the sequence’s admissibility bounds cn = c do not converge to 0. We now show that Definition 27 of NFLVR is equivalent to D&S.

74

The Fundamental Theorems

Theorem 24 (Equivalence of NFLVR Definitions) NFLVR is equivalent to D&S Proof (Step 1) Show a FLVR implies a D&S. Let (α0 , α)i ∈ A (x) be a FLVR, i.e. (α0 , α)i ∈ A (x) with initial value x ≥ 0, value processes Xti , lower admissibility bounds 0 ≥ ci where ∃c ≤ 0 with ci ≥ c for all i, and a FT -measurable random variable χ ≥ x with P(χ > x) > 0 such that XTi → χ in probability. Now, we have 0 ≥ ci ≥ c for all i. Thus, there exists a subsequence n such that ci → c∗ ∈ [0, c] with c ≤ 0. This implies 0 ≥ c∗ . Consider (α0 , α)n ∈ A (x). Define (α˜ 0 , α) ˜ n = (α0 − c∗ , α)n . This trading strategy has initial wealth x˜ = x − c∗ ≥ 0. The value process is X˜ t = Xt − c∗ . It is self-financing because this trading strategy just adds −c∗ ≥ 0 additional dollars to the mma in the original s.f.t.s. for all t and holds this new position for all t. The admissibility bounds are c˜n = min{cn − c∗ , 0} for all n with c˜n = min{cn − ∗ c , 0} → 0. Finally, we have that X˜ Tn → χ˜ = χ − c∗ in probability and χ˜ = χ − c∗ ≥ x − c∗ = x˜ with P(χ˜ = χ − c∗ > x − c∗ = x) ˜ > 0. Hence, this is a D&S. (Step 2) Show a D&S implies a FLVR. (Case a) Let (α0 , α)n ∈ A (x) be a simple arbitrage opportunity with admissibility bound c ≤ 0 and value process Xt that satisfies X0 = 0, XT ≥ 0, and P(XT > 0) > 0. Then, define the (constant) sequence of admissible s.f.t.s.’s using the simple arbitrage opportunity, i.e. (α0 , α)n = (α0 , α) whose time T value process XTn = XT approaches (equals) χ = XT where χ ≥ 0 with P(χ > 0) > 0 with lower admissibility bounds cn = c ≥ c for all n. This is a FLVR. (Case b) Let (α0 , α)n ∈ A (0) with value processes Xtn , lower admissibility bounds cn ≤ 0, and a FT -measurable random variable χ ≥ 0 with P(χ > 0) > 0 such that cn → 0 and XTn → χ in probability. Note that since cn → 0, there exists a N and c < 0 such that for all n ≥ N, cn ≥ c. Consider the sequence of admissible s.f.t.s.’s (α0 , α)∞ n=N ∈ A (0). This is a FLVR. This completes the proof. We can restate this theorem as NF LV R ⇐⇒ NA + DS(ii). From Theorem 12, we have N F LV R ⇐⇒ NA + NU BP R. Consequently, NU BP R ⇐⇒ DS(ii). This completes the characterization.

Chapter 3

Asset Price Bubbles

An important recent development in the asset pricing literature is an understanding of asset price bubbles. This chapter discusses these new insights. They are motivated by the first and third fundamental theorems which show that NFLVR only implies the existence of a local martingale measure and not a martingale measure. Asset price bubbles clarify the economic meaning of this difference. The material in this chapter is based on the papers by Jarrow et al. [114, 115].

3.1 The Set-Up Given is a normalized market (S, F, P) where the money market account Bt = 1 for all t ≥ 0 and there is only one risky asset trading denoted St . The restriction to one risky asset is without loss of generality and it is imposed to simplify the notation (to avoid subscripts for the j th risky asset). We assume that the market satisfies NFLVR, hence by the First Fundamental Theorem 13 of asset pricing in Chap. 2, there exists a probability measure Q ∈ Ml where Ml = {Q ∼ P : S is a Q local martingale}. The subsequent analysis applies to both complete and incomplete markets. Since we do not assume that the market is complete, there can exist many such equivalent local martingale measures. To study asset price bubbles, we need to determine a unique Q ∈ Ml . If the market is complete, then by the Second Fundamental Theorem 14 of asset pricing in Chap. 2, the local martingale measure is uniquely determined, and we are done. However, we do not assume that the market is complete. If the market is incomplete, to identify a unique equivalent local martingale measure the following approach can be employed. One assumes that the market studied is embedded in a larger market that includes the trading of derivatives on the traded risky assets (e.g. call and put options with different strikes and maturities). This is called the extended market. There are two possible cases in © Springer Nature Switzerland AG 2021 R. A. Jarrow, Continuous-Time Asset Pricing Theory, Springer Finance, https://doi.org/10.1007/978-3-030-74410-6_3

75

76

3 Asset Price Bubbles

the extended market. One, is that these traded derivatives complete the market and therefore the equivalent local martingale measure is again uniquely determined by the second fundamental theorem of asset pricing. The papers by Eisenberg and Jarrow [56], Dengler and Jarrow [48], Schwartz [174] provide models with markets where this is the case. Two, is that even with the traded derivatives, the extended market is incomplete. However, in this case under certain conditions, the traded derivatives’ price processes may provide enough information to identify a unique local martingale measure consistent with these prices. The papers by Jacod and Protter [77] and Schweizer and Wissel [175] study such extended but incomplete markets. For subsequent use, we say that the unique Q ∈ Ml determined in either a complete or incomplete extended market is the Q ∈ Ml chosen by the market.

3.2 The Market Price and Fundamental Value Fix a particular equivalent local martingale measure Q ∈ Ml chosen by the market. This section characterizes the risky asset’s market price in terms of its resale value on or before the risky asset’s liquidation date, time T . Theorem 25 (The Market Price) Given NFLVR, St = ess sup E Q [Sτ |Ft ] τ ∈[t,T ]

(3.1)

where τ ≥ t is a stopping time for all t ∈ [0, T ]. Proof Since St is a nonnegative local martingale under Q, it is a supermartingale (Lemma 4 in Chap. 1). This implies that St ≥ E Q [Sτ |Ft ] for any stopping time τ . Hence, St ≥ ess sup E Q [Sτ |Ft ]. τ ∈[t,T ]

Conversely, consider a localizing sequence of stopping times τ n → T with τ n < T for all n. Under this sequence, Sτ n ∧t is a martingale under Q for all n, hence Sτ n ∧s = E Q [Sτ n ∧t |Fs ] for s ≤ t ≤ T . Assume s < τ n ∧ t, then Ss = E Q [Sτ n ∧t |Fs ] . For t = T , we have Ss = E Q [Sτ n ∧T |Fs ] = E Q [Sτ n |Fs ]. Hence, for the stopping time τ n ∈ [t, T ] we have St = E Q [Sτ n |Ft ]. This implies ess sup E Q [Sτ |Ft ] ≥ St , which completes the proof. τ ∈[t,T ]

In this theorem, the stopping times can be interpreted as optimal “selling” times. If you already own the risky asset at time t, it states that the market price reflects the present value of the price received if sold at time τ . Note that E Q [ST |Ft ] represents the time t present value of selling the asset at time T because it is the (discounted) expected payoff adjusted for risk by using the local martingale measure Q ∈ Ml . The following corollary follows from the theorem’s proof.

3.2 The Market Price and Fundamental Value

77

Corollary 4 (Optimal Selling Times) Given NFLVR, (i) the optimal selling times in expression (3.1) are not unique, and they include the current time t and the localizing sequence for the Q local martingale St , (τn )∞ n=1 . (ii) T is an optimal stopping time if and only if St is a Q martingale. As stated, if one owns the risky asset at time t, then the optimal selling times are the current date t and any of the stopping times τn in the localizing sequence that makes S a Q local martingale. Further, if a bubble exists and St is a strict local martingale, then St > E Q [ST |Ft ] . Hence, it is not optimal to hold the asset until time T . If one holds the asset until time T , then the time t present value of the trading strategy is E Q [ST |Ft ], which is strictly less than the current market price. The intuition behind this result is that one needs to sell before the bubble bursts, which it does for sure by time T (see Theorem 26 below). If there is a bubble and one buys the risky asset at time t, then it is purchased to be resold before time T . Indeed, at any optimal selling time τn > t, the same time  t present value, St = E Q Sτn |Ft , is received. Also note that there are an infinite number of stopping times where selling is optimal, and they approach T but never reach T . It is never optimal to sell at T , and it is always possible (in probability) to sell before time T because there always exists another selling time τn+1 > τn for which τn+1 < T . This is the “greater fool” theory. That is, if someone buys the risky asset with a bubble at time t, then she is buying to resell at time τn > t to a “greater fool” before the bubble bursts. Indeed, there is always another selling time τn+1 > τn where the time τn purchaser can resell themselves before time T . This explains why in finite horizon discrete time models there can be no bubbles; in such models there are only a finite number of selling dates and the “greater fool” argument fails. In summary, expression (3.1) states that the market price of the risky asset is the value computed by considering retrading and selling the asset across all possible selling times before or at time T . If retrading has no value and the optimal selling time is τ = T , then the asset is valued as if it is held until the liquidation date. In this case the asset is worth the present value of its liquidation value, i.e. St = E Q [ST |Ft ]

(3.2)

for all t ∈ [0, T ]. This observation justifies the definition of the risky asset’s fundamental value to be the present value of buying and holding the asset until time T . Definition 33 (Fundamental Value) Given NFLVR, the asset’s fundamental value is E Q [ST |Ft ] for all t ∈ [0, T ]. This definition is consistent with that used in the economics literature (see Jarrow [91] for a review of this literature).

78

3 Asset Price Bubbles

3.3 The Asset Price Bubble We can now define the asset’s price bubble and explore some of its consequences in arbitrage free markets. Definition 34 (Asset Price Bubble) Given NFLVR, the asset’s price bubble βt is defined by βt = St − E Q [ST |Ft ]

(3.3)

for all t ∈ [0, T ]. We have the following theorem characterizing various properties of asset price bubbles. Theorem 26 (Properties of Price Bubbles) Given NFLVR, 1. βt ≥ 0. 2. βT = 0. 3. If βs = 0 for some s ∈ [0, T ], it is zero thereafter. Proof 1. This follows since St is a supermartingale. 2. Since ST = E Q [ST |FT ] + βT , βT = 0. 3. Suppose for some s ∈ [0, T ], βs = 0. Then 0 = βs ≥ EsQ [βt |Fs ] ≥ EsQ [βT |Fs ] = 0. The bubble is, thus, a nonnegative martingale after time s. Hence, it is identically zero after time s. This completes the proof. Condition (1) states that asset price bubbles are always nonnegative, βt ≥ 0. This is because one can always choose not to retrade and hold the asset until time T . Hence, the risky asset is always worth at least its fundamental value. Condition (2) states that the price bubble may burst before, but no later than time T . This is because at time T , the stock is worth its liquidation value, which also equals its fundamental value at that time. Last, condition (3) states that if the price bubble disappears before time T , it is never reborn.

3.3.1 Complete Markets Asset price bubbles can exist in a complete market, as the next two examples illustrate. Example 7 (A Complete Market with an Asset Price Bubble) This example gives a risky asset’s market price process with an asset price bubble, where the market is

3.3 The Asset Price Bubble

79

complete with respect to Q ∈ Ml . Consider the risky asset price that evolves under P as dSt = (b(St ) − rt )dt + σ (St )dWt

(3.4)

for all t ∈ [0, T ] where Wt is a standard Brownian motion with W0 = 0 and (b(St ), σ (St )) have appropriate measurability and boundedness conditions so that this stochastic differential equation is well defined with a solution (see Protter [158, Chapter V]). Using Theorem 19 in Chap. 2, we have that the market satisfies NFLVR if and only if there exists a predictable process θt satisfying T 2    T 0 θt dt 1 0 for all t ∈ [0, T ]. We assume that this condition is also satisfied so the market is complete with respect to Q, implying that the equivalent local martingale measure is unique by the second fundamental theorem of asset pricing. Finally, for this asset price process, we have the following result. S is a strict local martingale  ∞ x under Q (a bubble exists) if and only if 1 σ (x)2 dx < ∞. The proof is contained in Protter [159]. The above result characterizes conditions on the asset’s volatility σ (·) such that there is a price bubble. The condition is that the asset’s local σ (St ) must  ∞ volatility x increase fast enough as St increases so that the integral 1 σ (x) 2 dx is finite. This “explosion” in the asset’s local volatility is consistent with the fact that bubbles exist only when an asset is purchased to resell, and not to hold until liquidation. This completes the example.

80

3 Asset Price Bubbles

Example 8 (CEV Process) A special case of Example 7 is the constant elasticity of variance (CEV) process. This is obtained by letting the volatility function in Example 7 satisfy σ 2 (x) = β 2 x 2α for α, β > 0. We have the following three cases. 1. 0 < α < 1. (No bubble) 1 β2



∞

x 1−2α dx =

1

1 x 2−2α

∞ = ∞. β 2 2 − 2α 1

2. α = 1 (No bubble—Geometric Brownian Motion) 

1 β2

∞

x −1 dx =

1

1 ln(x) ∞ 1 = ∞. 2 β

3. α > 1. (Bubble) 1 β2



∞

x 1−2α dx =

1

1 x 2−2α

∞ < ∞. β 2 2 − 2α 1

This completes the example. Now, assume that the market is complete with respect to Q, and consider admissible s.f.t.s.’s involving the mma and the risky asset with price process S. We assume that S is a strict Q local martingale so that it S contains a price bubble β > 0. We claim there are multiple admissible s.f.t.s.’s that attain ST . Indeed, the first is a buy and hold trading strategy in S (with a unit holding) that generates the value process  Xt = 1 · St = S0 +

t

1 · dSu

0

for all t ∈ [0, T ]. Note that XT = ST and X0 = S0 . This value process Xt is not a Q martingale because although XT = ST , E Q [XT ] = E Q [ST ] < S0 = X0 . Hence, is a strict supermartingale under Q. Since the market is complete with respect to Q ∈ Ml , there exists an alternative admissible s.f.t.s. (α˜ 0 , α) ˜ ∈ A (x) ˜ with value process X˜ that also attains ST where X˜ t = α˜ 0 (t) + α˜ t St = x˜ +



t 0

α˜ u dSu = E Q [ST |Ft ]

3.3 The Asset Price Bubble

81

for all t ∈ [0, T ]. This value process X˜ is a martingale under Q, and it must be dynamic with (α˜ 0 , α) ˜ changing across time because the value process for a buy and hold trading strategy in the risky asset (shorting) was shown not to be a Q martingale. Algebra shows that the bubble’s magnitude is equal to the value process of the admissible s.f.t.s. (α˜ 0 , 1 − α) ˜ ∈ A (S0 − x), ˜ i.e. βt = St − E Q [ST |Ft ] = Xt − X˜ t 

t

= S0 − x˜ +

(1 − α˜ u ) dSu

0

for all t ∈ [0, T ]. Recall that the bubble process satisfies the following properties: (i) β0 > 0, (ii) βt ≥ 0 for all t ∈ [0, T ], and (iii) Q(βT = 0) = 1. Combined, these observations imply that the bubble’s magnitude is equal to the value process of a suicide strategy that loses β0 > 0 dollars for sure by time T . We formalize this observation in the following theorem. Theorem 27 (Complete Market Bubbles and Suicide Strategies) Assume NFLVR and that the market is complete with respect to Q ∈ Ml . If the risky asset S exhibits a bubble under Q, then the bubble’s magnitude is equal to the value process of a suicide strategy in the risky asset and mma. This implies that the stock is a dominated security, being dominated by the admissible s.f.t.s. (α˜ 0 , α) ˜ ∈ A (x) ˜ with value process X˜ t = E Q [ST |Ft ] for all t ∈ [0, T ]; proving the following result. Theorem 28 (Strict Local Martingales and Dominance) Assume NFLVR and that the market is complete with respect to Q ∈ Ml . Let S be a strict Q local martingale. Then, there exists an admissible s.f.t.s. (α˜ 0 , α) ˜ ∈ A (x) ˜ with value process X˜ t = α˜ 0 (t) + α(t)S ˜ that attains S with t T X˜ t = x˜ +



t

α˜ u · dSu = E Q [ST |Ft ]

0

where X˜ 0 = x˜ = E Q [ST ] < S0 for all 0 < t < T . This admissible s.f.t.s. dominates buying and holding St for all t ∈ [0, T ]. To determine the exact admissible s.f.t.s. (α˜ 0 , α) ˜ ∈ A (x), ˜ the method used in Sect. 2.7.2 in Chap. 2 in the synthetic construction of a derivative can be employed with the “derivative’s” time t value being X˜ t = E Q [ST |Ft ]. At this time, however, we can still say something about the trading strategy ˜ and how it differs from buying and holding the risky asset. For the typical (α˜ 0 , α) suicide strategy, see Sect. 2.1.3 in Chap. 2, the risky asset is shorted until time τ = inf {t > 0 : βt = 0}, the first time the bubble disappears. This implies that the

82

3 Asset Price Bubbles

admissible s.f.t.s. (α˜ 0 , 1 − α) ˜ ∈ A (S0 − x) ˜ generating β satisfies 1 − α˜ t < 0 for t ∈ [0, τ ), which implies α˜ t > 1 for t ∈ [0, τ ). Given α˜ 0 (t)+ α˜ t St = E Q [ST |Ft ] < St for t ∈ [0, τ ) as well, this implies that the position in the mma is strictly negative α˜ 0 (t) < 0 for t ∈ [0, τ ). Remark 32 (Bubbles and NFLVR) It is natural to wonder whether shorting the risky asset and going long the admissible s.f.t.s. (α˜ 0 , α) ˜ ∈ A (x) ˜ generates a FLVR. In general, if the risky asset’s price process is unbounded above, then the answer is no due to the admissibility condition on the combined s.f.t.s. Indeed, the combined s.f.t.s. would necessarily be unbounded below because of the short position in the risky asset, thereby violating the admissibility condition (see Remark 14 in Chap. 2). This completes the remark. Remark 33 (Alternative Definitions of an Asset Price Bubble) Alternative definitions of an asset price bubble appear in the literature, and one is worth mentioning. First, consider all admissible s.f.t.s. (α0x , α x ) ∈ A (x) that attain ST with value process Xx . This set is non-empty because the buy and hold trading strategy is within it. Because the value processes Xx are supermartingales under Q, we know that X0x = x ≥ E Q (ST ) for all x which attain ST . All of these admissible s.f.t.s. are said to super-replicate the risky asset’s payoff. We can now give the alternative definition of an asset price bubble (see Herdegen and Schweizer [70] and Loewenstein and Willard [141]). The super-replication bubble at time t is defined as β˜t = St − S¯t

where

S˜t = ess inf {x ≥ E Q (ST |Ft ) : (α0 , α)s∈[t,T ] ∈ A (x) attains ST }. S˜t is called the super-replication cost. In a complete market, we know the infimum in this expression is attained with the admissible s.f.t.s. (α˜ 0 , α) ˜ s∈[t,T ] ∈ A (x) ˜ in Theorem 28. In this case, the superreplication price S˜t = E Q [ST |Ft ], which equals the fundamental value and the two definitions coincide. In an incomplete market, however, it is possible that S˜t > E Q [ST |Ft ], and the two definitions differ. In this case β˜t ≤ βt with strict inequality possible. Given the connection of asset price bubbles to a trader’s optimal supermartingale deflator as discussed in the trader’s portfolio optimization problem (see Part II below), we prefer and use the definition based on an asset’s fundamental value. This completes the remark.

3.3 The Asset Price Bubble

83

3.3.2 Incomplete Markets We note that when the market is incomplete, an admissible s.f.t.s. that dominates buying and holding the risky asset as characterized in Theorem 28 may not exist. This section explores the study of bubbles in incomplete markets. Remark 34 (Discontinuous Price Processes with Bubbles) We note that most discontinuous sample path price processes imply that the market is incomplete. This occurs because when a jumps occurs, the distribution for the change in the price process is usually not just discrete with a finite number of jump amplitudes, where the number of jump amplitudes is less then the number of traded assets (see Cont and Tankov [34, Chapter 9.2]). In this case the number of traded assets will be insufficient to hedge the different jump magnitudes possible (in mathematical terms, the martingale representation property fails for the value processes of admissible s.f.t.s.). The above examples are for price processes that have continuous sample paths. For examples of discontinuous sample path price processes with bubbles, one can take the continuous sample path examples given above and add a Q-martingale jump process (because a strict local martingale plus a martingale is again a strict local martingale). For other discontinuous sample path price processes that are strict local martingales see Protter [160]. This completes the remark. We know a number of sufficient conditions for a price process not to exhibit a price bubble. These apply in both complete and incomplete markets. The first of these is given in the next theorem. Theorem 29 (Asset Price Processes with Independent Increments) Given NFLVR, choose a Q ∈ Ml . If St = S0 eLt for all t ∈ [0, T ] where Lt is a process with independent increments under Q , then there is no price bubble. Proof This proof is based on Medvegyev [143, Example 1.146, p. 103]. Note that St a local Q martingale implies that SS0t ≥ 0 is a Q local martingale. By Lemma 4

in Chap. 1, Xt = SS0t = eLt > 0 is a Q supermartingale. Given L0 = 0, X0 = 1 ≥ E Q [Xt ] for all t ∈ [0, T ]. Because Lt has independent increments,       E Q [Xt ] = E Q eLt −Ls eLs = E Q eLt −Ls Xs = E Q eLt −Ls E Q [Xs ] . Hence, EQ



  Lt −Ls 

    Xs Xt

e Xs Xs

Fs = E Q Fs = Q E Q eLt −Ls Fs = Q E Q eLt −Ls .

Q E [Xt ] E [Xt ] E [Xt ] [Xt ]

EQ

84

3 Asset Price Bubbles

Using both of the above expressions yields EQ Thus,

Xt E Q [Xt ]



 Xt

Xs Fs = Q .

Q E [Xt ] E [Xs ]

is a Q martingale, where 0 ≤ Xt ≤

By Lemma 5 in Chap. 1, Xt = This completes the proof.

St S0

Xt E Q [Xt ]

for all t ∈ [0, T ].

is a Q martingale. Hence, St is a Q martingale.

Remark 35 (Examples of No Bubble Price Processes) If Lt is a Levy process under Q, it has independent increments, and therefore St = S0 eLt has no bubbles. A special case of such a Levy process is where the normalized asset price process St under Q is a geometric Brownian motion process, i.e. dSt = σ dWtQ . This underlies the Black-Scholes-Merton option pricing model discussed in Chap. 5. Hence, by construction, the Black-Scholes-Merton market has no asset price bubbles. This completes the remark. Another sufficient condition is provided by using Remark 4 following Lemma 5 in Chap. 1. Theorem 30 (Bounded Asset Price Processes) Given NFLVR, if St ≥ 0 is bounded above for all t ∈ [0, T ], then there is no price bubble. Using the previous theorem, we can prove the following collection of facts related to asset price bubbles useful in subsequent Chapters. Remark 36 (Bonds Have No Bubbles) Consider a default-free zero-coupon bond that pays $1 at time T . If we assume that default-free forward rates are bounded below by a negative constant for all t ∈ [0, T ], the default-free zero-coupon bond price process p(t, T ) is bounded above for all t (this follows from the definition of a forward rate in Chap. 6). Then, by Theorem 30, default-free zero-coupon bond prices can have no bubbles. By extension, no default-free coupon bond, or credit risky zero- or coupon bonds can have price bubbles as well as long as ND holds, because credit risky bond prices will be less than or equal to default-free bond prices under ND. Chapter 7 studies credit risky bonds. Remark 37 (Put-Call Parity Need Not Hold) European puts and calls are derivatives whose payoffs depend on the value of the risky asset at some future date. Calls give the right, but not the obligation, to buy the risky asset at the future date for a fixed price. Puts give the right to sell. Let the market prices for the European put and call be denoted P utt and Callt , respectively. These are in units of the mma, just like the risky asset. Since we are in the normalized market and the risky asset’s price is denominated in units of the mma, but the options’ payoff are in dollars, we need to introduce the money market account’s value in dollars, Bt for t ∈ [0, T ] with B0 = 1 to transfer the options’ payoffs into units of the mma. For this purpose, we assume that Bt ≥ 1 for all t > 0, which means that interest rates are nonnegative. Let

3.3 The Asset Price Bubble

85

p(t, T ) denote the price of a zero-coupon bond paying a sure dollar at time T . Under NFLVR, itwill be shown using the Heath-Jarrow-Morton model in Chap. 6  ) Q 1 |F for all t ∈ [0, T ]. = E that p(t,T t Bt BT The time T payoffs to a European call and put on the risky asset with maturity date T and strike price K, in units of the mma, are   K CallT = max ST − ,0 BT

 and

P utT = max

 K − ST , 0 . BT

Note that the options’ payoffs depend on the market price of the risky asset, ST , which may contain a bubble. The payoff does not explicitly depend on the asset’s fundamental value. We define the fundamental value of the European call and put option as     K CalltF V = E Q max ST − , 0 |Ft BT

and

    K P uttF V = E Q max − ST , 0 |Ft . BT

Any difference between an option’s market price and fundamental value is the option’s bubble, which may be different from the underlying asset’s price bubble, if it exists. Put-call parity is based on the following identity     K K K ST − = max ST − , 0 − max − ST , 0 . BT BT BT The fundamental value of this expression is obtained by taking expectations with respect to the equivalent local martingale measure Q ∈ Ml chosen by the market. This gives E Q [ST |Ft ]−

        K p(t, T ) K K = E Q max ST − , 0 |Ft −E Q max − ST , 0 |Ft . Bt BT BT

Suppose that both the put and call options have no bubbles, i.e. P utt =

P uttF V

    K = E max − ST , 0 |Ft BT

Callt =

CalltF V

    K = E max ST − , 0 |Ft . BT

Q

and Q

86

3 Asset Price Bubbles

Suppose further that the risky asset has a price bubble, i.e. St > E Q [ST |Ft ], which is possible under NFLVR. Then, St −

p(t, T ) K > Callt − P utt Bt

and put-call parity is violated. For an example where put-call parity is violated and both the call and put options also have price bubbles see Protter [159]. This completes the remark. Remark 38 (Bounded at Time T Doesn’t Exclude Bubbles) Theorem 30 is not true if we replace the hypothesis St ≥ 0 is bounded above for all t ∈ [0, T ] with the weaker hypothesis that only ST ≥ 0 is bounded at time T . Indeed, an example is the following, which is based on Protter [159]. Consider the stock price 

t

St = 1 + 0

Ss dWs T −s

for all t ∈ [0, T ] where Ws is a Brownian motion initialized at W0 = 0 under Q. By construction, the process St is a strict Q local martingale. Hence, the market satisfies NFLVR, i.e. Q ∈ Ml . It can be shown that St → 0 as t → T a.s. Q. This implies that ST is bounded at time T because ST ≤ 0, but there is still an asset price bubble. This completes the remark.

3.4 Theorems Under NFLVR and ND This section adds the assumption of no dominance (ND) on the market and explores the implications for the existence of asset price bubbles. An interesting theorem results. Theorem 31 (No Bubbles) Let NFLVR and ND hold. Suppose the market is complete with respect to Q ∈ M where M = {Q ∼ P : S is a Q martingale}. Then, there is no asset price bubble. Proof From the Third Fundamental Theorem 16 of asset pricing in Chap. 2, we have NFLVR and ND imply the existence of a martingale measure Q ∈ M. By the Second Fundamental Theorem 14 of asset pricing in Chap. 2, completeness implies the martingale measure is unique. Hence, S is a Q martingale, and not a strict local martingale. This completes the proof. For bubbles to exist in a market that satisfies ND, this theorem shows that the market must be incomplete. In such an incomplete market satisfying ND, the local martingale Q ∈ Ml can still be uniquely identified if enough derivatives trade, see the discussion of an extended incomplete market in Sect. 3.1 above.

3.4 Theorems Under NFLVR and ND

87

Remark 39 (Incomplete Markets) In a market satisfying NFLVR and ND, when the market is incomplete, an asset price bubble can exist. Recall that in an incomplete market there may be an infinite number of local martingale measures. By the Third Fundamental Theorem 16 of asset pricing in Chap. 2, one of these is an equivalent martingale measure. But, it may not be the Q chosen by the market. There are examples where both the set of martingale and local martingale measures are nonempty for a given market, see Jarrow et al. [115]. This completes the remark. Remark 40 (Trading Constraints and Bubbles) It will be shown in Part IV of this book that trading constraints transform an otherwise complete market into an incomplete market. Hence, trading constraints can transform a market without any price bubbles to a market with bubbles. Unlike the bubbles under NFLVR, in a trading constrained market, bubbles do not arise because the retrade value exceeds the fundamental value of the asset. They arise due to the trading constraint itself, see Chap. 19, Sect. 19.3. This completes the remark. Theorem 32 (Bounded Asset Price Processes) Let NFLVR and ND hold. If ST ≥ 0 is bounded above at time T , then there is no price bubble. Proof Let ST ≤ k a.s. P for a constant k > 0. Next, suppose St > k for some t with positive probability. Then, buying the risky asset at time 0 and selling it and investing the St dollars in the mma at time t in this event generates St dollars for sure at time T . This dynamic s.f.t.s. dominates buying and holding the risky asset until time T , contradicting ND. Hence, St ≤ k for all t. An application of Theorem 30 completes the proof. Remark 41 (Put Options Have No Bubbles) For this remark, note that the payoffs are in units of the mma, except p(t, T ) and Bt , which are in dollars (see Remark 37 above). Consider a European put option on the risky asset with time T payoff   P utT = max BKT − ST , 0 ≤ BKT < K in units of the mma. Note that the time T payoff to the put option is bounded above, hence by Theorem 32, if we consider a market with both the risky asset and the European put option trading, then put options can have no price bubbles, i.e. P utt = P uttF V . But interestingly, the underlying risky asset can! This completes the remark. The next theorem is useful for pricing calls and puts in a market satisfying NFLVR and ND. Recall that the payoffs are in units of the mma, except p(t, T ) and Bt , which are in dollars (see Remark 41 above). Theorem 33 (Put-Call Parity) Let NFLVR and ND hold. Given European call and put options on the risky asset with identical strikes and maturity dates trade, then St − for all t ∈ [0, T ].

p(t, T ) = Callt − P utt Bt

(3.5)

88

3 Asset Price Bubbles

Proof If put-call parity is violated, then one can create a portfolio which dominates one of the component securities. This violates ND, which gives a contradiction. This is Merton’s [145] original argument. This completes the proof. Finally, one can determine the arbitrage free value of a traded call option, in a market where the underlying asset exhibits a price bubble. Using the Q ∈ Ml chosen by the market, in conjunction with put-call parity, we get the following theorem. Corollary 5 (European Call Price with Price Bubbles) Let NFLVR and ND hold. Choose a Q ∈ Ml . If St has a price bubble βt > 0, then     K Callt = E Q max ST − , 0 |Ft + βt BT

(3.6)

for all t ∈ [0, T ]. Proof By a math identity,   E Q [ST |Ft ] − E Q BKT |Ft         = E Q max ST − BKT , 0 |Ft − E Q max BKT − ST , 0 |Ft .

Q Recall  that βt = St − E [ST |Ft ], and by Remark 41, P utt = E Q max BKT − ST , 0 |Ft .     ) Q max S − K , 0 |F Substitution gives St − βt − p(t,T − P utt . = E T t Bt BT Expression (3.5) implies the result. This completes the proof.

American options behave differently in markets where the underlying risky asset exhibits a bubble. An American option gives the right to purchase the risky asset for K dollars from the date the option is purchased until the option’s maturity date, time T . Let AmCallt denote the American call’s option price in units of the mma, and define its fundamental value as     K FV Q AmCallt = ess sup E max Sτ − , 0 |Ft Bτ τ ∈[t,T ] for all t ∈ [0, T ]. Note that this is a hypothetical optimal stopping time based on the   K fundamental value of the payoff max Sτ − Bτ , 0 . The stopping time τ does not refer to the exercise time of the traded American call option, unless the American call’s fundamental value equals its market price. Theorem 34 Let NFLVR and ND hold for the options as well as the risky asset and mma. Choose a Q ∈ Ml .

3.4 Theorems Under NFLVR and ND

89

Assume there exists f and a martingale Y such that  a function  Su ≤ Su− + f

sup Ss (1 + Yu ) for all u ∈ [t, T ] and t ∈ [0, T ]. Then,

t≤s 0, note that by the previous observation and Theorem 34 we have that AmCalltF V = Callt . Suppose that τ = T . Then, CalltF V = AmCalltF V . But, we have Callt = CalltF V + βt . Hence, AmCalltF V > CalltF V , yielding a contradiction, which proves that τ < T . The reason why τ < T in the American call’s fundamental value is easy to understand. When there is a bubble, the market price of the risky asset is expected to decline over time. Hence, the bubble acts analogous to a continuous (random)

90

3 Asset Price Bubbles

cash outflow (dividend) from the risky asset. And, it is well-known that in a market with no bubbles, dividends can result in the early exercise of American calls (see Jarrow and Chatterjea [97]). This completes the remark.

3.5 Notes Much more is known about the local martingale theory of bubbles. For example, the impact of asset price bubbles on derivatives, foreign currencies, and how to test for the existence of asset price bubbles is well studied, however, no textbooks exist on this topic. Recent reviews include Jarrow and Protter [106], Protter [159], and Jarrow [91]. For an empirical study that tests for asset price bubbles using historical asset price time series data see Jarrow et al. [116]. This is a new and exciting area for future research.

Chapter 4

Basis Assets, Multiple-Factor Beta Models, and Systematic Risk

This chapter studies basis assets, the multiple-factor beta model, and characterizes systematic risk. This is done for an incomplete market where asset prices can have discontinuous sample paths. Multiple-factor beta models are used for active portfolio management and the determination of positive alphas trading strategies. These models can be derived using only the Third Fundamental Theorem 16 of asset pricing in Chap. 2. A special case of this chapter is Ross’s APT, which illustrates the notion of portfolio diversification. This chapter is based on Jarrow and Protter [108].

4.1 The Set-Up Given is a non-normalized market ((B, S), F, P). This set-up starts with the following assumption. Assumption (Existence of an  Equivalent Martingale Measure)  M = ∅ where M = Q ∼ P : BS is a Q martingale , i.e. there exists an equivalent martingale measure. By the Third Fundamental Theorem 16 of asset pricing in Chap. 2, this implies both NFLVR and ND hold in the market. We do not assume that the market is complete with respect to Q ∈ M, hence, there can exist many local martingale and martingale measures. Fix a Q ∈ M and let YT = dQ dP > 0 be the density function with respect to P. The subsequent analysis uses this equivalent martingale measure.

© Springer Nature Switzerland AG 2021 R. A. Jarrow, Continuous-Time Asset Pricing Theory, Springer Finance, https://doi.org/10.1007/978-3-030-74410-6_4

91

92

4 Basis Assets, Multiple-Factor Beta Models, and Systematic Risk

Remark 43 (NFLVR and Not ND) The subsequent analysis can be generalized to assuming only NFLVR, i.e. the existence of an equivalent local martingale measure so that Ml = ∅ where   S Ml = Q ∼ P : is a Q local martingale . B In this case, however, asset price bubbles are introduced into the subsequent results. This extension can be found in Jarrow [92]. This completes the remark. Recall that A (x) = {(α0 , α) ∈ (L(B), L (S)) : Xt = α0 (t)Bt + αt · St , ∃c ≤ 0,  Xt = x + 0

t

 α0 (u)dBu +

t

 αu · dSu ≥ c, ∀t ∈ [0, T ]

0

is the set of admissible s.f.t.s. and    t  t 0 α0 (u)dBu + αu · dSu = XT C (x) = XT ∈ L+ : ∃(α0 , α) ∈ A (x), x + 0

0

is the set of time T payoffs generated by all admissible s.f.t.s. with initial value x. This chapter focuses on the time T payoffs from all admissible s.f.t.s. with nonnegative initial values at time 0 and whose value processes are Q martingales, i.e.   Xt X = XT ∈ ∪ C (x) : is a Q martingale . x≥0 Bt Often, we will restrict consideration to strictly positive initial values so that returns on the admissible s.f.t.s. are well defined. To understand this set of payoffs X note the following. If BS are H 1 martingales  1  2 < ∞ for i = 1, . . . , n (see Protter [158, p. 193]), then under Q, i.e. E Q SBi , SBi T  1  X X 2 Q < ∞. Indeed, under this condition X includes those XT such that E B, B T

X B

is also a H martingale under Q (see the proof of Theorem 17 in Chap. 2), hence such an X B ∈ X . For our analysis, however, the set X is larger because it also includes Q martingales that are not H 1 martingales under Q. For this chapter, we change our terminology slightly. Rather than viewing X as the payoffs to derivatives, we view these as the liquidation values of traded portfolios of the risky assets, perhaps created by financial institutions such as hedge funds, mutual funds, and exchange traded funds (ETFs). This interpretation 1

4.2 Basis Assets

93

is realistic because such financial institutions’ holdings are dynamic and well represented by an element XT ∈ X . For later use, we note the properties of an arbitrary traded portfolio XT ∈ X . By definition, there exists an initial investment x = X0 ≥ 0 and an admissible s.f.t.s. (α0 , α) ∈ A (x) such that 

T

XT = x +



0

α(t) · dSt

where

(4.1)

0



t

Xt = α0 (t)Bt + α(t) · St = x +





XT BT

t

α0 (s)dBs +

0

Xt = EQ Bt

T

α0 (t)dBt +



Ft

α(s) · dSs

and

(4.2)

0

for all t ∈ [0, T ].

(4.3)

In expression (4.2), the first equality is the definition of the value process and the second equality is the s.f.t.s. condition. It is important to note that since the traded portfolio’s time T values in X are attainable securities generated by uniformly integrable martingales with the same initial values under any Q ∈ M, the value processes will have identical time t values under any Q ∈ M (see the discussion following Theorem 17 in Chap. 2). This observation is important because the set of equivalent martingale measures need not be a singleton since the market may be incomplete.

4.2 Basis Assets This section generates a set of basis assets, a necessary step in generating a multiplefactor beta model. We claim that due to the admissibility condition, the set of portfolio values X is not a linear subspace, but a convex cone in the space of FT −measurable random variables. To prove this claim, we need to give some definitions. Let L0 denote the space of FT −measurable random variables. A subset C ⊂ L0 is a cone if it has the property that if x ∈ C , then rx ∈ C for all r ≥ 0 where r ∈ R. A subset C ⊂ L0 is convex if it has the property that if x, y ∈ C , then rx + (1 − r)y ∈ C for all r ∈ [0, 1] ⊂ R. Finally, a subset C ⊂ L0 is a linear subspace if it has the property that if x, y ∈ C , then r1 x +r2 y ∈ C for all r1 , r2 ∈ R. Note that a linear subspace is both convex and a cone. We can now prove the claim. To show that X is a convex cone, consider two random variables X1T , X2T ∈ X . Let (α0i , α i ) ∈ A (x i ) be the admissible s.f.t.s.’s for i = 1, 2 that generate X1T , X2T ∈ X . Then, X is a cone because rX1T ∈ X for any

94

4 Basis Assets, Multiple-Factor Beta Models, and Systematic Risk

r ≥ 0, since (rα01 , rα 1 ) ∈ A (rx 1 ). And, X is convex because rX1T + (1 − r)X2T ∈ X for any r ∈ [0, 1], since (rα01 +(1−r)α 1 , rα02 +(1−r)α 2 ) ∈ A (rx 1 +(1−r)x 2 ). The set X is not a linear subspace because if X1T ∈ X is unbounded above, then / X because the admissibility condition is violated, i.e. there is no uniform −X1T ∈ lower bound for the value process X1t . This completes the proof of the claim. Given this observation, consider the smallest linear subspace in L0 containing X , denoted XL . Since X is a convex cone containing zero, this linear subspace XL = X ⊕ {−X }, the “positive” and “negative” traded portfolios. Next, since XL is a linear space, there always exists a Hamel basis (see Friedman [64, p. 130], Simmons [178, p. 196], Taylor and Lay [182, p. 41]). A Hamel basis is a possibly uncountably infinite collection of portfolios in XL such that any finite subcollection of this set are linearly independent and the entire set spans XL . In this case, the Hamel basis will also be a direct sum of “positive” traded portfolios satisfying the admissibility condition, and “negative” or short positions in traded portfolios that do not. Nonetheless, since the traded portfolios of concern in X are “positive,” the “positive” subset of this Hamel basis is sufficient to span the traded portfolios in X . Let this subset of the Hamel basis be denoted {Vi (T ) : i ∈ H } ⊂ X where the index set H labels the elements in the basis. This collection can be uncountably infinite. We call the traded portfolios in {Vi (T ) : i ∈ H } the basis assets. This is the minimal set of linearly independent elements in the set of traded portfolios X from which all of X can be obtained. More precisely, given an arbitrary traded portfolio XT ∈ X , there exists a finite set of basis assets ΦX ⊂ H depending upon XT such that (4.4) XT = j ∈ΦX ηXj Vj (T ) where ηXj = 0 are non-zero constants (F0 −measurable random variables). Different XT ∈ X will, in general, be spanned by different basis assets. But, the set of basis assets needed to span any particular traded portfolio is always finite. The number of basis assets needed to span the entire space X can be uncountably infinite. We note that since the value of the money market account BT is random at time T , it is treated similarly to any of the other traded securities. We first characterize the set of basis assets for subsequent use. Given a basis

asset Vk (T ) ∈ X , for each k ∈ H there exists an admissible s.f.t.s. α0k (t), α k (t) t∈[0,T ] depending on k such that  T T Vk (T ) = Vk (0) + α0k (t)dBt + 0 αtk · dS(t), (4.5) 0

 Vk (t) = α0k (t)Bt +αtk ·St = Vk (0)+ Vk (t) = EQ Bt



t 0

α0k (s)dBs +

 Vk (T )

Ft BT

t

k 0 αs ·dS(s),

for all t ∈ [0, T ].

and

(4.6) (4.7)

4.2 Basis Assets

95

Next, consider an arbitrary traded portfolio XT ∈ X . As noted above, by the definition of the set H , there exists a finite subset of basis assets ΦX ⊂ H such that XT =



j ∈ΦX ηXj Vj (T )

(4.8)

where ηXj = 0 are non-zero constants. Hence, we see that the time T payoff to any traded portfolio XT can be represented as a finite portfolio of the basis assets. Dividing both sides of this expression by BT , and taking conditional expectations under Q yields the following lemma. This lemma is essential to all of the subsequent results. Lemma 15 (Values of Traded Portfolios) Given XT η j ∈ΦX Xj Vj (T ), Xt =

∈ X where XT



j ∈ΦX ηXj Vj (t)

=

(4.9)

for t ∈ [0, T ]. This result states that the value of an arbitrary traded portfolio at time t can be written as the same linear combination of the same finite collection of the basis assets that hold at time T . This follows directly from the existence of an equivalent martingale measure Q ∈ M. As a special case of this lemma, we can consider the traded assets Si (T ) for i = 1, . . . , n. Note that each Si (T ) ∈ X because buying and holding one unit of risky asset i is an admissible s.f.t.s. that is a Q martingale, after normalization by the mma. Hence, the Lemma implies that for the traded risky assets themselves, there exists a finite set of basis assets Φi ⊂ H and constants ηij = 0 such that Si (t) =



j ∈Φi ηij Vj (t)

for all i = 1, . . . , n and t ∈ [0, T ].

(4.10)

The above discussion proves the following mutual fund theorem. Theorem 35 (Mutual Fund Theorem) Any investor is indifferent between (i) holding the portfolio XT , which is an admissible s.f.t.s. in the money market account and the risky assets (B, S) and (ii) holding the payoff XT generated by the admissible buy and hold(therefore self- financing) trading strategy constructed from the basis assets Vj : j ∈ ΦX   with share holdings ηXj : j ∈ ΦX as in expression (4.9). This theorem explains the existence of the plethora (thousands) of traded mutual funds and ETFs in current financial markets. These correspond to the collection of basis assets {Vi (T ) : i ∈ H }. As seen from the above setup, without additional restrictions on the model’s structure, an infinite number of these basis assets are needed to span any traded portfolio. Transaction costs, although outside of our

96

4 Basis Assets, Multiple-Factor Beta Models, and Systematic Risk

model structure, would imply that traders would prefer trading in the basis assets rather than in the assets themselves to form a desired portfolio X.

4.3 The Multiple-Factor Beta Model Given the preceding framework, we can now derive the multiple-factor beta model. To derive the multiple-factor model, we need to consider the return on a traded portfolio XT ∈ X over an arbitrary time interval within the trading horizon. First, partition [0, T ] into a collection of subintervals of length Δ > 0. Then, fix a time interval [t, t + Δ] ⊂ [0, T ] where t ≥ 0 aligns with one of these partitions. The return on the traded portfolio XT ∈ X over this time interval is RX (t) =

X(t + Δ) − X(t) X(t)

where we assume that Xt = 0 for all t = 0, . . . , T − Δ, so that these returns are well defined. Then, identifying the portfolio with its basis assets as in expression (4.9), simple algebra yields RX (t) =



Vj (t) j ∈ΦX ηXj X(t) =





Vj (t + Δ) − Vj (t) Vj (t)

j ∈ΦX βXj (t)rj (t)



(4.11)

where βXj (t) = ηXj

Vj (t + Δ) − Vj (t) Vj (t) , rj (t) = and j ∈ΦX βXj (t) = 1. X(t) Vj (t)

The coefficients of the basis assets βXj = 0 are called the portfolio’s betas. Although the share holdings ηXj are constant, the portfolio’s betas βXj (t) are stochastic due to both Vj (t) and X(t) being stochastic. Although stochastic, the betas are Ft −measurable, hence their values are known at time t. To simplify the subsequent expressions, we add the following assumption. Assumption (Traded Default-Free Zero-Coupon Bonds) The vector of risky assets S contains default-free zero-coupon bonds paying $1 at times t = Δ, . . . , T . Consider the default-free zero-coupon bond that matures at time t + Δ, with time t price denoted p(t, t + Δ). The return on this zero-coupon bond over [t, t + Δ] 1 is denoted r0 (t) = p(t,t+Δ) − 1. By its definition, r0 (t) is Ft −measurable, i.e. its return over [t, t + Δ] is known at time t. r0 (t) is called the default-free spot rate of interest over [t, t + Δ].

4.3 The Multiple-Factor Beta Model

97

Using these traded default-free zero-coupon bonds, one can construct the following dynamic portfolio. The portfolio starts at time 0 with a dollar invested in the shortest maturity of these zero-coupon bonds (the zero-coupon bond maturing at time Δ). When this zero-coupon bond matures, the value of the portfolio is reinvested in the (then) shortest maturity zero-coupon bond (the zero-coupon bond maturing at time 2Δ). This process is continued / until time T − Δ. This trading strategy’s value at times t = Δ, . . . , T equals ts=Δ [1 + r0 (s − Δ)]. This dynamic portfolio is a discrete money market account (mma) whose return over the time interval [t, t + Δ] is the spot rate r0 (t). Without loss of generality, we can include this discrete mma in the set of basis assets (see Simmons [178, Theorem A, p. 197]). Given this observation, expression (4.11) yields the next theorem. We note, however, that adding the discrete time mma in the set of basis assets implies that β0 may be equal to zero in expression (4.12) below. Theorem 36 (Multiple-Factor Beta Model) Given an arbitrary XT ∈ X , RX (t) = β0 r0 (t) +



j ∈ΦX βXj (t)rj (t)

β0 +



j ∈ΦX βXj (t)

where

(4.12)

=1

or equivalently, RX (t) − r0 (t) =



j ∈ΦX βXj (t)



rj (t) − r0 (t)

(4.13)

for all t = 0, . . . , T − Δ with βXj (t) = 0 all j. Remark 44 (Instantaneous mma Not Riskless over [t, t + Δ]) We note that conditioned on the information at time t, the instantaneous mma’s (Bt s) return is only “riskless” over [t, t + dt]. Over the discrete time interval [t, t + Δ] its return is random and similar to any other portfolio X ∈ X . This is why we assumed that a term structure of default-free zero-coupon bonds trade in the market. These defaultfree zero-coupon bonds guarantee the existence of a default-free spot rate over [t, t + Δ] for all t (in the partition of [0, T ]) and an admissible s.f.t.s. that earns this spot rate over this time interval. This admissible s.f.t.s. is the discrete time mma discussed above. This completes the remark. Remark 45 (Expectation Form of Multiple-Factor Model) Taking conditional expectations of expression (4.13) under P gives E[RX (t) |Ft ] − r0 (t) =



j ∈ΦX βXj (t)



E[rj (t) |Ft ] − r0 (t) .

(4.14)

This is a generalization of the representation of a multiple-factor beta model seen in the asset pricing literature. This completes the remark.

98

4 Basis Assets, Multiple-Factor Beta Models, and Systematic Risk

Remark 46 (Empirical Form of Multiple-Factor Model) To obtain an empirical form of the multiple-factor beta model, observation error is added to expression (4.13) to account for noise in the data and/or model error, i.e. RX (t) = r0 (t) +



j ∈ΦX βXj (t)



rj (t) − r0 (t) + εX (t)

where

(4.15)

βXj (t) = 0 all j . In empirical testing, the model is “accepted” if εX (t) is “white noise,” i.e. 1. E[εX (t) |Ft ] = 0, 2. εX (t)  with respect to F, and  is a P martingale 3. cov εX (t), rj (t) |Ft = 0 for all j ∈ ΦX . In addition, from an economic point of view, the model will be practically useful if the R 2 (goodness of fit) is large, which implies that εX (t) is small. This completes the remark.

4.4 Positive Alphas An important tool in portfolio evaluation and stock selection is identifying a risky asset’s alpha. This section defines an asset’s alpha and studies its use in portfolio management. Recall that expression (4.15), using the Third Fundamental Theorem 16 of asset pricing in Chap. 2, is derived under the assumptions of NFLVR and ND. Consequently, a violation of NFLVR or ND corresponds to a violation of expression (4.13). This can be captured by adding a non-zero alpha (an F optional process) to the right side of expression (4.13), i.e. RX (t) − r0 (t) = αX (t) +



j ∈ΦX βXj (t)



rj (t) − r0 (t) .

(4.16)

Definition 35 (A Traded Portfolio’s Alpha) The traded portfolio X s alpha is defined to be the αX (t) in expression (4.16). To be consistent with the existing literature on this topic, we use the notation αX (t) for a traded portfolio’s alpha. This alpha, however, is unrelated to the notation for an admissible s.f.t.s. (α0 , α) ∈ A (x) generating X. This double use of the notation “α” should cause no confusion in the subsequent exposition because the notation for a portfolio’s alpha is only temporarily used in this section of this chapter. The importance of a traded portfolio’s alpha is due to the next theorem. Theorem 37 (Positive Alphas) A non-zero αX (t) on [0, T ] implies there does not exist an equivalent martingale measure for the risky assets S.

4.5 The State Price Density

99

By a non zero alpha on [0, T ] we mean that αX (t) is an optional process that is not the zero process, i.e. there exists a (possibly and usually non unique) stopping time τ such that |αX (τ )1{τ ≤T } | > 0 with positive probability. Proof Suppose we have a non zero and F optional alpha process in expression (4.16), and that there exists an equivalent martingale measure. Given the existence of an equivalent martingale measure, we have that expression (4.13) holds for all times t. This means that the process αX must be equal to 0 under Q, a contradiction, which completes the proof. This result is important for active portfolio management because it implies that since arbitrage opportunities or dominated assets are rare in well-functioning markets, positive alpha trading strategies are less common than currently believed to be true by active portfolio managers, see Jarrow [89] for an elaboration.

4.5 The State Price Density This section revisits the notion of a state price density (stochastic discount factor). Given NFLVR and ND, let YT = dQ dP > 0 be an equivalent martingale measure’s density function. Recall that the state price density is given by Ht =

Yt Bt

(4.17)

where Yt = E [YT |Ft ] for all t ∈ [0, T ]. Also recall that under NFLVR, we could only prove that (St Ht ) is a local martingale with respect to P (see Lemma 14 in Chap. 2). Here with ND, however, we can prove that (St Ht ) is a martingale with respect to P, i.e.

  HT

Ft St = E ST Ht for all t ∈ [0, T ].

  

Proof St = E Q BSTT Ft Bt = E BSTT completes the proof.



YT E[YT |Ft ] Ft

(4.18)



 Bt = E ST

HT Ht



Ft . This

In particular, at time 0 we obtain S0 = E [ST HT ] .

(4.19)

This expression shows that multiplying a time T payoff by the state price density generates the “certainty equivalent” of the payoff. By definition, the current value of a certainty equivalent is its (discounted) expected value. We can, therefore, interpret

100

4 Basis Assets, Multiple-Factor Beta Models, and Systematic Risk

the state price density as containing an adjustment for risk. The quantity HHTt is sometimes called a stochastic discount factor. For later use, we apply expression (4.18) at time t to the traded zero-coupon bond that matures at time t + Δ, i.e.

  Ht+Δ

Ft . (4.20) p(t, t + Δ) = E 1 · Ht Finally, recall that expression (4.3) for a traded portfolio is Xt = EQ Bt





  XT

YT XT

F Ft . = E t BT Yt B T

Using the state price density notation, this is equivalent to 

HT Xt = E XT Ht



Ft ,

(4.21)

which implies that Xt Ht is also a P martingale.

4.6 Arrow Debreu Securities This section introduces abstract Arrow Debreu securities. Consider the indicator random variable 1A = {1 if ω ∈ A, 0 otherwise} for some A ∈ FT . This indicator variable is called an Arrow Debreu security. It represents a security paying one dollar if event A ∈ FT occurs, zero otherwise. If 1A ∈ X , using the state price density, we can compute the time t price of this security, 

HT pA (t) = E 1A Ht



 

Ft = E Q 1A Ft Bt

B

(4.22)

T

for all t ∈ [0, T ]. Note that we have already encountered an example of such an Arrow Debreu security earlier in this section, a zero-coupon bond paying a sure dollar at time T . Indeed, here the event A = Ω, so that



   HT

HT

F Ft p(t, T ) = pΩ (t) = E 1Ω = E (4.23) t Ht Ht for all 0 ≤ t ≤ T . Arrow Debreu securities enable us to “rank” events A ∈ FT based on their time 0 Arrow Debreu prices. Indeed, given two events A1 and A2 with P(A1 ) = P(A2 ), we say that the event A1 ∈ FT is more valuable than event A2 ∈ FT if pA1 > pA2 . An

4.7 Systematic Risk

101

event with the same probability of occuring is more valuable when its Arrow Debreu price is larger, and it is less valuable when its Arrow Debreu price is smaller.

4.7 Systematic Risk We can now introduce the concept of systematic risk. To do this, we need the following theorem. Theorem 38 (The Risk Return Relation) A traded portfolio’s expected return over [t, t + Δ] satisfies



Ht+Δ E [RX (t) |Ft ] = r0 (t) − cov RX (t), (1 + r0 (t))

Ft Ht 

(4.24)

for t = 0, . . . , T − Δ. Proof To simplify the notation in the proof, we write E[·] = E[· |Ft ] and we omit the t argument in RX (t). By expression (4.21),  since Xt Ht is a P martingale,  Xt+Δ Ht+Δ 1 = E X t Ht       Ht+Δ Ht+Δ = E (1 + RX ) HHt+Δ = E R + E . X Ht Ht t   Ht+Δ But, E Ht = p(t, t + Δ), and       Ht+Δ Ht+Δ E RX HHt+Δ = cov R + E[R . , ]E X X Ht Ht t Substitution yields   1 − p(t, t + Δ) = cov RX , HHt+Δ + E [RX ] p(t, t + Δ). t Divide by p(t, t + Δ) and use the fact that result. This completes the proof.

1−p(t,t+Δ) p(t,t+Δ)

= r0 (t) to get the final

In this theorem, expression (4.24) characterizes the concept of systematic risk. An asset’s return RX (t) often contains systematic risk, which requires an expected return that exceeds  the default-free spot rate if and only if  Ht+Δ < 0. Note the minus sign in front of the cov RX (t), Ht (1 + r0 (t)) |Ft covariance term in expression (4.24). To understand this statement recall that • the state price density’s value Ht+Δ (ω) is large when a payoff in the state ω ∈ Ω is valuable, Debreu price for 1{ω} is large.   i.e. when the Arrow

|Ft > 0, the asset X’s returns are high when the Arrow • If corr RX (t), HHt+Δ t Debreu price for 1{ω} is large—it is “anti -risky,” and 

|Ft < 0, the asset X’s returns are low when the Arrow • if corr RX (t), HHt+Δ t Debreu price for 1{ω} is large—it is “risky.”

102

4 Basis Assets, Multiple-Factor Beta Models, and Systematic Risk

Risk Factors In the multiple-factor beta model of Theorem 36, it is important to know which basis assets have non-zero risk premium (equivalently, non-zero excess expected returns). These are called risk factors because they can be used to construct any traded security and they determine the required compensation for systematic risk within the traded securities. To understand why this identification is important, as an illustration, let us consider Ross’s APT (the model in Sect. 4.8 below). In this model, there are K + 1 basis assets identified with each primary traded security. K of these basis assets are associated with all primary securities, and their risks are not diversifiable in a large portfolio. Hence, they have nonzero risk premium. These are the risk factors. One of these basis assets is idiosyncratic and unique to each primary security. This basis asset’s risk is diversifiable in a large portfolio, and has no risk premium. Hence, it is not a risk factor. The implication that idiosyncratic risk is not a risk factor is the key insight of Ross’s APT. Here, in a more general setting, we can determine which basis assets are risk factors and have nonzero risk premium by applying expression (4.24) to the basis assets {ri (t) : i ∈ H } themselves. This yields

 

Ht+Δ (1 + r0 (t))

Ft . E [ri (t) |Ft ] − r0 (t) = −cov ri (t), Ht

(4.25)

The last expression identifies which basis assets’ excess expected returns are equal to zero or not, i.e. a basis asset is a risk factor if and only if its conditional covariance with the state price density is non-zero. The Beta Model In general, the state price density’s time T value HT is not the value of a traded portfolio. If the state price density is a portfolio’s value, then we can obtain an alternative risk return relation, the standard beta model. We now derive this relation. First, if HT ∈ X , then there exists a finite set of basis assets ΦH ⊂ H such that the state price density’s time T value is the payoff to the traded portfolio {αHj (t) : j ∈ ΦH }, i.e. HT = j ∈ΦH αHj Vj (T ) with αHj = 0 all j ∈ ΦH . Using Lemma 15, the time t value of this traded portfolio is Ht =



j ∈ΦH αHj Vj (t).

Denote the state price density’s return over [t, t + Δ] as RH (t) = that expression (4.21) applied to HT ∈ X gives

  HT

Ft , Ht = E HT Ht which implies that Ht2 is a P martingale.

(4.26) Ht+Δ −Ht . We note Ht

4.8 Diversification

103

Theorem 39 (The Beta Model) Assume that all the quantities in expression (4.27) are squared integrable with respect to P. Then, cov [RH (t), RX (t) |Ft ] (E [RH (t) |Ft ] − r0 (t)) var [RH (t) |Ft ] (4.27) where E [RH (t) |Ft ] − r0 (t) < 0. E [RX (t) |Ft ] − r0 (t) =

Proof To simplify the notation we write E[·] = E[· |Ft ] and we drop the t arguments in RH (t), RX (t), r0 (t). 1 + RH = HHt+Δ . Using expression (4.24) for RH we have t   Ht+Δ , (1 + r ) E [1 + RH ] = 1 + r0 − cov HHt+Δ 0 . Or, Ht t   (1 + r0 ). This proves E [RH ] − r0 < 0. Next, E [RH ] = r0 − var HHt+Δ t E[RH ]−r0 H

= −(1 + r0 ). Using expression (4.24) for RX gives   . Or, E[RX ] = r0 − (1 + r0 )cov RX , HHt+Δ t var

t+Δ Ht

E[RX ] = r0 (t) +

E[RH ]−r0 cov [RX , RH ]. H

var

t+Δ Ht

This completes the proof.

If the state price density is traded, then because E [RH (t) |Ft ] − r0 (t) < 0, it can be considered as a risk factor. Furthermore, as shown in expression (4.27), the standard beta model implies that only this single risk factor’s return, RH (t), is sufficient to determine any primary securities expected return, i.e. its compensation for systematic risk. This is an important simplification when it applies. Last, we also know that this single risk factor consists of only a finite number of basis asset returns rj (t) for j ∈ ΦH . If the state price density trades, then the same finite collection of basis assets determine any traded asset’s expected return. And, this collection of basis assets forms the minimal set of risk factors needed to determine any security’s expected return. This is true despite the fact that an uncountable infinity of basis assets are needed to characterize a traded asset’s realized return, as distinct from its expected return. We point out that for many purposes, the realized return is the relevant object of investigation, e.g. synthetic construction of a derivative (hedging), and the entire set of basis assets is relevant, not just its expected return.

4.8 Diversification This section introduces the notion of diversification, i.e. “don’t put all your eggs in one basket.” The notion of diversification was historically introduced in finance via the use of the mean variance efficient frontier. In this context, a diversified portfolio is one that has minimum variance for a given expected return. In its most pure sense,

104

4 Basis Assets, Multiple-Factor Beta Models, and Systematic Risk

however, the notion of diversification is an application of the law of large numbers— the cancelling of uncorrelated risks—in a large portfolio. With respect to pricing, the intuition is that if a basis asset’s risk can be diversified away in a large portfolio, then this basis asset should not be priced, i.e. it’s expected excess return should be zero, and it is not a risk factor. We can show this intuition is true using a modest extension of our existing arbitrage-free market with the insights from Ross [165]. The model presented is the generalization of Ross’s Arbitrage Pricing Theory (APT) to our continuous-time and continuous trading market. The modest extension needed to our market is that the primary traded risky assets are a countably infinite set (Si (t))∞ i=1 ≥ 0. This extension is needed so that diversification can take place (in the limit). We consider an arbitrary time interval [t, t + Δ] (as above). All of the previous results apply unchanged to this extended market with the exception that instead of assuming NFLVR and ND, we need to directly assume the existence of a martingale measure (it is no longer implied by NFLVR and ND because the Third Fundamental Theorem 16 of asset pricing in Chap. 2 was only proven for a finite asset market). Recall that the existence of a martingale measure implies NFLVR and ND, even with trading in an infinite number of risky assets. We note that by Theorem 36, the risky asset returns {RSi (t)}∞ i=1 can be spanned by a countable infinite set of basis assets, denoted {rj (t) : j = 0, 1, . . . , ∞}. To investigate the importance of diversification on expected returns, we need to add an additional assumption that decomposes a risky asset’s return into “systematic risk” and “idiosyncratic risk,” idiosyncratic risk being that risk which can be diversified away. Assumption (Common Basis Assets) {rj (t) : j = 0, 1, . . . , ∞} can be written as {f1 (t), . . . , fK (t), u1 (t), u2 (t), .....} where for each risky asset expression (4.13) is written as RSi (t) = r0 (t) +

K

k=1 βik (t)[fk (t) − r0 (t)] + βiu (t)[ui (t) − r0 (t)]

(4.28)

where   cov ui (t), uj (t) |Ft = 0 f or i = j, var [βiu (t)ui (t) |Ft ] ≤ σt2 , and βiu (t) is Ft − measurable, β0k (t) = β0u (t) = 0 f or k = 1, . . . , K, for all i = 1, 2, . . . , ∞ and t = 0, . . . , T − Δ. This assumption states that the risky assets’ basis assets can be decomposed into two sets: (1) common basis assets {f1 (t), . . . , fK (t)} plus (2) an infinite number of idiosyncratic basis assets {u1 (t), u2 (t), . . . ..}. The K basis assets are common across all primary assets Si . The basis asset ui (t) is called idiosyncratic because it is uniquely identified with a particular risky asset Si . In addition, the covariances

4.8 Diversification

105

of the idiosyncratic basis assets across the primary traded assets are zero and their variances over [t, t + Δ] are assumed to be bounded. The condition that β0k (t) = β0u (t) = 0 for k = 1, . . . , K just identifies the zeroth asset as the discrete mma, i.e. RS0 (t) = r0 (t). Under this assumption, taking conditional expectations yields E[RSi (t) |Ft ] = r0 (t) + K k=1 βik (t) (E[fk (t) |Ft ] − r0 (t)) +βiu (t) (E[ui (t) |Ft ] − r0 (t)) .

(4.29)

Of course, the intuition is that idiosyncratic risk is diversifiable and it should not be priced, implying that the last term in the above expression should be approximately zero. In addition, the K common basis assets should correspond to the risk factors determining any traded asset’s systematic risk and expected return. One additional assumption gives us this result. We need to exclude arbitrage opportunities in the limit. Assumption (No Limiting Arbitrage Opportunities) nGivenn a sequence of portfolios of the primary assets {w0n , w1n , . . . , wnn }∞ i=0 wi = 1, which satisfies n=1 with lim var

 n

n i=0 wi RSi (t)

n→∞



=0

(4.30)

then lim E

n→∞

 n

n i=0 wi RSi (t)



= r0 (t).

(4.31)

This is true for all t = 0, . . . , T − Δ. This assumption states that there is no sequence of portfolios of the primary assets that approaches a limiting arbitrage opportunity in the sense of Definition 25 in Chap. 2. Indeed, this limiting portfolio has no variance, hence it is riskless. Then, it must earn the default-free spot rate in the limit; or, arbitrage in the limit is possible. It can be shown that under this assumption lim

n

n→∞

i=1 (βiu (t) (E[ui (t) |Ft

] − r0 (t)))2 < ∞

(4.32)

for t = 0, . . . , T − Δ. Proof The proof is based on Jarrow [82]. It is a proof by contradiction. For convenience we write E[· |Ft ] = E [·] and we omit the t argument. Assume that lim ni=1 (βiu (E[ui ] − r0 ))2 = ∞. n→∞ Without loss of generality, we let assets i = 1, . . . , K have returns RSi = fi , since we can always increase the traded risky assets to include these securities.

106

4 Basis Assets, Multiple-Factor Beta Models, and Systematic Risk

Let n > K + 1 and form the portfolio f or i = K + 1, . . . , n, win = n βiu (E[ui ]−r0 ) (βiu (E[ui ]−r0 ))2 i=K+1 wkn = − ni=K+1 βik win f or k = 1, . . . , K, and w0n = − ni=1 win + 1. Note by the definition of w0n , w0n + ni=1 win = 1. We show that this portfolio is a limiting arbitrage opportunity, yielding the contradiction. Now, using RSi = fi for i = 1, . . . , K, n

n n n 1 r0 + K wkn fk + ni=K+1 win RSi i=0 w i=1 wi + k=1 i RS i = −

n n n = r0 + K k=1 wk (fk − r0 ) + i=K+1 wi RSi − r0 . n Using the definition of wk for k = 1, . . . , K gives n



n n n = r0 − K k=1 i=K+1 βik wi (fk − r0 ) + i=K+1 wi RSi − r0 . Changing the order of  summation 

n n = r0 + ni=K+1 win − K k=1 βik (fk − r0 ) + i=K+1 wi RSi − r0 . Combining terms and using expression (4.28), we get the final result n n nR = r + n w 0 S i i=0 i i=K+1 wi βiu (ui − r0 ). First, taking the variance gives  n n 2  n n var [βiu (ui − r0 )]. var i=0 wi RSi = i=K+1 wi 2 But, var(βiu ui ) ≤ σt for all i, thus  n  n 2 n n 2 0 ≤ var i=0 wi RSi ≤ σt i=K+1 wi 2 1 2  → 0 = σt2 ni=K+1  n [βiu (E[ui ]−r0 )] 2 = σt  n 2 i=K+1 (βiu (E[ui ]−r0 ))

2

i=K+1 (βiu (E[ui ]−r0 ))

as n → ∞. taking expectations yields  n nSecond, nR n = r w + E 0 S i i=0 i i=K+1 wi βiu (E [ui ] − r0 ). n , we have that Substituting  in the definition of wi  n n βiu (E[ui ]−r0 ) βiu (E [ui ] − r0 ) = 1. i=K+1 2  i ]−r0 ))  ni=K+1n (βiu (E[u Hence, E i=0 wi RSi = r0 + 1 > r0 for all n. This shows that the portfolio is a limiting arbitrage opportunity, which completes the proof. Alternatively stated, this result implies that for any  > 0, except for a finite number of the risky assets (i = 1, · · · , ∞), |βiu (t) (E[ui (t) |Ft ] − r0 (t))| < . Furthermore, if there exists a constant β > 0 such that βiu (t) ≥ β for all i, then the expected excess returns on the idiosyncratic basis assets are approximately zero for all but a finite number of the risky assets (i = 1, · · · , ∞), i.e. E[ui (t) |Ft ] − r0 (t) ≈ 0.

4.9 Notes

107

Using expression (4.25), since ui is a basis asset for each i, this implies that

  Ht+Δ

Ft ≈ 0. cov ui (t), Ht One can view Ross’s APT as providing a set of sufficient conditions on an asset’s return process under which the previous expression is valid, i.e. idiosyncratic risk is not priced. Equivalently, Ross’s APT shows that the K basis assets f1 , . . . , fK are the risk factors that determine any traded security’s expected compensation for systematic risk, i.e. E[RSi (t) |Ft ]) ≈ r0 (t) +

K

k=1 βik (t) (E[fk (t) |Ft

] − r0 (t)) .

(4.33)

4.9 Notes Testing of the general multi-factor model presented in this chapter can be found in Zhu et al. [186]. An important special case of the multi-factor model, Ross’s APT, has already seen significant empirical testing, see Jagannathan et al. [79] for a review. Recommended books for understanding the empirical methodology for testing asset pricing models include Campbell, et al. [25], Cochrane [33], and Ruppert [168].

Chapter 5

The Black Scholes Merton Model

This chapter presents the seminal Black-Scholes-Merton (BSM) model for pricing options. Since this chapter is a special case of the material contained in Sect. 2.8 in Chap. 2, the presentation will be brief. In addition, as an application of the BSM model, Merton’s structural model [146] for credit risk is included herein. We are given a normalized market (S, F, P) where there is only one risky asset trading. The money market account (mma) and risky asset’s evolutions are Bt = 1,

(5.1)

dSt = St ((b − r)dt + σ dWt ), or 1

St = S0 e(b−r)t− 2 σ

2 t+σ W t

(5.2)

for all t ∈ [0, T ] where b, r, σ are strictly positive constants and Wt is a standard Brownian motion with W0 = 0 that generates the filtration F. This evolution is called geometric Brownian motion. Here, the constant r corresponds to the defaultfree spot rate of interest, see the non-normalized market in Sect. 5.2 below. We note that this is a subcase of the evolution given in expression (2.29) in Chap. 2. Hence, we can apply the theorems from Sect. 2.8 in Chap. 2 to prove all the desired results.

5.1 NFLVR, Complete Markets, and ND First, we want to show that the risky asset price process in expression (5.2) satisfies NFLVR. In this regards, define θ=

(b − r) σ

© Springer Nature Switzerland AG 2021 R. A. Jarrow, Continuous-Time Asset Pricing Theory, Springer Finance, https://doi.org/10.1007/978-3-030-74410-6_5

(5.3) 109

110

5 The Black Scholes Merton Model

and the equivalent probability measure Qθ by 1 2 dQθ = e−θ·WT − 2 θ T > 0. dP

Note that T

θ 2 dt = θ 2 T < ∞ and  T     1 T 1 2 2 E e− 0 θdWt − 2 0 θt  dt = E e−θWT − 2 θ T = 1. 0

The second condition follows from the characteristic function of the normally distributed random variable WT . By Theorem 19 in Chap. 2, the evolution satisfies NFLVR. By the First Fundamental Theorem 13 of asset pricing in Chap. 2, this implies that Qθ ∈ Ml where Ml = {Q ∼ P : S is a Q local martingale}, i.e. it is an equivalent local martingale measure. For subsequent use, we note that the evolution of the risky asset price under Qθ is given by dSt = St σ dWtθ 1

St = S0 e− 2 σ

or (5.4)

2 t+σ W θ t

where dW θ (t) = dW (t) + θ (t)dt is a Brownian motion under Qθ . Taking natural logarithms, we get that 1 log(St ) = log(S0 ) − σ 2 t + σ Wtθ . 2

(5.5)

In this form, it is easy to see that ! Q

θ

log(St /S0 ) + 12 σ 2 t ≤x √ σ t

"

 =Q

θ

Wtθ √ ≤x t



 =

x −∞

1 2 1 √ e− 2 z dz 2π (5.6)

where the right side of this expression is the standard cumulative normal distribution function, denoted N(x). Second, we want to show that the market is complete with respect to Qθ ∈ Ml . Since σ > 0 is a scalar, a direct application of Theorem 22 in Chap. 2 gives this result. This implies by the Second Fundamental Theorem 14 of asset pricing in Chap. 2 that Qθ ∈ Ml is unique.

5.2 The BSM Call Option Formula

111

Third, we want to show that the market satisfies ND. Again, a trivial application of Theorem 23 in Chap. 2 gives the result since 1

e2σ

2T

0, a constant, i.e. Bt = ert

(5.7)

and, the risky asset’s price is 1

St = S0 ebt− 2 σ

2 t+σ W t

(5.8)

for all t ∈ [0, T ]. A European call option on the risky asset with maturity T and strike price K is a financial contract that gives the owner the right, but not the obligation, to purchase the underlying asset at time T for K dollars. To maximize the payoff of the call option on the maturity date, the owner should exercise the option and buy the asset if and only if the time T asset price ST exceeds the strike price K. The call’s time T payoff is therefore ZT = max[ST − K, 0] ∈ L0+ .

(5.9)

112

5 The Black Scholes Merton Model

Normalizing, we get   ZT max[ST − K, 0] = = max ST − Ke−rT , 0 . BT BT

ZT =

  θ θ  θ Since E Q [ZT ] = E Q max ST − Ke−rT , 0 ≤ E Q [ST ] = S0 < ∞, we have ZT ∈ L1+ (Qθ ). Using the risk neutral valuation Theorem 18 in Chap. 2, the option’s time t value is     θ θ

Xt = E Q [ZT |Ft ] = E Q max ST − Ke−rT , 0 Ft . By substitution    

ST Xt θ = E Q max − Ke−rT , 0

Ft . Bt BT Under the martingale measure Qθ , the evolution of the risky asset’s price is 1

St = S0 e(r− 2 σ

2 )t+σ W θ t

(5.10)

.

Algebra yields

 1 2 θ θ

max[St e− 2 σ (T −t)+σ (WT −Wt ) − Ke−r(T −t) , 0] Ft   √ 1 2 = E z max[St e− 2 σ (T −t)+σ T −t·z − Ke−r(T −t) , 0]

Xt = E Q

θ



where the random variable z has the standard cumulative normal distribution N(z). Computing this expectation gives the Black-Scholes-Merton (BSM) formula for a European call option Xt = St N(d1 ) − Ke−r(T −t) N(d2 )

(5.11)

where d1 =

log(St /K) + (r + 12 σ 2 (T − t)) √ σ T −t

and

√ d2 = d1 − σ T − t.   √ 1 2 Proof Xt = E z max[St e− 2 σ (T −t)+σ T −t·z − Ke−r(T −t) , 0]

  √ √ 1 2 1 2

= E z St e− 2 σ (T −t)+σ T −t·z − Ke−r(T −t) St e− 2 σ (T −t)+σ T −t·z ≥ Ke−r(T −t) .

5.2 The BSM Call Option Formula 1

Note that St e− 2 σ

2 (T −t)+σ

√

T −t·z ≥ Ke−r(T −t)  log SKt +r(T −t)− 12 σ 2 (T −t) √ − = −d2 . σ T −t 

z≥

113

if and only if

(first term in expectation)

  √ √ 1 2 1 2

E z St e− 2 σ (T −t)+σ T −t·z St e− 2 σ (T −t)+σ T −t·z ≥ Ke−r(T −t)

  √ 1 2

= E z St e− 2 σ (T −t)+σ T −t·z z ≥ −d2  − 1 z2 √ ∞  1 2 = St −d2 e− 2 σ (T −t)+σ T −t·z e√ 2 dz 2π  ∞  − 1 z−σ √T −t 2  1 √ dz = St −d2 e 2 2π √ A change of variable y = z − σ T − t yields  1 2 ∞ = St −d2 −σ √T −t e− 2 y √1 dy 2π   √ = St P rob y ≥ −d2 − σ T − t = St P rob [y ≥ −d1 ] = St P rob [y ≤ d1 ]. (second term in expectation)   √

1 2 E z Ke−r(T −t) St e− 2 σ (T −t)+σ T −t·z ≥ Ke−r(T −t) = Ke−r(T −t) E z [ 1| z ≥ −d2 ] = Ke−r(T −t) P rob [z ≥ −d2 ] = Ke−r(T −t) P rob [z ≤ d2 ]. Substitution completes the proof. Remark 48 (BSM Formula Independence of the Asset’s Expected Return) As seen in expression (5.11), the BSM formula does not depend on the stock’s expected return

  dSt

Ft = b · dt. E St This independence makes the BSM formula usable in practice because estimating the stock’s expected return per unit time, b, or equivalently estimating the stock’s risk premium θ is a very difficult exercise (see Jarrow and Chatterjea [97], Chapters 19 and 20 for a discussion of these statements). This completes the remark.

114

5 The Black Scholes Merton Model

5.3 The Synthetic Call Option For hedging risk, one needs to construct the synthetic call option. The synthetic call is obtained from the admissible s.f.t.s. that generates the call option’s time T payoff. This section shows how to construct the synthetic call option in the BSM market. Since the market is complete with respect to Qθ , by the definition of a complete market, we know there exists an initial investment  t x > 0 and an admissible s.f.t.s. (α0 , α) ∈ A (x) with value process Xt = x + 0 αs · dSs = α0 (t) + αt St , a Qθ martingale, such that  x+

T

αt · dSt = max[ST − Ke−rT , 0].

(5.12)

0

We need to determine both x and (α0 (t), αt ). First, taking expectations under Qθ we get that the initial investment is x = X0 = E Q

θ

  max[ST − Ke−rT , 0] .

(5.13)

This is the BSM formula’s value at time 0. This observation makes sense since the initial cost of the synthetic call in an arbitrage-free market should be the arbitragefree value of the call. Second, we know from the self-financing condition that Xt = α0 (t) + αt St .

(5.14)

Hence, once we know αt , then we know α0 (t) by solving this last expression. Finally, to get αt , we apply Ito’s formula to the BSM formula (5.11). This gives

x+

 T  ∂Xt 0

∂t

∂Xt ∂Xt 1 ∂ 2 Xt ∂t dt + 2 ∂St2 dSt , dSt  + ∂St dSt   T 1 ∂ 2 Xt 2 2 t dt + 0 ∂X 2 ∂St2 St σ ∂St dSt = max[ST

dXt = +

or − Ke−rT , 0] (5.15)

where the last expression uses the fact that dSt , dSt  = St2 σ 2 dt. Identifying the integrands of dSt across equations (5.12) and (5.15) gives the final result αt =

∂Xt ∂Xt = = N(d1 ). ∂St ∂St

(5.16)

5.4 Original Derivation of the BSM Formula

115 

Proof The second equality follows because 

∂

Xt Bt ∂St





=

   St t ∂ X Bt ∂ Bt   ∂St . ∂ St

∂

Xt Bt ∂St





=

∂Xt 1 ∂St Bt

,

∂

St Bt ∂St



=

1 Bt ,

and

Bt

To obtain the third equality, note that ∂Xt ∂St

But,

−r(T −t) ∂N (d2 ) ∂d2 . 1 = N (d1 ) + St ∂N∂d(d11 ) ∂d ∂St − Ke ∂d2 ∂St ∂d1 ∂St

∂N (d2 ) ∂d2

= = =

=

∂d2 ∂St

and

√ 1 1 2 2 √1 e− 2 (d2 ) = √1 e− 2 (d1 −σ T −t) 2π 2π √ − 12 (d1 )2 +d1 σ T −t− 12 σ 2 (T −t)

=

√1 e 2π √ ∂N (d1 ) d1 σ T −t− 12 σ 2 (T −t) ∂d1 e ∂N (d1 ) St ∂d1 Ke−r(T −t) .

Substitution yields ∂Xt ∂St

∂N (d1 ) ∂d1 1 = N (d1 ) + St ∂N∂d(d11 ) ∂d ∂St − St ∂d1 ∂St = N(d1 ).

This completes the proof. The number of units held in the risky asset αt to create a synthetic call option is called the option’s delta. We note that since there is only one risky asset trading in this market, the market contains no redundant assets (see Sect. 2.4 in Chap. 2), hence the admissible s.f.t.s. attaining the call’s payoff is unique.

5.4 Original Derivation of the BSM Formula It is useful from a historical and pedagogical perspective to study the original derivation of the BSM model as contained in Merton [145]. Given is a frictionless and competitive, non-normalized market ((B, S), F, P). Consider a European call option with payoff denoted CT = max[ST − K, 0] at time T . Let Ct denote its time t value. We start by assuming that Ct = C(t, St ) is a deterministic function of time t and the underlying risky asset’s price St that is continuously differentiable in t and twice continuously differentiable in St . Next, standing at time t, we want to construct a portfolio of the option and underlying risky asset that will riskless over [t, t + dt], and then compare its return to the return on the mma over [t, t + dt], which is also riskless. No arbitrage must then imply that the two returns are equal, and from this, we can obtain an implicit formula for C(t, St ).

116

5 The Black Scholes Merton Model

Let us begin. Consider the portfolio with value process Xt = C(t, St ) + nt St where nt is the number of shares in the risky asset at time t. Using Ito’s formula (see Chap. 1), the change in the value of the portfolio over [t, t + dt] is dXt = dC(t, St ) + nt dSt =

∂C(t, St ) ∂C(t, St ) 1 ∂ 2 C(t, St ) 2 dt + dSt + σ dt + nt dSt . ∂t ∂St 2 ∂S2t

(5.17)

t) Choosing nt = − ∂C(t,S yields a riskless portfolio over [t, t + dt] because the dSt ∂St terms disappear. This implies, to avoid arbitrage, that dXt = rXt dt, i.e.

  ∂C(t, St ) 1 ∂ 2 C(t, St ) 2 ∂C(t, St ) + St = σ . r C(t, St ) − ∂St ∂t 2 ∂S2t This is a partial differential equation subject to the boundary condition C(T , ST ) = max[ST − K, 0], whose solution is the call option’s time t price C(t, St ), and is given by the BSM formula. This completes the original derivation. The economic reasoning underlying this argument is compelling, but the argument is imprecise from a mathematical perspective in three ways. 1. How does this hedged portfolio change from [t, t + dt] to [t + dt, t + 2dt] to stay riskless, with no cash inflows or outflows? 2. What do we precisely mean by no arbitrage? 3. How do we know the call value can be written as the function C(t, St ) with the required smoothness properties? The quest for the answers to these questions was the genesis of the field of mathematical finance whose answers are the content of Chap. 2.

5.5 Merton’s Structural Model As a direct application of the Black-Scholes-Merton model we can get Merton’s structural model for pricing and hedging risky debt. Given is a frictionless and competitive, non-normalized market ((B, S), F, P) where S now represents the traded assets of a firm with the time t balance sheet given in the following Table. Assets St

Liabilities Zero-coupon bond (maturity T , face value K) Dt Equity Et

5.5 Merton’s Structural Model

117

A useful accounting identity is St = Dt + Et which holds for all t ∈ [0, T ]. The equity holders own the firm. The liabilities consist of a single zero-coupon bond with maturity T and face value K. The debt is risky because at its maturity date, if the value of the firm’s assets is not large enough, the firm may not be able to pay back the face value of the debt K. Rational equity holders will decide to repay the debt at maturity if the firm’s assets are worth more than what is owed, i.e. ST ≥ K. Hence, the time T value of the equity and debt are given by the following expressions: ET = max [ST − K, 0] ,

and

DT = min [K, ST ] = K − max [K − ST , 0] . These payoffs are called the absolute priority conditions, determined by the debt’s covenants and the limited liability of equity. We see that equity is equal to a European call option on the firm’s assets with maturity T and strike K. And, the debt’s value is equal to the payoff to a default-free zero-coupon bond with face value K and short a European put option on the firm’s assets with maturity T and strike K. Given the BSM model from the previous sections, we can value these securities using risk neutral valuation. The value of the equity and debt are given by Et = B t E Q

θ



 max

 

ST − Ke−rT , 0

Ft = St N(d1 ) − Ke−r(T −t) N(d2 ) BT

and Dt = Bt E Q

θ



  

ST max Ke−rT − , 0

Ft = Ke−r(T −t) N(d2 ) + St N(−d1 ) BT

where d1 =

log(St /K) + (r + 12 σ 2 (T − t)) √ σ T −t

and

√ d2 = d1 − σ T − t. Merton’s structural model has four limitations when applied in practice. 1. 2. 3. 4.

Interest rates are assumed constant. The firm’s liability structure is too simple. In default, the bankruptcy courts do not follow absolute priority. The assets of the firm do not trade.

118

5 The Black Scholes Merton Model

Limitations 1 and 2 are easily handled by generalizing the model. Limitation 3 requires a different payoff structure for the debt and equity, but this also can be handled by generalizing the model. Limitation 4, however, is more problematic. Limitation 4, that the firm’s assets don’t trade, makes the synthetic construction of the debt’s payoff impossible, implying that the risk neutral valuation methodology not longer applies. Correcting this limitation is the motivation for the reduced form model for the pricing and hedging of credit risk discussed in Chap. 7.

5.6 Notes There is a vast literature on the BSM formula and its application in practice. An introductory guide to derivative pricing is Jarrow and Chatterjea [97]. Good reference books include the classics by Cox and Rubinstein [35] and Jarrow and Rudd [109]. More recent textbooks include Back [4], Baxter and Rennie [10], Jarrow and Turnbull [112], Musiela and Rutkowski [152], and Shreve [176, 177].

Chapter 6

The Heath Jarrow Morton Model

This chapter presents the Heath, Jarrow et al. [72] model for pricing interest rate derivatives. Given frictionless and competitive markets, and assuming a complete market, this is the most general arbitrage-free pricing model possible with a stochastic term structure of interest rates. This model, with appropriate modifications, can also be used to price derivatives whose values depend on a term structure of underlying assets, examples include exotic equity derivatives where the underlyings are call and put options, commodity options where the underlyings are futures prices, and credit derivatives where the underlyings are risky zero-coupon bond prices, see Carr and Jarrow [29], Carmona [26], Carmona and Nadtochiy [27], and Kallsen and Kruhner [120]. HJM generalizes the BSM model in two important ways. One, the BSM model assumes deterministic interest rates. Two, the BSM has only one traded risky asset. The HJM model has stochastic interest rates and a continuum of traded risky assets. In some sense, the issue to be solved is that the market contains a continuum of redundant risky assets, more than are needed to complete the market. This chapter is based on Jarrow [87].

6.1 The Set-Up Given is a non-normalized market ((B, S), F, P). The money market account (mma) t Bt = e 0 rs ds , B0 = 1, is continuous in time t,and of finite variation. rt is the defaultt free spot rate of interest, adapted to Ft with 0 |rs |ds < ∞. Here we modify the risky asset price vector. Instead of a finite number of risky assets St = (S1 (t), . . . , Sn (t)) , we let the traded risky assets correspond to the term structure of default-free zero-coupon bonds of all maturities, with time t prices p(t, T ) for a sure dollar paid at time T for all 0 ≤ t ≤ T ≤ T . That is, p(t, t) = 1 for all t ≥ 0. The zero-coupon price process p(t, T ) is adapted to F for all T . We © Springer Nature Switzerland AG 2021 R. A. Jarrow, Continuous-Time Asset Pricing Theory, Springer Finance, https://doi.org/10.1007/978-3-030-74410-6_6

119

120

6 The Heath Jarrow Morton Model

also assume that zero-coupon bond prices are strictly positive, i.e. p(t, T ) > 0 for all t, T . This last assumption excludes trivial arbitrage opportunities from the market. Note that there are a continuum of risky assets trading in this market. The forward rate is implicitly defined by the expression p(t, T ) = e−

T t

f (t,u)du

.

(6.1)

Taking natural logarithms and differentiating with respect to T (assuming the derivative exists) yields the expression f (t, T ) = −

∂p(t, T ) 1 ∂ log p(t, T ) =− . ∂T ∂T p(t, T )

(6.2)

The forward rate at time t corresponds to default-free borrowing and lending rate implicit in the zero-coupon bonds over the future time period [T , T + dt]. To see this note that for a small time interval Δ, f (t, T )Δ ≈ −

p(t, T + Δ) − p(t, T ) Δ p(t, T + Δ) = − 1. Δ p(t, T ) p(t, T )

This expression shows that the forward rate at time t is the implicit rate earned on p(t, T + Δ) in addition to that paid by p(t, T ) over the last time period in its life [T , T + Δ]. The default-free spot rate is rt = f (t, t). This is the rate for immediate borrowing or lending over the time interval [t, t + dt]. Remark 49 (Self-Financing Trading Strategies) Although there are a continuum of traded risky assets, we assume that an admissible s.f.t.s. in an HJM model consists of the mma and only a finite number of zero-coupon bonds over the trading horizon [0, T ]. This enables us to use the fact that the existence of an equivalent martingale measure implies both NFLVR and ND, even with trading in an infinite number of risky assets. The existence of an equivalent martingale measure is no longer implied by NFLVR and ND because the Third Fundamental Theorem 16 of asset pricing was only proven for a finite asset market, see Chap. 2 for further elaboration. This completes the remark.

6.2 Term Structure Evolution The HJM model starts by assuming an evolution for the default-free term structure of interest rates. Although unnecessary, for simplicity of computation, we select the forward rate curve to represent the term structure of interest rates. Alternative possibilities are the yield curve or the zero-coupon bond price curve.

6.2 Term Structure Evolution

121

In particular, the HJM model assumes a given initial forward rate curve f (0, T ) for 0 ≤ T ≤ T T where 0 |f (0, T )| dT < ∞, and a stochastic process for the evolution of the forward rate curve  t D  t  f (t, T ) = f (0, T )+ μ(s, T )ds+ σi (s, T )dWi (s) for 0 ≤ t ≤ T ≤ T 0

i=1

0

(6.3) where Wi (t) for i = 1, . . . , D are standard independent Brownian motions with Wi (0) = 0 for all i that generate the filtration F, μ(t, T ) and σi (t, T ) for i = 1, . . . , D are Ft -measurable and satisfy the regularity conditions a.s. P 

T



0

t

 |μ(v, t)| dv dt < ∞,

0

 t 

2

t

dv < ∞

σi (v, y)dy 0

v

 t !

"2

T

σi (v, y)dy 0

dv < ∞

t

for all t ∈ [0, T ], T ∈ [0, T ], i = 1, · · · , D. In stochastic differential equation form this is df (t, T ) = μ(t, T )dt +

D 

σi (t, T )dWi (t).

(6.4)

i=1

This is called a D-factor model for the forward rate evolution. The D-factors represent distinct economic forces affecting forward rates (e.g. inflation, unemployment, economic growth, monetary policy, fiscal policy). Expression (6.3) represents a very general stochastic process. The only substantive economic restriction in expression (6.3) is that the process has continuous sample paths, however, even this restriction can be relaxed (see Jarrow and Madan [98, 99], Bjork, et al. [15], Bjork et al. [16], and Eberlein and Raible [55]). We do not pursue this relaxation here. The evolution need not be Markov (in a finite number of state variables) and it can be path dependent. In addition, given that the market does not have currency trading (only zero-coupon bonds and a mma), forward rates can be negative. To exclude negative forward rates, an additional restriction can be imposed on the evolution in expression (6.3). We do not impose such a restriction here.

122

6 The Heath Jarrow Morton Model

Under this evolution, the spot rate process is 

t

rt = r0 +

∂f (0, s) ds + ∂T

0



t

μ(s, t)ds +

D  

0

i=1

t

σi (s, t)dWi (s).

(6.5)

0

In stochastic differential equation form this is  drt =

 D  ∂f (0, t) + μ(t, t) dt + σi (t, t)dWi (t). ∂T

(6.6)

i=1

Proof Using thedefinition rt = f (t, t),  t expression (6.3) gives t rt = f (0, t) + 0 μ(s, t)ds + D i=1 0 σi (s, t)dWi (s). But, f (0, t) − f (0, 0) =  t ∂f (0,s) 0 ∂T ds. Substitution, along with the fact that r0 = f (0, 0), completes the proof. Note that in this expression as time t changes, the second argument in the last two integrands also change. This makes the evolution of the spot rate more complex than that of the forward rate process itself. From expression (6.3), we can deduce the evolution of the zero-coupon bond price curve p(t, T ) = p(0, T )e

t

0 (rs +b(s,T

))ds− 12

D  t i=1 0

t ai (s,T )2 ds+ D i=1 0 ai (s,T )dWi (s)

(6.7)  b(s, T ) = −

where

1 μ(s, u)du + ai (s, T )2 and 2

T

D

s



i=1

T

ai (s, T ) = −

σi (s, u)du for all i = 1, . . . , D.

s

Proof First, p(t, T ) = e− − ln p(t, T ) =

 T t

T t

f (t,u)du

f (0, u)du +

. Substitution of expression (6.3) gives

 T  t t

0

μ(s, u)dsdu +

D  T  t  i=1 t

0

σi (s, u)dWi (s)du.

A stochastic Fubini’s theorem (see Heath et al. [72, appendix]) gives   t

T



T

t

μ(s, u)dsdu =

0

t t 0



σi (s, u)dWi (s)du =

 t

T

μ(s, u)duds, and 0

t

 t !

T

" σi (s, u)du dWi (s).

0

t

6.2 Term Structure Evolution

123

Substitution yields 

T

− ln p(t, T ) =

f (0, u)du +

 t ! 0

t

T

" μ(s, u)du ds +

t

D  t  i=1

0

T

σi (s, u)dudWi (s).

t

Rewriting yields  t  t  T f (0, u)du − 0 f (0, u)du + 0 s μ(s, u)du ds   t  t t T − 0 s μ(s, u)du ds + D i=1 0 s σi (s, u)dudWi (s) D  t  t − i=1 0 s σi (s, u)dudWi (s).

− ln p(t, T ) =

T 0

But,  t  0

 μ(s, u)ds du =

0

 t 0

u

u

 t  0

σi (s, u)dWi (s)du =

0

T 0

and

t

σi (s, u)dudWi (s). s

f (0, u)du and using expression (6.5) yields

− ln p(t, T ) = − ln p(0, T ) +

 t !

D  t  i=1

0

T

"

T

μ(s, u)du ds 0

+

 μ(s, u)du ds

s

 t 0

Noting that − ln p(0, T ) =

t

s

 σi (s, u)dudWi (s) −

t

ru du. 0

s

Simple algebra completes the proof. As a stochastic differential, we have that the zero-coupon bond price evolves as  dp(t, T ) = (rt + b(t, T )) dt + ai (t, T )dWi (t). p(t, T ) D

(6.8)

i=1

The zero-coupon bond’s instantaneous return consists of a drift plus D-random shocks. Within the drift, one can interpret b(s, T ) as the risk premium on the T maturity zero-coupon bond in excess of the default-free spot rate of interest rt . The bond’s volatilities are {a1 (t, T ), . . . , aD (t, T )}. We add the following assumption.

124

6 The Heath Jarrow Morton Model

Assumption (Nonsingular Volatility Matrix) For any T1 , . . . , TD ∈ [0, T ] such that 0 < T1 < . . . < TD ≤ T , ⎤ a1 (t, T1 ) · · · aD (t, T1 ) ⎥ ⎢ .. .. ⎦ ⎣ . . ⎡

a1 (t, TD ) · · · aD (t, TD ) is nonsingular a.s. P × Λ where Λ is Lebesgue measure. Later, given the existence of a probability measure Q ∈ M where   p(t, T ) M= Q∼P: for all T ∈ [0, T ] are Q martingales , Bt this assumption will guarantee that the market is complete with respect to Q. In fact, since this condition holds for all D subsets of the traded zero-coupon bonds, and there is an infinite collection of such D subsets, the market has a continuum of redundant assets.

6.3 Arbitrage-Free Conditions This section provides the conditions that must be imposed on the evolution of the forward rate process such that there exists an equivalent martingale measure. Theorem 40 (HJM Arbitrage-Free   Drift Condition) An equivalent martingale p(t,T ) measure Q exists such that are Q martingales for all 0 ≤ T ≤ T Bt if and only if there exist risk premium processes φi (t) f or i = 1, . . . , D Ft -measurable with

T 0

φi (t)2 dt < ∞ for all i such that

T D  T 1 D dQ 2 = e i=1 0 φi (s)dWi (s)− 2 i=1 0 φi (s) ds dP

is a probability measure,  D  T  T 1 D 2 E e i=1 0 (ai (s,T )+φi (s))dWi (s)− 2 i=1 0 (ai (s,T )+φi (s)) ds = 1,

6.3 Arbitrage-Free Conditions

125

and μ(t, T ) = −

D 

&



'

T

σi (t, T ) φi (t) −

(6.9)

σi (t, v)dv t

i=1

for all 0 ≤ t ≤ T ≤ T . Proof All equivalent martingale measures have the form T n  T 1 n 2 dQ i=1 0 φi (s)dWi (s)− 2 i=1 0 φi (s) ds for suitably measurable and inte= e dP grable {φ1 (t), . . . , φD (t)}, see Theorem 9 in Chap. 1. By Girsanov’s theorem (see Theorem 5 in Chap. 1), dWiQ (t) = dWi (t) − φi (t)dt for i = 1, . . . , D are standard independent Brownian motions under Q. Given expression (6.8), we have that  dp(t, T ) = (rt + b(t, T )) dt + ai (t, T )dWi (t) p(t, T ) D

i=1

= (rt + b(t, T )) dt +

D 

  ai (t, T ) φi (t)dt + dWiQ (t)

i=1

= rt dt +

D 

ai (t, T

!

)dWiQ (t) +

b(t, T ) +

i=1

D 

" φi (t)ai (t, T ) dt.

i=1

Using the change of numeraire, expression (2.6) in Chap. 2, we have that   ) d p(t,T B(t) dp(t, T ) − rt dt = p(t,T ) , p(t, T ) B(t)

which implies that  d

p(t,T ) B(t)

p(t,T ) B(t)

Thus,

p(t,T ) B(t)

 =

D 

! ai (t, T

)dWiQ (t) +

b(t, T ) +

i=1

D 

" φi (t)ai (t, T ) dt.

i=1

is a local Q martingale if and only if b(t, T ) +

D 

φi (t)ai (t, T ) = 0. Or,

i=1



T

− s

1 μ(s, u)du + 2 D

i=1

!

T

"2 σi (s, u)du

s

−

D  i=1

 φi (t) s

T

σi (s, u)du = 0.

126

6 The Heath Jarrow Morton Model

Differentiate in T , and simplify to obtain the equivalent expression μ(s, T ) = −

&

D 

!

"'

T

σi (s, T ) φi (t) −

σi (s, u)du

.

s

i=1

) To complete the proof, we need to show that p(t,T B(t) is a Q martingale, and not just a local This follows from   Dmartingale. T T 1 D 2 (a (s,T )+φ i i (s))dWi (s)− 2 i=1 0 (ai (s,T )+φi (s)) ds = 1, which is both a E e i=1 0 necessary and sufficient condition. Indeed, expression (6.7) implies that under the probability measure Q, p(T ,T ) B(T )

= p(0, T )e

T 0

T 1 D 2 (b(s,T )+ D i=1 φi (s)ai (s,T ))ds− 2 i=1 0 ai (s,T ) ds D  T + i=1 0 ai (s,T )dWiQ (s)

×e

.

Using the Q local martingale condition that b(t, T ) + all t, this simplifies to T 1 D p(T , T ) = p(0, T )e− 2 i=1 0 B(T ) p(t,T ) B(t)

D

i=1 φi (t)ai (t, T

) = 0 for

T Q ai (s,T )2 ds+ D i=1 0 ai (s,T )dWi (s)

.

is a Q martingale if and only if  1 D  T E Q e− 2 i=1 0

 T Q ai (s,T )2 ds+ D i=1 0 ai (s,T )dWi (s)

see Protter [158, p. 138]. Using  1 D  T E e− 2 i=1 0

dQ dP ,

= 1,

this is equivalent to

 T T D  T Q 1 D 2 ai (s,T )2 ds+ D i=1 0 ai (s,T )dWi (s) i=1 0 φi (s)dWi (s)− 2 i=1 0 φi (s) ds e

= 1.

Combining terms, and substituting dWiQ (t) = dWi (t) − φi (t)dt, yields  1 D  T  D  T 2 2 E e− 2 i=1 0 (ai (s,T ) +2ai (s,T )φi (s)+φi (s) )ds+ i=1 0 (ai (s,T )+φi (s))dWi (s) = 1. Algebra completes the proof. We assume the existence of an equivalent martingale measure Q ∈ M for the subsequent analysis. This implies, of course, by the Third Fundamental Theorem 16 of asset pricing in Chap. 2 that the market satisfies NFLVR and ND. Expression (6.9) is known as the arbitrage-free HJM drift restriction. In the HJM drift restriction, (φ1 (t), . . . , φD (t)) are interest rate risk premium corresponding to the D-Brownian motions. It is important to emphasize that a characterization of these risk premium

6.3 Arbitrage-Free Conditions

127

is not determined by the arbitrage-free restrictions imposed by NFLVR and ND on the evolution of the term structure of interest rates, the theorem only assures their existence. Theorem 41 (Market Completeness) Given an equivalent martingale measure Q ∈ M exists and the non-singular volatility matrix assumption, the market is complete with respect any Q ∈ M and M is the singleton set, i.e. Q is unique. As stated in this theorem, given the existence of an equivalent martingale measure Q ∈ M, the non-singularity assumption for the volatility matrix of an arbitrary collection of zero-coupon bonds with maturities T1 , . . . , TD satisfying 0 < T1 < . . . < TD ≤ T , the market is complete. The proof follows by applying Theorem 22 in Chap. 2 for the restricted Brownian motion market consisting of just these D zero-coupon bonds and the mma proving that this restricted market is complete with respect to Q ∈ M. Of course, this also implies that the larger market consisting of all the traded zero-coupon bonds is complete with respect to Q ∈ M. Finally, by the Second Fundamental Theorem 14 of asset pricing in Chap. 2, we have that Q ∈ M is unique, completing the proof. These results imply, of course, that the standard hedging and valuation methodology in Chap. 2 can now be applied to value and hedge interest rate derivatives. Remark 50 (The Bond Market and Redundant Assets) As just discussed, given Q ∈ M, the non-singularity assumption for the volatility matrix of an arbitrary collection of zero-coupon bonds with maturities T1 , . . . , TD satisfying 0 < T1 < . . . < TD ≤ T implies that the market is complete with respect to Q ∈ M for any such subset. This implies that the bond market has redundant assets as defined in Sect. 2.4 of Chap. 2. Indeed, any collection of zero-coupon bonds with maturities T1 , . . . , TD satisfying 0 < T1 < . . . < TD ≤ T completes the market with respect to Q ∈ M. This means, of course, that the admissible s.f.t.s. that attains a derivative’s payoffs such that the value process is a Q martingale will not be unique. The non-uniqueness is due to the ability to use different redundant assets to do the synthetic construction. This is an advantage when using the HJM model in practice because it offers multiple ways to hedge derivatives using different maturity bonds. The difficulty in proving Theorem 40 was to guarantee that the market is “arbitrage-free” given there are redundant bonds trading because with redundant assets, more arbitrage opportunities are possible. The key insight of the theorem is that the market is “arbitrage-free” if and only if the risk premiums are independent of the collection of zero-coupon bonds maturing at T1 , . . . , TD selected to complete the market, i.e. φi (t) for i = 1, · · · , D are independent of T1 , . . . , TD . This independence ensures that the redundant assets do not introduce additional FLVR trading strategies. This completes the remark. The next two corollaries formalize the implications of the previous insights for valuing zero-coupon bonds and interest rate derivatives.

128

6 The Heath Jarrow Morton Model

Corollary 6 (Zero-Coupon Bond Formula) Given an equivalent martingale measure Q ∈ M exists,  T p(t, T ) = E Q e− t



Ft

rs ds

(6.10)

for all 0 ≤ t ≤ T ≤ T .   ) being Q martingales for all 0 ≤ T ≤ T implies Proof p(t,T Bt   p(t,T ) Q p(T ,T ) |F . Substitution of the definition for the mma and = E t Bt BT algebra gives the result. This completes the proof. This corollary gives the intuitive result that the price of a zero-coupon bond equals its expected discounted payoff, where the discount rates correspond to the default-free spot rate of interest at each intermediate date over the life of the bond, and the expectation is computed using the unique equivalent martingale measure. The equivalent martingale measure includes an adjustment for the risk of the payoff since the discount rate used in expression (6.10) is the default-free spot rate of interest. Corollary 7 (Risk Neutral Valuation) Given an equivalent martingale measure Q ∈ M exists and the non-singular volatility matrix assumption, any derivative XT 1 BT ∈ L+ (Q) satisfies  T Xt = E Q XT e− t

rs ds

 |Ft .

(6.11)

Proof This is a direct application of Theorem 18 in Chap. 2, given the existence of an equivalent martingale measure Q ∈ M and the fact that the market is complete with respect to Q. This completes the proof. This corollary provides the justification for using risk neutral valuation to deter  XT mine the arbitrage-free price of an arbitrary interest rate derivative BT ∈ L1+ (Q) . Applying the insights from Chap. 2 Sect. 2.7.2 with respect to the synthetic construction of any traded derivative, the standard hedging methodology can now also be applied. Remark 51 (Assuming Only Q ∈ Ml Exists) Instead of assuming that there exists an equivalent martingale measure Q ∈ M (by the Third Fundamental Theorem 16 of asset pricing in Chap. 2 this implies NFLVR and ND), one can relax the previous structure slightly and only assume that an equivalent local martingale measure Q ∈ Ml exists where Ml =  ) for all T ∈ [0, T ] are local Q martingales (by the First FunQ ∼ P : p(t,T Bt damental Theorem 13 of asset pricing in Chap. 2 this implies NFLVR). In this case, the market is still complete with respect to Q by Theorem 22 from Chap. 2 and

6.4 Examples

129

the above assumption about the non-singularity of the zero-coupon bond volatility matrix.   ) are Q martingales, we need to exclude price To get the condition that p(t,T Bt bubbles in zero-coupon bonds. This follows if we assume that forward rates are bounded below by some negative constant and then apply Theorem 30 from Chap. 3. This completes the remark.

6.4 Examples To characterize a HJM model, one needs to specify the forward rate curve evolution under the equivalent martingale measure Q. This is because in a complete market with respect to Q ∈ M, the interest rate risk premium are not restricted by the arbitrage-free conditions NFLVR and ND, and therefore the risk premium can be arbitrarily specified. However, to uniquely specify the evolution of the forward rate curve under the statistical probability measure P, the interest rate risk premium need to be identified. Conceptually, interest rate risk premium are determined by additional (above those imposed by NFLVR and ND) restrictions that an equilibrium imposes on an market (see Part III of this book). Although the drift of the forward rate curve’s evolution changes under the equivalent martingale measure Q (due to Girsanov’s Theorem 5 in Chap. 1), the volatility structure remains unchanged. This is seen by noting that the evolutions of the forward rate curve, spot price, and bond price under the equivalent martingale measure are given by f (t, T ) = f (0, T ) +

! D  t  0

i=1

σi (s, T )

 T s

" σi (s, u)du ds +

D  t  i=1 0

Q

σi (s, T )dWi (s),

(6.12)

  t  t D  t D  t   ∂f (0, s) Q ds + rt = r0 + σi (s, t) σi (s, u)du ds + σi (s, t)dWi (s), ∂T 0 0 s 0 i=1

p(t, T ) = p(0, T )e

t

1 0 rs ds− 2

i=1

D  t i=1 0

ai (s,T

t )2 ds+ D i=1 0

(6.13) ai (s,T

Q )dWi (s)

(6.14)

where WiQ (t) for i = 1, . . . , D are independent standard Brownian motions under Q ∈ M. Given these evolutions, one can easily price and hedge any interest rate derivative using the standard techniques as presented in Sect. 2.7 in Chap. 2. For this reason, hedging and valuation will not be discussed further in this chapter with the exception of Sect. 6.6 below, which prices caps and floors using the HJM methodology. Instead, we will provide various examples for the evolution of the term structure of interest rates useful in empirical applications of this methodology. We note that for the purposes of valuation and hedging, a particular HJM model is uniquely

130

6 The Heath Jarrow Morton Model

identified by an initial forward rate curve {f (0, T ) : T ∈ [0, T ]} and a collection of volatilities {σi (t, T ) : all 0 ≤ t ≤ T ≤ T }D i=1 . Consequently, all of the subsequent examples are specified in this manner.

6.4.1 Ho and Lee Model The Ho and Lee model, in the HJM framework is represented by an arbitrary initial forward rate curve {f (0, T ) : T ∈ [0, T ]} and the evolution of the forward rate curve under the equivalent martingale measure Q as given by df (t, T ) = σ 2 (T − t)dt + σ dW Q (t)

(6.15)

for all t ∈ [0, T ] where σ > 0 is a constant. This is a single-factor model. Integration yields t f (t, T ) = f (0, T ) + σ 2 t (T − ) + σ W Q (t). 2 This expression shows that the forward rate curve drifts across time in a non-parallel fashion, but with parallel random shocks. The spot rate process implied by this evolution is  t ∂f (0, s) t2 ds + σ 2 + σ W Q (t). rt = r0 + ∂T 2 0 This is called an affine model for the spot rate because the spot rate is linear in the state variable process W Q (t). The zero-coupon bond price evolution is p(t, T ) =

p(0, T ) − σ 2 T t (T −t)−σ (T −t)W Q (t) e 2 . p(0, t)

The natural logarithm of the zero-coupon bond’s price is also seen to be linear in the state variable process W Q (t).

6.4.2 Lognormally Distributed Forward Rates Given an arbitrary initial forward rate curve {f (0, T ) : T ∈ [0, T ]}, the forward rate evolution implying lognormally distributed forward rates under the equivalent martingale measure Q, if a solution exists to this stochastic differential equation, is given by !



df (t, T ) = σf (t, T )

T

" σf (t, u)du dt + σf (t, T )dW Q (t)

t

for all t ∈ [0, T ] where σ > 0 is a constant.

6.4 Examples

131

Unfortunately, as shown in Heath et al. [72], the solution to this stochastic differential equation does not exist. It can be shown that under this evolution, forward rates explode (become infinite) with positive probability in finite time. When forward rates explode, zero-coupon bond prices become zero, implying the existence of arbitrage opportunities. This implies that a martingale measure Q does not exist for this evolution. Alternatively stated, lognormally distributed (continuously compounded) forward rates are inconsistent with a market satisfying NFLVR and ND.

6.4.3 Vasicek Model The Vasicek model, in the HJM framework, is represented by the evolution of the spot rate process under the equivalent martingale measure Q as given by drt = k(θ − rt )dt + σ dW Q (t) for all t ∈ [0, T ] where r0 , k, θ , and σ are positive constants (see Brigo and Mercurio [24, p. 50] for the derivation). Integrating, we get  t   rt = rs e−k(t−s) + θ 1 − e−k(t−s) + σ e−k(t−s) dW Q (t). s

Using expression (6.7), we have p(t, T ) = A(T − t)e−C(T −t)rt where 

A(T − t) = e

 2 2 θ− σ 2 (C(T −t)−(T −t))− σ4k C(T −t)2 2k

 1 1 − e−k(T −t) . C(T − t) = k

and

The forward rate curve evolution implied by this process is    σ2  f (t, T ) = 1 − e−k(T −t) θ − 2 1 − e−k(T −t) + e−k(T −t) rt with 2k 

f (0, T ) = 1 − e

−kT

  σ2  −kT θ − 2 1−e + e−kT r0 . 2k

Not all initial forward rate curves {f (0, T ) : T ∈ [0, T ]} can be fit by this model. To match an arbitrary initial forward rate curve, the Vasicek model needs to be

132

6 The Heath Jarrow Morton Model

extended with θ a function of time, and then the function θt needs to be calibrated to the initial forward rate curve’s values.

6.4.4 Cox Ingersoll Ross Model The Cox, Ingersoll, and Ross (CIR) model, in the HJM framework under the equivalent martingale measure Q, is represented by the spot rate process √ drt = k(θ − rt )dt + σ rt dW Q (t) for all t ∈ [0, T ] where r0 , θ , k, and σ are positive constants with 2kθ > σ 2 (see Brigo and Mercurio [24], p. 56 for the derivation). Using expression (6.7), we have p(t, T ) = A(T − t)e−C(T −t)rt where &

' 2kθ2 (T −t) σ 2he(k+h) 2

A(T − t) = , 2h + (k + h) e(T −t)h − 1

2 e(T −t)h − 1

, and C(T − t) = 2h + (k + h) e(T −t)h − 1 0 h = k 2 + 2σ 2 . The forward rate curve evolution implied by this process is f (t, T ) = −

d ln A(T − t) dC(T − t) − rt with dT dT

f (0, T ) =

d ln A(T ) dC(T ) − r0 . dT dT

Not all initial forward rate curves {f (0, T ) : T ∈ [0, T ]} can be fit by this model. To nearly match an arbitrary initial forward rate curve, the CIR model needs to be extended with θ a function of time. However, even in this circumstance, not all forward rate curves can be attained by calibrating θt to market prices (see Heath et al. [72]).

6.5 Forward and Futures Contracts

133

6.4.5 Affine Model This section studies the affine model of Duffie and Kan [53] and Dai and Singleton [41]. The affine models are a multi-factor extension of the spot rate models presented above. This class of models is called affine because both the spot rate and the natural logarithm of the zero-coupon bond’s price are assumed to be affine functions of a k-dimensional vector Xt of state variables, i.e. rt = ρ0 + ρ 1 Xt and p(t, T ) = eA(T −t)+C(T −t)Xt where dXt = (K0 + K1 Xt ) dt + Σ(t, Xt )dWQ (t) and Σ(t, Xt )Σ(t, Xt )T = H0 + H1 Xt for all t ∈ [0, T ] with ρ0 a constant, ρ 1 a k-vector, Q the equivalent martingale measure, WQ (t) = (W1Q (t), . . . , WDQ (t)), Σ(t, Xt ) is a k × D matrix, K0 , H0 are k-vectors, H1 , K1 are k × k matrices, A(T − t) is a scalar function, and C(T − t) is a k-dimensional vector function. Duffie and Kan [53] show that such a system exists if and only if A(s), C(s) satisfy the following system of differential equations dC(s) 1 T = ρ 1 − KT 1 C(s) − C(s) H1 C(s) ds 2 dA(s) 1 = ρ0 − K0 C(s) − C(s)T H0 C(s) ds 2 with C(T ) = 0 and A(T ) = 0. The forward rate evolution implied by this process is f (t, T ) = −

d ln A(T − t) dC(T − t) − Xt with dT dT

f (0, T ) =

d ln A(T ) dC(T ) − X0 . dT dT

6.5 Forward and Futures Contracts A stochastic term structure of interest rate model is essential for understanding commodity derivatives, especially forward and futures contracts because there is no economic difference between these two contracts unless interest rates are stochastic. This observation was first proven in the classical literature by Jarrow and Oldfield

134

6 The Heath Jarrow Morton Model

[105] and Cox et al. [36]. This section studies forward and futures contracts in an HJM model. Consider an asset with time t price S(t) that is adapted to the filtration F. This assumption implies that the asset’s randomness is generated by the Brownian motions underlying the evolution of the term structure of interest rates. For example, the asset S(t) could be a zero-coupon bond. Although this restriction can easily be relaxed, it is sufficient for our purpose (see Amin and Jarrow [1] for a generalization). As before, we assume that S(t) is a semimartingale and that this asset has no cash flows over [0, T ). To embed this asset into the HJM model, we ) 1 consider this asset as a derivative with time T payoff S(T BT ∈ L+ (Q). Then, we use Corollary 7 which implies that the value process S(t), when normalized by the mma’s value, is a Q martingale.

6.5.1 Forward Contracts A forward contract obligates the owner (long position) to buy the asset on the delivery date T for a predetermined price. This predetermined price is set, by market convention, such that the value of the forward contract at initiation (time t) is zero. This market clearing price is called the forward price and denoted K(t, T ). Given the underlying asset has no cash flows over the life of the forward contract, the arbitrage-free forward price must equal K(t, T ) =

S(t) . p(t, T )

(6.16)

Proof The payoff to a forward contract at time T is [S(T ) − K(t, T )]. By market convention, the forward price makes the present value of this payoff equal to zero, i.e.   T 0 = E Q [S(T ) − K(t, T )] e− t rs ds |Ft    T  T = E Q S(T )e− t rs ds |Ft − K(t, T )E Q e− t rs ds |Ft = S(t) − K(t, T )p(t, T ). The second equality uses the fact that martingale. Algebra completes the proof.

S(t) B(t)

is a Q

This expression shows that the forward price is the future value of the asset’s time t price. Indeed, if at time t one invests S(t) dollars in a zero-coupon bond maturing at time T , the time t value of this investment is expression (6.16).

6.5.1.1

The Forward Price Measure

To facilitate understanding, it is convenient to introduce another equivalent probability measure. This equivalent probability measure makes asset payoffs at some

6.5 Forward and Futures Contracts

135

future date T martingales when discounted by the T -maturity zero-coupon bond price. This equivalent martingale measure was first discovered by Jarrow [81] and later independently again by Geman [65]. Fix T ∈ [0, T ], the forward price measure QT is defined by dQT 1 > 0. = dQ p(0, T )BT Proof Note that

dQT dQ

 1 1 Q = p(0,T E ) BT completes the proof.

6.5.1.2

= 

1 p(0,T )BT

> 0 with E Q



dQT dQ



(6.17) = EQ



1 p(0,T )BT



= 1. So, it is an equivalent probability measure. This

An Alternative Characterization of Q ∈ M

We can now give an alternative characterization of Q ∈ M in an HJM model. Using T 1 the forward price measure dQ dQ = p(0,T )BT > 0, we can prove the following characterization. Crucial in this characterization is the fact that the maturity of the zero-coupon bond used for the forward price measure is larger than all of the other traded zero-coupon bonds’ maturities. Otherwise, after the bond used in the denominator of the forward price measure ceases to exist, nothing can be said with respect to NFLVR and ND for the longer maturity zero-coupon bonds over their remaining lives. Theorem 42 (Characterization of NFLVR and ND) There exists a Q ∼ P such ) that p(t,T are Q martingales for all 0 ≤ T ≤ T Bt if and only if ) T there exists a QT ∼ P such that p(t,T p(t,T ) are Q martingales for all 0 ≤ T ≤ T . Proof To prove the theorem, we need the following facts. (1) At time T < T , p(T , T ) = 1. Investing this dollar into the mma at T and holding over [T , T ] results in the value BBTT at time T . For clarity, denote this value as p(T , T ) = BBTT when T < T .  T    1 Q |F |F = E = (2) Define Zt = E Q dQ t t dQ p(0,T )BT

1 Q p(0,T ) E



1 BT

|Ft



p(t,T ) = p(0,T )Bt . Given a suitably integrable FT - measurable X, then using Shreve [177, Lemma 5.2.2, p. 212],     T T Bt Q X 1 |F , and E Q [X |Ft ] = Z1t E Q X dQ t dQ |Ft = p(t,T ) E BT   T ) QT dQ |F E Q [X |Ft ] = Zt E Q X dQ = p(t,T [XBT |Ft ]. t T Bt E   T ,T ) ) |F = p(t,T ("⇒) Given Q satisfying the hypothesis, show E Q p(T t p(T ,T ) p(t,T ) .

136

6 The Heath Jarrow Morton Model

Now, by fact (2) we have       T ,T ) Bt 1 1 1 QT Q |F |F |F = E = E Q p(T E t t t p(T ,T ) p(T ,T ) p(t,T ) p(T ,T ) BT       p(T ,T ) Bt Bt 1 1 Q Q 1 |F Q |F |F = p(t,T = E E E t t T ) p(T ,T ) BT p(t,T ) p(T ,T ) BT   p(t,T ) Bt 1 Q = p(t,T ) E BT |Ft = p(t,T ) .   ) (⇐") Given QT satisfying the hypothesis, show E Q p(TBT,T ) |Ft = p(t,T Bt . Now, by facts (1) and (2) we have       ) QT 1 E Q p(TBT,T ) |Ft = E Q B1T |Ft = p(t,T Bt E BT BT |Ft  T   ) QT 1 Q |F |F = p(t,T E E B T t T Bt BT  T  p(t,T ) QT Q = Bt E E [p(T , T ) |FT ] |Ft  T     ) QT p(T ,T ) p(t,T ) QT p(T ,T ) Q |F |F |F = p(t,T E = E E t t T Bt p(T ,T ) Bt p(T ,T ) =

p(t,T ) p(t,T ) Bt p(t,T )

=

p(t,T ) Bt .

This completes the proof.

6.5.1.3

Risk Neutral Valuation (Revisited)

Using this forward price measure, for any suitably measurable and integrable Xt 1 T T random payoff X BT ∈ L+ (Q) received at time T , p(t,T ) is a Q martingale, which enables us to prove the following lemma. Lemma 16 (Alternative Risk Neutral Valuation Formula) Given

XT BT

T

Xt = p(t, T )E Q [XT |Ft ] .

∈ L1+ (Q), (6.18)

Proof First, note that   T 1 Q |F |F = E E Q dQ t t dQ p(0,T )BT   p(t,T ) 1 1 = p(0,T ) E Q BT |Ft = p(0,T )Bt .     T Then, given Xt = E Q XT e− t rs ds |Ft = E Q XT BBTt |Ft , we have using Shreve [177, Lemma 5.2.2, p. 212], that   T T  E Q XT dQ |Ft p(t, T )E Q [XT |Ft ] = p(t, T )  dQ1T dQ  )Bt Q XT E = p(t, T ) p(0,T p(t,T ) completes the proof.

EQ

1 p(0,T )BT

dQ

|Ft

|Ft



 = E Q XT

Bt BT

|Ft



= Xt . This

6.5 Forward and Futures Contracts

137

Expression (6.18) gives a useful alternative procedure for computing present values. This procedure is to take the payoff’s expectation under the forward price measure QT and discount the expectation to time t using the appropriate zerocoupon bond’s price. Note that the default-free spot rate of interest process does not explicitly appear in this expression. It is implicit, however, in the forward price measure QT . It is important to emphasize that unlike the martingale probability Q, the forward price measure depends on a particular future date T corresponding to the maturity of the zero-coupon bond used in its definition. Note that the payoff date for the derivative T must match the bond’s maturity used in the definition of the forward price measure QT . Applying this result to the time T price of the asset S(T ) yields an equivalent expression for the forward price T

K(t, T ) = E Q [S(T ) |Ft ] ,

(6.19)

which explains the name for this probability measure. T

Proof S(t) = p(t, T )E Q [S(T ) |Ft ]. Equating this to expression (6.16) completes the proof. For subsequent usage, it is easy to show that the forward rate equals the expected time T spot rate under the forward price measure, i.e. f (t, T ) = E Q

T

  rT |Ft .

(6.20)

 T  Proof Given expression (6.10), p(t, T ) = E Q e− t rs ds |Ft . Differentiating yields   T  T  ) Q r e− t rs ds |F − ∂p(t,T = p(t, T )E Q rT |Ft where the second = E t T ∂T equality follows from expression (6.18). Dividing by p(t, T ) completes the proof. This implies, of course, that the forward price is not an unbiased estimate of the future spot price (under the statistical probability P).

6.5.2 Futures Contracts A futures contract is similar to a forward contract. It is a financial contract, written on the asset S(t), with a fixed maturity T . It represents the purchase of the underlying asset at time T via a prearranged payment procedure. The prearranged payment procedure is called marking-to-market. Marking-to-market obligates the purchaser (long position) to accept a cash flow stream when holding the contract equal to the continuous changes in the futures prices for the same maturity futures contract.

138

6 The Heath Jarrow Morton Model

The time t futures prices, denoted k(t, T ), are set (by market convention) such that newly issued futures contracts (at time t) on the same underlying asset with the same maturity date T , have zero market value. Hence, futures contracts (by construction) have zero market value at all times, and a continuous cash flow stream equal to dk(t, T ). At maturity, the last futures price must equal the underlying asset’s price k(T , T ) = S(T ). This follows by the definition of the contract. Indeed, it represents the purchase price for the asset for immediate delivery. It can be shown that the arbitrage-free futures price must satisfy k(t, T ) = E Q [S(T ) |Ft ] .

(6.21)

Proof The accumulated value from buying and holding a futures contract over [t, T ] and investing all proceeds into the mma is  BT

T

1 dk(s, T ). Bs

t

Hence, the futures price k(t, T ) solves ⎡ EQ ⎣

BT

T t

1 Bs dk(s, T

BT

⎤

)

Ft ⎦ Bt = 0 for all t and k(T , T ) = S(T ).

  t T This implies E Q t B1s dk(s, T ) |Ft = 0, i.e. Mt = 0 B1s dk(s, T ) is a Q martingale. Using of the stochastic integral (Protter [158, p. 165]),  t the associativity t we have that 0 Bs dMs = 0 Bs B1s dk(s, T ) = k(t, T )−k(0, T ) is a Q martingale, or k(t, T ) = E Q [k(T , T ) |Ft ] = E Q [S(T ) |Ft ] . This completes the proof. This expression shows that the futures price is not the expected value of the asset’s time T payoff under the statistical probability measure P, unless the equivalent martingale measure equals the statistical probability, i.e. Q = P. Recall that these two probability measures are equal, if and only if, interest rate risk premia are identically zero. Hence, in general, the futures price will not be an unbiased predictor of the future spot price of an asset. This expression shows the difference between forward and futures prices for the same asset S(t) with the same delivery date T . The forward price is the conditional expectation of the spot price at delivery S(T ) using the forward price measure QT (see expression (6.19)) while the futures price uses the equivalent martingale measure Q (see expression (6.21)). We can alternatively quantify the relation between forward and futures prices as follows. K(t, T ) = k(t, T ) + cov

Q



1 S(T ), BT



Bt

Ft .

p(t, T )

(6.22)

6.6 The Libor Model

139

Proof Using expression (6.16), we get   K(t, T )p(t, T ) = St = E Q BSTT |Ft Bt     = E Q [ST |Ft ] E Q B1T |Ft Bt +cov Q S(T ), B1T |Ft Bt . Using expression (6.21) and algebra completes the proof. Expression (6.22) shows that forward and futures prices are equal if and only if, under the martingale measure Q, the covariance between the asset’s spot price and the reciprocal of the mma’s value is zero. A sufficient condition for this covariance to equal zero, and hence for the equivalence of forward and futures prices, is that interest rates are deterministic (the classical result).

6.6 The Libor Model This section presents the Libor model for pricing caps and floors. Caps and floors are interest rate derivatives written on a future realized spot interest rate. Caps are a portfolio of European caplets, where a caplet is a European call option on a future realized spot interest rate, and floors are a portfolio of floorlets, where a floorlet is a European put option on a future realized spot interest rate. These financial instruments historically were based on LIBOR (London InterBank Offer Rates), an index of Eurodollar borrowing rates (see Jarrow and Chatterjea [97, Chapter 2 ], for more explanation), hence, the name for the model. Here, we assume that the spot interest rate corresponds to the default-free interest rate that can be earned from investing a dollar for a fixed time period, say δ units of a year (e.g. 14 of a year), computed as a discrete spot rate. This is distinct from the continuously compounded and instantaneous default-free spot rate rt . Just as with the continuously compounded and instantaneous forward rates, there is an analogous discrete forward rate associated with this discrete spot rate. Consider the future time interval [T , T + δ] where δ corresponds to the “earning” interval. The discrete forward rate at time t for the time interval [T , T + δ] is defined by 1 + δL(t, T ) =

p(t, T ) . p(t, T + δ)

(6.23)

The right side of this expression isolates the implicit interest embedded in the zerocoupon bonds over [T , T + δ]. The discrete spot rate is L(t, t) for [t, t + δ]. Given the evolution for a 1-factor (continuous) forward rate process as in expression (6.3) and the definition (6.23) of the discrete forward rate involving zerocoupon bond prices, the evolution of the discrete forward rate is  dL(t, T ) =

 1 + δL(t, T ) QT +δ (t) (a1 (t, T ) − a1 (t, T + δ)) dW1 δ

(6.24)

140

6 The Heath Jarrow Morton Model T +δ

for all t ∈ [0, T ] where W1Q (t) is a standard independent Brownian motion under the forward price measure QT +δ . Proof For ease of notation we drop the subscript 1. From expression (6.14), we have that p(t, T ) = p(0, T )e

t

 1 t 0 rs ds− 2 0

p(t, T + δ) = p(0, T + δ)e Thus, p(t,T ) p(t,T +δ)

=

p(0,T ) − 12 p(0,T +δ) e

t 0

t a(s,T )2 ds+ 0 a(s,T )dW Q (s) and t  t 1 t 2 Q 0 rs ds− 2 0 a(s,T +δ) ds+ 0 a(s,T +δ)dW (s)

a(s,T )2 ds+ 12

t 0

.

t a(s,T +δ)2 ds+ 0 [a(s,T )−a(s,T +δ)]dW Q (s)

.

Define an equivalent change of measure dQx dQ

= e−

T 0

T 1 2 2 xs ds+ 0

xs dW Q (s)

> 0.

By Girsanov’s Theorem 5 in Chap. 1, x dW Q (s) = −xs ds + dW Q (s) is a Brownian motion under Qx . p(t,T ) x We want to determine xs such that p(t,T +δ) is a Q martingale for all T . Given the market is complete with respect to Q, this implies the market is complete with respect to QT +δ when using p(t, T +δ) as the numeraire. Hence, by the definition of the forward price measure QT +δ , this implies that Qx = QT +δ by uniqueness of the equivalent martingale measure (using the numeraire p(t, T + δ)) due to the Second Fundamental Theorem 14 of asset pricing in Chap. 2. We use this observation in the last step of the proof. By substitution, p(t,T ) p(t,T +δ)

=

   p(0,T ) − 12 0t a(s,T )2 ds+ 12 0t a(s,T +δ)2 ds+ 0t [a(s,T )−a(s,T +δ)]xs ds e p(0,T +δ) t x + 0 [a(s,T )−a(s,T +δ)]dW Q (s)

·e . Using the Dolean Dades exponential (see Medvegyev [143, p. 412]), 1 if eZ− 2 [Z,Z] is a Qx local martingale. Z is a Qx local martingale if and only  t Hence, adding and subtracting + 12 0 [a(s, T ) − a(s, T + δ)]2 ds in the exponent yields t t t − 12 0 a(s, T )2 ds + 12 0 a(s, T + δ)2 ds + 0 [a(s, T ) − a(s, T + δ)] xs ds  t + 12 0 [a(s, T ) − a(s, T + δ)]2 t x t + 0 [a(s, T ) − a(s, T + δ)] dW Q (s) − 12 0 [a(s, T ) − a(s, T + δ)]2 ds is a Qx local martingale if and  tonly if t t − 12 0 a(s, T )2 ds + 12  0 a(s, T + δ)2 ds + 0 [a(s, T ) − a(s, T + δ)] xs ds t + 12 0 [a(s, T ) − a(s, T + δ)]2 ds = 0 for all t

6.6 The Libor Model

141

if and only if − 12 a(s, T )2 + 12 a(s, T + δ)2 + [a(s, T ) − a(s, T + δ)] xs for all s + 12 [a(s, T ) − a(s, T + δ)]2 = 0 if and only if − 12 a(s, T )2 + 12 a(s, T + δ)2 + [a(s, T ) − a(s, T + δ)] xs + 12 a(s, T )2 − a(s, T )a(s, T + δ) + 12 a(s, T + δ)2 = 0 for all s if and only if a(s, T + δ)2 − a(s, T )a(s, T + δ) = − [a(s, T ) − a(s, T + δ)] xs

for all s

if and only if − [a(s, T ) − a(s, T + δ)] a(s, T + δ) = − [a(s, T ) − a(s, T + δ)] xs all s. Hence, xs = a(s, T + δ) for all s. x By substitution, we get using W Q (s) that p(t,T ) p(t,T +δ)

=

p(0,T ) p(0,T +δ) e

t

0 [a(s,T

x

)−a(s,T +δ)]dW Q (s)− 12

t

0 [a(s,T

)−a(s,T +δ)]2 ds

for

.

Using the definition of L(t, T ) yields 1 + δL(t, T ) = [1 + δL(0, T )] t

Qx

1

t

e 0 [a(s,T )−a(s,T +δ)]dW (s)− 2 0 [a(s,T )−a(s,T +δ)] ds . This is the unique solution to (see Medvegyev [143, p. 412]), 2

d [1 + δL(s, T )] = [1 + δL(s, T )] [a(s, T ) − a(s, T + δ)] dW Q (s). x

Algebra gives x [1+δL(s,T )] [a(s, T ) − a(s, T + δ)] dW Q (s). δ fact that Qx = QT +δ completes the proof.

dL(s, T ) = Using the

This discrete forward rate evolution is consistent with no-arbitrage (by construction), since it is derived from an the arbitrage free (continuously compounded) forward rate evolution in expression (6.12), after a change of probability measure from Q to QT +δ . To facilitate an analytic formula for a cap’s value, the LIBOR model assumes that   1 + δL(t, T ) γ (t, T ) = (6.25) (a1 (t, T ) − a1 (t, T + δ)) δL(t, T ) is a deterministic function of time. Hence, under this assumption, the discrete forward rate evolves as dL(t, T ) = γ (t, T )L(t, T )dW1Q

T +δ

(t)

(6.26)

142

6 The Heath Jarrow Morton Model

for all t ∈ [0, T ]. This is a (generalized) geometric Brownian motion process. Note the analogy to the evolution of the risky asset price process in the BSM model in Chap. 5. We now value caps and floors under this evolution. But first, some definitions are needed. A European caplet is a European call option with maturity T + δ and strike k on the discrete spot rate at time T . Caps are a portfolio of European caplets, with the same strike but with different and increasing maturity dates. The maturity dates of the included caplets are evenly spaced at fixed intervals (e.g. every 3 months) up to the maturity of the cap (e.g. 5 years). Hence, to value a cap, one only needs to value the constituent caplets. Similarly, a floorlet is a European put option on the forward interest rate. Floors are a portfolio of floorlets with the same strike and with increasing maturity dates. The sequence of maturity dates is similar to that of a cap. To value a floor, one needs only value the constituent floorlets. Using put-call parity (see Jarrow and Chatterjea [97, Chapter 16 ], for a discussion of put-call parity), if one prices European caplets, then the formula for European floorlets immediately follows. For this reason, this section only concentrates on pricing caplets. Now, consider a caplet on the discrete forward rate with maturity T + δ and strike k. The standard caplet pays off at time T + δ, based on the discrete forward rate at time T , i.e. the payoff is defined by XT +δ = max[(L(T , T ) − k) δ, 0] at time T + δ. This can be interpreted as the differential interest (L(T , T ) − k) δ earned on the principal of 1 dollar over the time period [T , T + δ], but only if the differential is positive. Note that the interest paid is prorated by the time period δ over which it is earned. Using expression (6.18), the caplet’s time 0 value is X0 = p(0, T + δ)E Q

T +δ

[max[(L(T , T ) − k) δ, 0]] .

(6.27)

Given that L(T , T ) follows a lognormal distribution under expression (6.26), it is straightforward to show that the caplet’s value satisfies the subsequent formula, which appears remarkably similar to the BSM formula (5.11) in Chap. 5. X0 = p(0, T + δ)δ[L(0, T )N (d1 ) − kN (d2 )] ⎛ d1 = ⎝

 log

L(0,T ) k



3 T 0

6 

T

d2 = d1 − 0

+

 1 T 2 0

γ (t, T

γ (t, T )2 dt )2 dt

γ (t, T )2 dt.

where ⎞ ⎠ and

(6.28)

6.7 Notes

143

Proof Using expression (6.27), we have T +δ X0 = p(0, T + δ)δE Q [max[(L(T , T ) − k) , 0]]. The same proof as used to prove the BSM formula (5.11) in Chap. 5 applies with the following identifications: S(t) = L(t, T ), T = T , K = k, r = 0, and σ 2 = the proof.

T 0

γ (t,T )2 dt . T

This completes

It is easy to generalize this 1-factor to an D-factor model. The lognormal evolution for forward Libor rates has also been extended to jump diffusions, Levy processes, and stochastic volatilities (see Jarrow et al. [113] and references therein). For textbooks expanding on the LIBOR model, as used in practice, see Rebanato [163] and Schoenmakers [173]. Remark 52 (Geometric Brownian Motion (Continuously Compounded) Forward Rates) It was shown in Sect. 6.4 above that (continuously compounded) forward rates f (t, T ) following a geometric Brownian motion process are inconsistent with a market satisfying NFLVR and ND. This fact implies that a formula for caplets analogous to the Black-Scholes-Merton formula, using continuously compounded forward rates, is also inconsistent with NFLVR and ND. This inconsistency was the motivation for deriving the LIBOR model (see Sandmann et al. [170], Miltersen et al. [149], and Brace et al. [20]), and it is the reason discrete forward rates L(t, T ) are used in the derivation of expression (6.28). This completes the remark.

6.7 Notes There is much more known about implementing the HJM model and its empirical testing, especially as related to fixed incomes securities. Excellent books in this regard include Brigo and Mercurio [24], Carmona and Tehranchi [28], Filipovic [60], Jarrow [85], Rebonato [163], Schoenmakers [173], and Zagst [185].

Chapter 7

Reduced Form Credit Risk Models

Credit risk arises whenever two counter parties engage in borrowing and lending. Borrowing can be in cash, which is the standard case, or it can be through the “shorting” of securities. Shorting a security is selling a security one does not own. To do this, the security must first be borrowed from an intermediate counter party, with an obligation to return the borrowed security at a later date. The borrowing part of this shorting transaction involves credit risk. Since the majority of transactions in financial and commodity markets involve some sort of borrowing, understanding the economics of credit risk is fundamental to the broader understanding of economics itself. There are two models for studying credit risk. The first is called the structural approach, which was introduced by Merton [146]. This model assumes that all of the assets of the firm trade, an unrealistic assumption. Consequently, this model is best used for conceptual understanding (see Jarrow [90] for a detailed discussion). Merton’s structural model was studied in Sect. 5.5 of Chap. 5. The second is called the reduced form model, which was introduced by Jarrow and Turnbull [110, 111]. This model assumes that only a subset of the firm’s liabilities trade, those that need to be priced and hedged. This is the model studied in this chapter. This chapter is based on Jarrow [88].

7.1 The Set-Up This chapter builds on the HJM model in Chap. 6 by adding trading in a risky zerocoupon bond. Given isa non-normalized market ((B, S), F, P). The money market t account (mma) Bt = e 0 rs ds , B0 = 1, is continuous in time t,  tand of finite variation. rt is the default-free spot rate of interest, adapted to F with 0 |rs | ds < ∞. As in the HJM model in Chap. 6 we modify the risky asset price vector. Instead of a finite number of risky assets St = (S1 (t), . . . , Sn (t)) , we let the traded risky © Springer Nature Switzerland AG 2021 R. A. Jarrow, Continuous-Time Asset Pricing Theory, Springer Finance, https://doi.org/10.1007/978-3-030-74410-6_7

145

146

7 Reduced Form Credit Risk Models

assets include the collection of default-free zero-coupon bonds of all maturities, with time t price p(t, T ) > 0 for a sure dollar paid at time T for all 0 ≤ t ≤ T ≤ T . The default-free zero-coupon price process is assumed to be a strictly positive continuous semimartingale adapted to F for all T . In addition, also traded is a risky zero-coupon bond promising to pay a dollar at time T with time t price denoted D(t, T ) ≥ 0. This bond may default, and the promised dollar not paid. The value of the risky zero-coupon bond at time T , D(T , T ), may be strictly less than one if default occurs. The risky zero-coupon price process is assumed to be a nonnegative semimartingale adapted to F. This price process, in general, is not continuous. More risky securities could be traded, but they are not needed for the subsequent presentation. Remark 53 (Self-Financing-Trading Strategies) As in the HJM model in Chap. 6, although there are a continuum of traded risky assets, we assume that an admissible s.f.t.s. consists of the mma and only a finite number of default-free and risky zerocoupon bonds over the trading horizon [0, T ]. This enables us to use the fact that the existence of an equivalent martingale measure Q ∈ M implies NFLVR and ND by the Third Fundamental Theorem 16 of asset pricing in Chap. 2, even with trading in an infinite number of risky assets. The existence of an equivalent martingale measure is no longer implied by NFLVR and ND because the first and third fundamental theorems were only proven for a finite asset market (see the discussion following the First Fundamental Theorem 13 in Chap. 2). This completes the remark.

7.2 The Risky Firm To provide an increased understanding of the pricing of the risky zero-coupon bond, we add some additional structure related to default and the recovery rate process of the risky zero-coupon bond. These are best characterized by considering the firm (more generally a credit entity) that issued this risky zero-coupon bond. The firm can have many outstanding liabilities in addition to this zero-coupon bond. All of these liabilities contain contractual payments to be made by the firm at various future dates over the contracts’ lives. The first time the firm does not make a contractual payment on any of its liabilities, the firm is in “default” on that liability’s payments. Due to cross-defaulting provisions, this means that all of the firm’s liabilities are in default too (usually all contractual payments—interest owed and principal outstanding— become due). We want to characterize the default time and the recovery rate on the risky zerocoupon bond, conditional upon default. Let Γt = (Γ1 (t), . . . , Γm (t)) ∈ Rm be a collection

of stochastic processes characterizing the state of the market at time t with FtΓ t∈[0,T ] representing the filtration generated by the state variables Γt up to and including time t ≥ 0. These state variables are assumed to be nonnegative semimartingales adapted to F, which implies that FtΓ t∈[0,T ] ⊂ F. We include the default-free spot rate of interest rt in this set of state variables, which implies that rt

7.2 The Risky Firm

147

is FtΓ - measurable. Examples of additional state variables that could be included in this set are the remaining default-free forward rates, unemployment rates, inflation rates, etc. Let λ : [0, T ] × Rm −→ [0, ∞), denoted λt = λt (Γt ) ≥ 0, be jointly Borel T measurable with 0 λt (t )dt < ∞ a.s. P. Let Nt ∈ {0, 1, 2, · · · } with N0 = 0 be a Cox process conditioned on FTΓ with λt (t ) its intensity process, i.e. for all 0 ≤ s < t and n = 0, 1, 2, · · · , ∞,



P Nt − Ns = n | FTΓ ∨ Fs =

e−

t s

λu (u )du

 t s

n λu (u )du (7.1)

n!

where FTΓ ∨ Fs is the smallest σ -algebra containing both FTΓ and Fs (see Definition 13 in Chap. 1). Finally, let τ ∈ [0, T ] be the stopping time adapted to the filtration F defined by τ = inf {t > 0 : Nt = 1} . The stopping time τ represents the firm’s default time. The function λt (t ) is the firm’s default intensity, which can be interpreted as the probability of default over a small time interval [t, t + Δ] conditional upon no default prior to time t. We note that the default time is totally inaccessible, i.e. it cannot be written as a predictable stopping time. A stopping time is predictable if there exists an increasing sequence of stopping times that approach τ from below (see Protter [158] for these definitions). For simplicity of notation, we often write λt (t ) as λt below. Intuitively, a Cox process is a point process which, conditional upon the information set generated by the state variables process Γt over the entire trading horizon [0, T ], behaves like a standard Poisson process. For subsequent use, we provide the next lemma (for a proof see Lando [136]). Lemma 17 (Cox Process Probabilities) For τ > t, T

  P(τ > T FTΓ ∨ Ft ) = E 1τ >T | FTΓ ∨ Ft = e− t

λu du

,

(7.2)

 T     P(τ > T |Ft ) = E E 1τ >T | FTΓ ∨ Ft |Ft = E e− t

λu du

 |Ft ,

(7.3)

 T P(t < τ ≤ T |Ft ) = 1 − E e− t P( τ ∈ [s, s

+ dt)| FTΓ

λu du

 |Ft ,

s dP(τ ≤ s FTΓ ∨ Ft ) = λs e− t λu du . ∨ Ft ) = ds

(7.4)

(7.5)

148

7 Reduced Form Credit Risk Models

If the firm defaults prior to the zero-coupon bond’s maturity date, we assume that at time T the bond receives a recovery payment less than or equal to the promised dollar, i.e. 7 R(τ,T )B(T ) ≤ 1 if 0 < τ ≤ T B(τ ) D(T , T ) = (7.6) 1 if 0 ≤ T < τ where 1 ≥ R(t, T ) ≥ 0 is FtΓ -measurable. Note that the recovery rate R(τ, T ) is realized at the default time τ . This recovery payoff is invested in the mma at time τ and held until time T . Thus, D(T , T ) represents the value of this mma investment at time T . In the literature, for analytic convenience, three recovery rate processes have been frequently used. 1. Recovery of Face Value. R(τ, T ) = δ where δ ∈ [0, 1] is a constant. This states that on the default date, the debt is worth some constant recovery rate between 0 and 1. 2. Recovery of Treasury. R(τ, T ) = δp(τ, T ) where δ ∈ [0, 1] is a constant. This states that on the default date, the debt is worth some constant percentage of an otherwise equivalent, but default-free zero-coupon bond. 3. Recovery of Market Value R(τ, T ) = δD(τ −, T ) where δ ∈ [0, 1] is a constant and D(τ −, T ) =

lim D(t, T ) is the value of the debt issue an instant

t→τ,t≤τ

before default. This states that on the default date, the debt is worth some constant fraction of its value an instant before default occurred, at time τ −. For this recovery rate process there is a discrete jump at the default time, i.e. D(τ, T ) = D(τ, T ) − D(τ −, T ) < 0.

7.3 Existence of an Equivalent Martingale Measure For pricing and hedging credit derivatives we need to assume that there exists an equivalent probability measure Q ∈ M where   p(t, T ) D(t, T ) M= Q∼P: for all T ∈ [0, T ] and are Q martingales . Bt Bt

7.3 Existence of an Equivalent Martingale Measure

149

Assumption (Existence of an Equivalent Martingale Measure) There exists an equivalent probability measure Q ∈ M such that p(t, T ) for all T ∈ [0, T ] and Bt

D(t, T ) Bt

are Q martingales. This assumption implies that the market satisfies NFLVR and ND by the Third Fundamental Theorem 16 of asset pricing in Chap. 2. Of course, as shown in Chap. 2, given there are an infinite number of traded risky assets, the converse of this theorem is not necessarily true. For the purposes of this chapter, we do not need to specify a particular evolution for the term structure of interest rates as done in the HJM model in Chap. 6. The previous assumption immediately implies the next corollary. Corollary 8 (Zero-Coupon Bond Formulas)  T p(t, T ) = E Q e− t

rs ds

|Ft

 (7.7)

for all 0 ≤ t ≤ T ≤ T and  T D(t, T ) = E Q D(T , T )e− t

rs ds

|Ft

 (7.8)

for all 0 ≤ t ≤ T . The zero-coupon bond prices are given by the expected discounted cash flows using the equivalent martingale measure Q ∈ M. As before, the risk of the payoffs is incorporated into these present value formulas because expectations are taken with respect to the equivalent martingale measure Q and not the statistical probability measure P. Expression (7.8) can be written as  τ T D(t, T ) = E Q R(τ, T )e− t rs ds 1{τ ≤T } + 1 · e− t

rs ds

 1{T s). Then, EQ

&

T

ys 1τ >s≥t e−

s t

' ru du

ds |Ft

= EQ

t

&

T

ys e−

s t

(ru +8λu )du

t

  s T Proof E Q t ys 1τ >s e− t ru du ds |Ft     s

T = E Q E Q t ys 1τ >s e− t ru du ds FTΓ ∨ Ft |Ft

'

ds Ft .

(7.14)

Notingthat rt is Ft adapted yields  s T   = E Q t ys e− t ru du E Q 1τ >s | FTΓ ∨ Ft ds |Ft   s s 8 T = E Q t ys e− t ru du e− t λu du ds |Ft . This completes the proof.

7.4.3 Cash Flow 3 Random cash flow Yτ which is Fτ —measurable at time τ , but the cash flow occurs only if default has occurred within [0, T ], i.e. (τ ≤ T ). Then, &   τ E Q Yτ 1t α2 . 5. Intermediate Value ρ ρ For any X1 , X2 , X3 ∈ L0+ , if X1 $ X2 $ X3 then there exists a unique ρ

α ∈ (0, 1) such that X2 ∼ αX1 + (1 − α)X3 . 6. Strong Independence ρ For any X1 , X2 ∈ L0+ , if X1 $ X2 then for any α ∈ (0, 1) and any X ∈ L0+ , ρ

αX1 + (1 − α)X $ αX2 + (1 − α)X. These properties are called rationality axioms because they reflect rationality in ρ

the trader’s choices. Properties (1)–(3) make # an equivalence relation. Essentially, as an equivalent relation the trader can decide likes and dislikes on all elements in the choice set L0+ and his choices are transitive. Property (4), order preserving, states that if X1 is preferred to X2 then combinations of X1 and X2 with more X1 are preferred to those with more X2 . Property (5), intermediate value, is a continuity restriction on preferences. It states that if X1 is preferred to X2 which is preferred to X3 then there is a unique combination of X1 and X3 that is indifferent to X2 . Last, property (5) is the strong independence axiom. It states that if X1 is preferred to X2 , then given any third choice X, any combination of X1 and X is preferred to a combination of X2 and X. Another important set of properties that a preference relation can exhibit depends on specifying a probability measure P : F → [0, 1], interpreted as the trader’s beliefs. For future use, we define the set of all non-trivial probability measures on F by P, i.e. all P such that there exists an A ∈ F with P(A) > 0 and P(Ac ) > 0 where A ∪ Ac = . There are four such properties. 7. Strictly Monotone ρ For any X1 , X2 ∈ L0+ , if X1 ≥ X2 a.s. P and P (X1 > X2 ) > 0 then X1 $ X2 . If a trader’s preferences are strictly monotone as in property (7), then the trader prefers more wealth to less.

9.1 Preference Relations

171

A trader’s aversion to risk with respect to a probability belief P is characterized in property (8). 8. Risk Aversion ρ

# is risk averse with respect to P if ρ

# is risk neutral with respect to P if ρ

ρ

1Ω (ω)E [X] $ X, ρ

1Ω (ω)E [X] ∼ X, ρ

# is risk loving with respect to P if 1Ω (ω)E [X] ≺ X for any X ∈ L0+ with E [X] < ∞. A trader is risk averse with respect to X if she prefers receiving the expected value of the terminal wealth E [X] instead of receiving the random wealth X itself. A trader is risk neutral if she is indifferent between the two. And, a trader is risk loving if she prefers the random wealth to the expected value of the wealth. A stronger notion of risk aversion is given by the following property. It is stronger because the preference relation must exhibit risk averse behavior with respect to all P ∈ P. 9. Uniform Risk Aversion ρ

ρ

# is uniformly risk averse if # is risk averse with respect to all P ∈ P. Uniformly risk neutral and uniformly risk loving are defined similarly. For the subsequent theory, the following property of a preference relation is crucial. 10. State Dependent Expected Utility Representation with respect to P Given a probability belief P and a utility function U : (0, ∞) × Ω → R such that U (x, ω) is B(0, ∞) ⊗ FT measurable where B(0, ∞) is the Borel σ -algebra ρ

on (0, ∞), we say that # has a state dependent EU representation with respect to P if ρ

X1 $ X2

⇐⇒

E [U (X1 , ω)] > E [U (X2 , ω)]

for all X1 , X2 ∈ L0+ with E[U (X1 , ω)] < ∞ and E [U (X2 , ω)] < ∞. The utility function is called state independent if U (x, ω) = U (x) does not depend on ω ∈ Ω. In this representation of a trader’s preferences, the probability measure P is interpreted as the trader’s beliefs and U as the trader’s utility function. The first important observation about this property is that the utility function representing the preference relation depends on the realized state ω ∈ Ω in addition to the terminal wealth. Hence, the “past” affects the trader’s “current” preferences. Remark 57 (Preferences Over Lotteries) An alternative choice set in the asset pricing literature is the set of all probability measures on B(0, ∞), called lotteries, which can be determined from the probability measure P : F → [0, 1] and the choice set L0+ . This set of probability measures PX : B(0, ∞) → [0, 1] for X ∈ L0+ is defined by PX (A) = P (X ∈ A) for all A ∈ B(0, ∞). Call this set

172

9 Utility Functions

UP . Preferences can be defined over UP instead of L0+ , and the rationality axioms imposed on these preferences, e.g. see Jarrow [80] and Follmer and Schied [63], Chap. 2 for such a formulation. This completes the remark.

9.2 State Dependent EU Representation This section studies the implications of the existence of a state dependent EU representation with respect to P. We first study the rationality axioms implied by such a state dependent EU representation. Afterwards, we study the concept of risk aversion for state dependent EU preferences.

9.2.1 Rationality Axioms We start with the most basic rationality axioms. ρ

Lemma 18 (Necessary Rationality Axioms) Let # have a state dependent EU representation with respect to P. ρ

Then, # satisfies reflexivity, comparability, and transitivity. Proof (Reflexivity) Let X ∈ L0+ , then E [U (X, ω)] ≥ E [U (X, ω)]. ρ

Or, X # X. This completes the proof. (comparability) Let X1 , X2 ∈ L0+ , then by the properties of R either E [U (X1 , ω)] ≥ E [U (X2 , ω)] or E [U (X2 , ω)] ≥ E [U (X1 , ω)]. ρ

ρ

Or, X1 # X2 or X2 # X1 . This completes the proof. ρ

ρ

(transitivity) Let X1 , X2 , X3 ∈ L0+ , if X1 # X2 and X2 # X3 , then E [U (X1 , ω)] ≥ E [U (X2 , ω)] and E [U (X2 , ω)] ≥ E [U (X3 , ω)]. This implies E [U (X1 , ω)] ≥ E [U (X3 , ω)]. ρ

Or, X1 # X3 . This completes the proof. A state dependent EU representation with respect to P does not imply prefρ

erences # satisfy the order preserving, the intermediate value, or the strong independence axioms. The next example illustrates this observation. Example 10 (EU Representation: Not Order Preserving, Not Intermediate Value, or Not Strong Independence Preferences) Assume that a state dependent EU ρ

representation with respect to P exists for #. Consider the following examples. They ρ

show that # need not satisfy the order preserving, the intermediate value, nor the strong independence properties. (Not Order Preserving) Choose any probability measure P ∈ P.

9.2 State Dependent EU Representation

173

Let U (x, ω) = x1x E [U (X2 , ω)] = 0. Consider α1 = 12 and α2 = 13 . α1 X1 +(1−α1 )X2 = 54 > 1, which implies E [U (α1 X1 + (1 − α1 )X2 , ω)] = 0, and α2 X1 +(1−α2 )X2 = 32 > 1, which implies E [U (α2 X1 + (1 − α2 )X2 , ω)] = 0. ρ

In conjunction, this gives an example where X1 $ X2 and there exist α1 , α2 ∈ ρ (0, 1) with α1 > α2 such that [α1 X1 + (1 − α1 )X2 ] ∼ [α2 X1 + (1 − α2 )X2 ], which violates the order preserving property. (Not Intermediate Value) Choose the discrete probability P ∈ P where for B ∈ F , P(B) = 14 and P(Bc ) = 34 . Let U (x, ω) = 1x≤1 + 2 · 1x>1 for all ω ∈ Ω. Consider X1 (ω) = 2, X2 (ω) = 12 1B (ω) + 2 · 1B c (ω). We have E [U (X1 , ω)] = 2 > E [U (X2 , ω)] = P(B) + 2P(Bc ) = 74 > E [U (X3 , ω)] = 1. Note that E [U (αX1 + (1 − α)X3 , ω)] = 2 for all α ∈ (0, 1), hence, E [U (αX1 + (1 − α)X3 , ω)] = E [U (X2 , ω)]. ρ ρ In conjunction, this gives an example where X1 $ X2 $ X3 , but there exists no ρ α ∈ (0, 1) such that X2 ∼ αX1 + (1 − α)X3 , which violates the intermediate value property. (Not Strong Independence) Choose any probability measure P ∈ P. Let U (x, ω) = x1x E [U (X2 , ω)] = 13 . Letting α = 12 , αX1 + (1 − α)X = 54 > 1, which implies E [U (αX1 + (1 − α)X, ω)] = 0, and αX2 + (1 − α)X = 32 > 1, which implies E [U (αX2 + (1 − α)X, ω)] = 0. ρ

In conjunction, this gives an example where X1 $ X2 and not for any α ∈ (0, 1), ρ αX1 + (1 − α)X $ αX2 + (1 − α)X, which violates the strong independence property. This completes the example. Remark 58 (Behavioral Finance) Behavioral finance studies financial markets where traders violate rationality. The definition of rationality in this literature is often taken to be very restrictive. Under a restrictive definition, a trader is defined to be rational if their utility function U (x) : (0, ∞) → R is state independent and has an EU representation. For a survey of this literature see Barberis and Thaler [7]. Using a more relaxed definition of rationality that includes state dependent utility functions with beliefs that can differ from statistical probabilities (as above), it can

174

9 Utility Functions

be shown that many of the “observed” behavioral biases are no longer irrational (see Kreps [134, Chapter 3]). The above violation of the strong independence axiom by a state dependent EU preference relation is one instance of this assertion. Violations of the laws of probability (e.g. Bayes law), however, remain inexplicable by the general utility framework used here. This completes the remark. Remark 59 (Sufficient Conditions for an EU Representation) Sufficient conditions ρ

on preferences # to obtain either a state dependent or a state independent EU representation are known. See Jarrow [80] and Follmer and Schied [63] for the sufficient conditions for a state independent EU representation where the choice set is over lotteries; see Kreps [134] and Mas-Colell, et al. [142] for a discussion of sufficient conditions for both state independent and state dependent EU representations over lotteries and realizations; and see Wakker and Zank [184] for sufficient conditions for state dependent EU representations. This completes the remark.

9.2.2 Additional Properties The next lemma explores some additional properties of a state dependent EU representation with respect to P. ρ

Lemma 19 (Utility Functions are Non-unique) Let # have a state dependent EU representation with respect to P. Then, for any F -measurable random variable a(ω) and constant b > 0, U ∗ (x, ω) = a(ω) + bU (x, ω)

(9.1)

is also an equivalent state dependent EU representation with respect to P. ρ

Proof (Step 1) Let EU represent # and assume that there exists a F -measurable random variable a(ω) and a constant b > 0 such that U ∗ (x, ω) = a(ω) + bU (x, ω). Then, ρ X1 $ X2 ⇐⇒ E [U (X1 , ω)] > E [U (X2 , ω)] ⇐⇒ E [a] + bE [U (X1 , ω)] > E [a] + bE [U (X2 , ω)] ⇐⇒ E [a  + U (X1 ,ω)] >E [a + U (X 2 , ω)] ⇐⇒ E U ∗ (X1 , ω) > E U ∗ (X2 , ω) . ρ

Thus, EU ∗ represents #. This completes the proof. Remark 60 (Converse of Lemma 19) The converse of Lemma 19 is true for any state ρ

ρ

independent EU representation of # where # is defined over the lottery choice set ρ

in Remark 57 and a(ω) is also a constant, i.e. if U and U ∗ represent # defined over the set of lotteries, then U ∗ (x) = a + bU (x) for some constants a ∈ R and b > 0

9.2 State Dependent EU Representation

175

gives equivalent EU representations (see Follmer and Schied [63, Chapter 2]). This completes the remark. ρ

The importance of Lemma 19 is that it implies for a given preference relation # one cannot compare utility levels across investors. That is, a utility level of 200 for trader A and 100 for trader B does not imply that trader A is twice as well off as trader B. Despite the non-uniqueness of the utility function in an EU representation, the optimal choice of a trader is invariant to the equivalent utility function used in the representation. That is, the solution Xˆ to sup E[U (X, ω)] subject to constraints X∈L0+

ρ

is the same as the solution Xˆ = arg max { # :

X ∈ L0+ } subject to the ρ

same constraints for all U satisfying the EU representation for # with respect to P. Hence, for the purposes of characterizing a trader’s optimal portfolio decision, therefore, the non-uniqueness is unimportant. Given the existence of a state dependent EU representation with respect to P, we now explore sufficient conditions on the utility function U such that the underlying ρ

preferences # are strictly monotone. A sufficient condition is that U (x, ω) is strictly increasing in x. ρ

Lemma 20 (Strictly Monotone) Suppose # has a state dependent EU representation with respect to P. Then, ρ

U strictly increasing in x implies # is strictly monotone. Proof Let X1 , X2 ∈ L0+ be such that X1 ≥ X2 a.s. P and X1 = X2 . Then, U strictly increasing in x implies that U (X1 , ω) ≥ U (X2 , ω) and U (X1 , ω) = U (X2 , ω). ρ Hence, E [U (X1 , ω)] > E [U (X2 , ω)], which implies X1 $ X2 . This completes the proof. Remark 61 (Sure Bets) With state dependent EU preferences with respect to P, a sure bet (a constant payoff) can be risky. Indeed, let X(ω) = k > 0 where k is a constant. Then, the utility function’s value for this payoff is U (k, ω), which (in general) is a random variable with var [U (k, ω)] > 0. Hence, the sure bet is a risky choice. For state dependent EU preferences, a riskless choice is any Y (ω) ∈ L0+ such that U (Y (ω), ω) = c > 0 where c is a constant. In essence, the random payoff must offset the state dependent induced randomness in the utility function. This completes the remark.

9.2.3 Risk Aversion We now explore state dependent EU representations and risk aversion.

176

9 Utility Functions ρ

Lemma 21 (Risk Aversion) Let # have a state dependent EU representation with respect to P. ρ The preference relation $ is risk averse with respect to P if and only if E [U (E [X] , ω)] > E [U (X, ω)] for all X ∈ L0+ with E [X] < ∞. ρ

Let # have a state dependent EU representation with respect to all P ∈ P. ρ The preference relation $ is uniformly risk averse if and only if E [U (E [X] , ω)] > E [U (X, ω)] for all X ∈ L0+ with E [X] < ∞ and all P ∈ P. Similar results hold for risk neutral and risk loving with “ = ” and “ < ” replacing “ > ”, respectively. ρ

Proof Given any X ∈ L0+ with E [X] < ∞ , # is risk averse if and only if ρ

ρ

1Ω (ω)E [X] $ X . Using the fact that # has a state dependent EU representation, this is equivalent to E [U (E [X] , ω)] > E [U (X, ω)] . The arguments for the uniformly risk averse, risk neutral, and risk loving cases follows similarly. This completes the proof. ρ

Next, we study the relation between risk averse preferences # with an EU representation and strictly concave utility functions. The next example shows that the state dependent utility function U being strictly increasing and strictly concave ρ

in x does not imply that preferences # are risk averse. This is in contrast to the well-known result for state independent preferences that strict concavity implies risk aversion (see Corollary 10 below). Example 11 (U (·) Strictly Increasing and Strictly Concave, but not Risk Averse with respect to P) This following is an example of a state dependent utility function U that is strictly ρ

increasing and strictly concave in x, but where preferences # are not risk averse with respect to P. Because risk aversion with respect to P is weaker than uniform risk aversion, this shows that strictly increasing and strict concavity in x does not imply it either. Let ω ∈ Ω = [0, 1], F = B[0, 1], and dP(ω) = dω, which is the uniform distribution. The mean E [ω] = 12 is equal to the median, i.e. P (ω ≤ E [ω]) = P (ω > E [ω]) = 12 . √ √ Let U (x, √ ω) = ε(2 x)1ω≤E[ω] + (2 x)1ω>E[ω]1 for ε > 0 a constant. The function 2 x is a power utility function with ρ = 2 . For all ω ∈ Ω, U (x, ω) is strictly increasing and strictly  We have  √ concave in x.

X ω ≤ E(ω) P (ω ≤ E [ω]) E [U (X, ω)] = 2εE √ 

+2E X ω > E(ω) P (ω > E [ω]).  √  √

= εE X ω ≤ E [ω] + E X ω > E [ω] . √ 

Thus, E [U (X, ω)] → E X ω > E [ω] as ε → 0.

9.2 State Dependent EU Representation

177

  √ Second, [X]E 1ω≤E[ω]  = 2ε E √ √ E [U (E [X] , ω)] +2 E [X]E 1ω>E P [ω] = (ε + 1) E [X]. √ And, E [U (E [X] , ω)] → E [X] as ε → 0. To obtain the result, take X(ω) = ω.Then, there exists a small ε > 0 such that √  √

E [U (X, ω)] = E X ω > E [ω] = E ω ω > 12 3 √ > 12 = E [X] = E [U (E [X] , ω)]. ρ

Hence, # is not risk averse with respect to P. This completes the example. The next theorem gives sufficient conditions on a state dependent utility function U ρ

such that the underlying preference relation # is risk averse with respect to P. We write the utility function’s derivative in x ∈ (0, ∞) for a given ω ∈ Ω as U  (x, ω). Theorem 45 (Sufficient Conditions for Risk Averse Preferences with Respect to ρ

P) Suppose # has a state dependent EU representation with respect to P where U is (1) continuously differentiable in x, (2) strictly increasing in x, (3) strictly concave in x, and (4)   dQX cov X, ≤0 dP

(9.2)

  is satisfied for all X ∈ L0+ with E [X] < ∞ and E U  (E [X] , ω) < ∞ where dQX U  (E [X] , ω) = > 0. dP E [U  (E [X] , ω)] ρ

Then, # is risk averse with respect to P. Proof By strict concavity (see Guler [66, p. 98]) U (X, ω) − U (E [X] , ω) < U  (E [X] , ω)(X − E [X]) Since U  > 0 is strictly increasing, U (X,ω)−U (E[X],ω) U  (E[X],ω) < E[U  (E[X],ω)] (X − E [X]). E[U  (E[X],ω)] Taking expectations yields   E[U (X,ω)]−E[U (E[X],ω)] U  (E[X],ω) < E (X − E [X])  E[U  (E[X],ω)]    E[U (E[X],ω)] dQX X = E [X] E dP [X] + cov X, dQ dP − E [X] ≤ 0   X since E dQ = 1 and hypothesis (4). dP This completes the proof.

178

9 Utility Functions

In the hypotheses of Theorem 45, condition (1) is a smoothness condition. ρ

Condition (2) guarantees that the preference relation # is strictly monotone by Lemma 20. Condition (3) was shown not to be sufficient for risk aversion. To ρ

guarantee the preference relation # is risk averse, condition (4) is needed. Condition (4), that the covariance in expression (9.2) is either negative or zero, implies that receiving the gamble X is risky to the trader. When the covariance condition is negative, the marginal utility U  (X, ω) of wealth from the gamble is large (on average) due to the state dependence on ω when the payoff to the gamble X is low  X (note that dQ dP is proportional to U (X, ω)). That is, the gamble does not pay off (on average) when wealth is needed. So it is viewed as risky to the trader. And, when the covariance condition is zero, the strict concavity of the utility function alone is sufficient to guarantee the gamble is risky. Now, since this is true for all possible gambles X, it means that all the decision maker’s choices are viewed as risky, even given the state-dependence. Thus, the trader is risk-averse for this choice set. We note that a strictly increasing utility function implying more wealth is preferred to less is also needed for Theorem 45, and it is this monotonicity in X wealth that enables the interpretation of dQ dP as the trader’s state price probability density. This condition is weaker than state independence of the utility function, which yields a similar conclusion when evaluating the risk of any possible wealth X. ρ

Corollary 10 (State Independent Utility) Suppose # has a state independent EU representation with respect to P where U is (1) continuously differentiable in x, (2) strictly increasing in x, (3) strictly concave in x. ρ

Then, # is risk averse respect to P.

  U  (E[X]) dQX X = 0. Proof Note that dQ dP = E[U  (E[X])] = 1 for all ω ∈ Ω. Hence, cov X, dP This completes the proof.

9.3 Strict Concavity and Risk Aversion This section characterizes the economic meaning of strict concavity for preferences with an EU representation with respect to P. To do this, we need to introduce some new definitions, in particular, the notion of independent gambles. This section is based on Jarrow and Li [104].

9.3 Strict Concavity and Risk Aversion

179

9.3.1 Independent Gambles To introduce independent gambles into the choice set, we expand the probability ˜ and let the set of non-trivial probability measures ˜ F × F˜ , P × P) space to (Ω × Ω, ˜ i.e. all P × P˜ such that there exists an A ∈ F × F˜ with be denoted by P × P,     ˜ Note that P × P˜ (A) > 0 and P × P˜ (Ac ) > 0 where A ∪ Ac =  × Ω. the product of the two probability measures that events in Ω and Ω˜ are  implies  ˜ independent, i.e. for A ∈ F and B ∈ F˜ , P × P˜ (A ∩ B) = P(A)P(B). ˜ F × F˜ ). Two subsets The choice set is also expanded to X(ω, ω) ˜ ∈ L0+ (× Ω, of this new choice set are importance: the original choice set X(ω) ∈ L0+ (, F ) ˜ F˜ ). These are the subsets where X(ω) = X(ω, ω) ˜ and the gambles Y (ω) ˜ ∈ L0+ (Ω, ˜ is constant across ω˜ ∈ Ω and Y (ω) ˜ = Y (ω, ω) ˜ is constant across ω ∈ Ω. ˜ F × F˜ ) be a nonnegative constant. This is a sure bet. The Let k ∈ L0+ ( × Ω, utility function’s value for this sure bet is U (k, ω), a random variable due to the ˜ F × F˜ ) is said to state dependence. An arbitrary payoff X(ω, ω) ˜ ∈ L0+ ( × Ω, be an independent gamble if X(ω, ω) ˜ and U (k, ω) are independent with respect to ˜ Intuitively, the gamble X is an independent gamble the probability measure P × P. ˜ F˜ ) are if it is independent of the utility function’s argument ω ∈ . The set L0+ (Ω, independent gambles. Remark 62 (State Independent Utility Functions) For state independent utility functions, the notion of an independent gamble trivializes. Here, the utility function U (k, ω) = U (k) is independent of all the elements in the choice set L0+ ( × ˜ F × F˜ ). Hence, the entire choice set consists of independent gambles. This Ω, completes the remark. All of the previous definitions for risk aversion and uniform risk aversion extend by replacing the probability space and choice set (Ω, F , P), L0+ (, F ) with (Ω × ˜ L0+ ( × Ω, ˜ F × F˜ , P × P), ˜ F × F˜ ). Ω,

9.3.2 Risk Aversion To understand the economic interpretation of strict concavity, we need to define a weaker version of risk aversion. ρ

Definition 37 (Risk Aversion for Independent Gambles) Let  have a state ˜ dependent expected utility representation with respect to P × P. ρ

The preference  is risk averse for independent gambles with respect to P × P˜ if ˜

ρ

E P [Y ] $ Y,

180

9 Utility Functions

˜ F˜ ), Y not a constant, with E P˜ [Y ] < ∞. for all Y ∈ L0+ (Ω, ρ

The preference  is uniformly risk averse for independent gambles if it is risk ˜ averse for independent gambles with respect to all P × P˜ ∈ P × P. This notion of risk aversion is weaker because the choice set is restricted to the set ˜ F˜ ). The following lemma follows directly from the of independent gambles L0+ (Ω, above definitions. ρ

Lemma 22 (Risk Aversion for Independent Gambles) Let  have a state depen˜ dent expected utility representation with respect to P × P. ρ

The preference relation  is risk averse for independent gambles with respect to P × P˜ if and only if ˜

˜

E[U (E P [Y ], ω)] > E P×P [U (Y (ω), ˜ ω)] ˜ F˜ ), Y not a constant, with E P˜ [Y ] < ∞ and E[U (E P˜ [Y ], ω)] < for all Y ∈ L0+ (Ω, ∞. ρ

The preference relation  is uniformly risk averse for independent gambles if the ˜ previous inequality holds for all P × P˜ ∈ P × P.

9.3.3 Characterization Theorems We can now prove a characterization theorem for strict concavity of the utility function. ρ

Theorem 46 (Characterization of Strict Concavity) Let  have a state depen˜ dent expected utility representation U with respect to P × P. The following statements are equivalent: (1) U (x, ω) is pointwise strictly concave; ρ

(2)  is uniformly risk averse for independent gambles. Proof (1) implies (2). ˜ F˜ ) with E P×P˜ [|U (Y (ω), For Y (ω) ˜ ∈ L0+ (Ω, ˜ ω)|] < ∞, we have 

 ˜ ˜ ˜  ˜ ω) F ˜ ω)] = E P×P E P×P U (Y (ω), E P×P [U (Y (ω),    ˜ ˜ = E P×P E P U (Y (ω), ˜ ω)   ˜ ˜ ˜ ω) < E P×P U (E P [Y (ω)],   ˜ ˜ ω) . = E U (E P [Y (ω)],

9.3 Strict Concavity and Risk Aversion

181

Note that the first equality is the tower rule for conditional expectations. The second equality follows by the independence of Y from F . The third inequality follows from pointwise strict concavity for a fixed ω ∈ Ω. And, the fourth equality is due to independence again. (2) implies (1). Let Y (ω) ˜ be F˜ -measurable. Let P × P˜ = λδω×ω˜ 1 + (1 − λ)δω×ω˜ 2 where δω×ω˜ i equals 1 if ω × ω˜ = ω × ω˜ i and zero elsewhere for i = 1, 2 and λ ∈ (0, 1). By uniform risk aversion for independent gambles over all non-trivial probability measures, ˜

˜

E[U (E P [Y (ω)], ˜ ω)] > E P×P [U (Y (ω), ˜ ω)]. This implies U (λY (ω˜ 1 ) + (1 − λ)Y (ω˜ 2 ), ω) > λU (Y (ω˜ 1 ), ω) + (1 − λ)U (Y (ω˜ 2 ), ω). Since Y (ω˜ 1 ) ≥ 0 and Y (ω˜ 2 ) ≥ 0 are arbitrary and ω × ω˜ 1 and ω × ω˜ 2 are also arbitrary, the state dependent utility function is pointwise strictly concave. This completes the proof. We can also characterize when state independent EU preferences arise. ρ

Theorem 47 (A Characterization of State Independent Preferences) Let  ˜ have a state dependent expected utility representation U (x, ω) with respect to P×P. Let U (x, ω) is continuously differentiable, strictly increasing, and pointwise strictly concave. The following statements are equivalent: (1) U (x, ω) satisfies the covariance condition (9.2) for all probability measures P × P˜ ∈ P; ρ

(2)  is uniformly risk averse; (3) U (x, ω) is state independent. Proof (1) implies (2) follows from the definition of uniform risk aversion and Theorem 45. (3) implies (1) follows from Corollary 10. We now show that (2) implies (3). The is a proof by contradiction. Suppose U (x, ω) is state dependent. For simplicity we restrict consideration to a gamble X ∈ L0+ (Ω, F ), hence, we need only consider the probability space (Ω, F , P). By the non-uniqueness of utility function, we can normalize it such that U (0, ω) = 0 for all ω ∈ Ω. By assumption, there exists ω1 = ω2 ∈ Ω such that U (x0 , ω1 ) > U (x0 , ω2 ) for some x0 > 0, and thus there exists xˆ ∈ (0, x0 ) such that ˆ ω1 ) > U  (x, ˆ ω2 ), U  (x,

182

9 Utility Functions

because if U  (x, ω1 ) ≤ U  (x, ω2 ) holds for all x ∈ (0, x0 ], then we get a contradiction from  x0  x0  U (x, ω1 )dx ≤ U  (x, ω2 )dx = U (x0 , ω2 ). U (x0 , ω1 ) = 0

0

Note that U (x, ω) is continuously differentiable in x. By the continuity of U  (x, ω), there exists an open ball Bε (x) ˆ such that for all x ∈ Bε (x), ˆ U  (x, ω1 ) > U  (x, ω2 ).

(9.3)

Now, take X(ω1 ) = xˆ +δ, X(ω2 ) = xˆ −δ, P(ω1 ) = P(ω2 ) = 12 and thus E[X] = x. ˆ Here, we take δ > 0 small enough so that xˆ ±δ ∈ Bε (x). ˆ A simple calculation shows E[U (E[X], ω)] =

1 1 U (x, ˆ ω1 ) + U (x, ˆ ω2 ), 2 2

and E[U (X(ω), ω)] =

1 1 U (xˆ + δ, ω1 ) + U (xˆ − δ, ω2 ), 2 2

which further yields

1 U (xˆ + δ, ω1 ) − U (x, ˆ ω1 ) 2

1 U (xˆ − δ, ω2 ) − U (x, + ˆ ω2 ) 2 1 1 = U  (x1 , ω1 )δ − U  (x2 , ω2 )δ 2 2

E[U (X(ω), ω)] − E[U (E[X], ω)] =

where x1 , x2 ∈ Bε (x) ˆ is guaranteed by the Mean Value theorem. By Eq. (9.3), this implies E[U (X(ω), ω)] > E[U (E[X], ω)].

(9.4)

Hence, the utility function is not risk averse for this P and X ∈ L0+ (Ω, F ) as defined above. This is the desired contradiction, which completes the proof.

9.4 Measures of Risk Aversion for Independent Gambles This section studies measures of risk aversion for state dependent utility functions. ˜ F × F˜ , P × P˜ ) and We continue to use the expanded probability space (Ω × Ω, ˜ F × F˜ ) from the previous section. For state dependent the choice set L0+ ( × Ω,

9.4 Measures of Risk Aversion for Independent Gambles

183

utility functions, as shown above, risk aversion only holds under strong hypotheses. In fact, Theorem 47 above indicates that these stronger measures of risk aversion really only apply to state independent utility functions. This is in contrast, however, to the weaker notions of risk aversion for independent gambles. Consequently, this section explores measures of risk aversion for independent gambles. The importance of risk aversion measures is that they are comparable across traders. ρ

Definition 38 (A Risk Aversion Measure for Independent Gambles) Let # have a state dependent EU representation with respect to P × P˜ with U continuously ˜ F˜ ) with E P˜ [Y ] < differentiable in x. For a given independent gamble Y ∈ L0+ (Ω, ˜

∞ and E P [U (Y, ω)] < ∞ for all ω ∈ Ω a.s. P, a measure of risk aversion for independent gambles given ω with respect to P × P˜ is ˜

R(E P [Y ] , ω) =

˜

˜

U (E P [Y ] , ω) − E P [U (Y, ω)] U  (E P˜ [Y ] , ω)

(9.5)

for all ω ∈ Ω a.s. P. This is a measure of risk aversion for independent gambles because the difference ˜ ˜ U (E P [Y ] , ω) − E P [U (Y, ω)] in the numerator, for a given ω ∈ Ω measures ˜ the difference in utility due to taking E P [Y ] for sure versus taking the gamble 0 ˜ F˜ ). The normalization by U  (E P˜ [Y ] , ω) in the denominator makes Y ∈ L+ (Ω, this measure invariant with respect to a positive affine transformation, i.e. for any U (x, ω) = a(ω) + bU ∗ (x, ω) with a(ω) F -measurable and b > 0. This implies ˜ that R(E P [Y ] , ω) is unique for any trader (it represents a single element for the class of equivalent state dependent utility functions), and its magnitude is therefore ˜ ˜ comparable across traders. That is, given ω, if R1 (E P [Y ] , ω) > R2 (E P [Y ] , ω) for trader 1 versus trader 2, then trader 1 is more risk averse with respect to independent gambles. We can relate this measure of risk aversion to others in the literature. We start with the absolute risk aversion coefficient. ρ

Lemma 23 (Absolute Risk Aversion) Suppose # has a state dependent EU representation with respect to P × P˜ where U is (1) twice continuously differentiable in x, (2) strictly increasing in x, and (3) strictly concave in x. Then, ARA(x, ω) = −

U  (x, ω) >0 U  (x, ω)

is another measure of risk aversion for independent gambles with respect to P × P˜ given ω ∈ Ω.

184

9 Utility Functions

Proof First note that ARA(x, ω) is invariant to a positive affine transformation as defined above. Let Y (ω) ˜ be an independent gamble. ˜ To simplify the notation, let E P [Y ] = x. Using a Taylor series expansion, ˜ − x) U (Y (ω), ˜ ω) = U (x, ω) + U  (x, ω)(Y (ω) 2 (Y (ω)−x) ˜  ˜ ω) for ξ(ω) ˜ ∈ [Y (ω), ˜ x]. +U (ξ(ω), 2 Rearranging terms and taking expectations with respect to P˜ yields   ˜ ˜  ˜  U (E P[Y ] , ω) − E P U (Y (ω), ˜ ω) = U  (x, ω)E P x − Y (ω) ˜ ˜

˜ . ˜ ω) (Y (ω)−x) +E P −U  (ξ(ω), 2   ˜ P ˜ = 0, implying But, E x − Y (ω)    2 ˜ ˜  ˜ ˜ P ˜ ω) = E P −U  (ξ(ω), ˜ ω) (Y (ω)−x) U (E [Y ] , ω) − E P U (Y (ω), 2   2 ˜ ˜ For small E P (Y (ω)−x) , one can replace ξ(ω) ˜ with x in the above expression 2 to obtain      2 ˜ ˜ ˜  ˜ ˜ E P U (E P [Y ] , ω) − E P U (Y (ω), ˜ ω) = −U  (x, ω)E P (Y (ω)−x) 2 2

Divide both sides by U  (x, ω), and using the definition of R(x, ω) gives    (x,ω) 1 P˜ Y (ω) R(x, ω) = − UU  (x,ω) ˜ . 2 var This completes the proof.

Another common measure of risk aversion is called relative risk aversion, and it is defined as RRA(x, ω) = x · ARA(x, ω). The inverse of the absolute risk aversion measure is called risk tolerance and denoted RT (x, ω) =

1 . ARA(x, ω)

Examples of utility functions satisfying the above assumptions include the following state independent utility functions. Example 12 (State Independent Utility Functions) (logarithmic) U (x) = ln(x) for x > 0. This implies U  (x) = x1 > 0 for x > 0 and (U  )−1 (y) = Note that U  (x) = − x12  (x) = x1 ARA(x) = − UU  (x)

< 0.

> 0. RT (x) = x > 0  (x) RRA(x) = − xU U  (x) = 1.

(power) U (x) =

xρ ρ

> 0 for x > 0 and ρ < 1, ρ = 0.

1 y

for y > 0.

9.5 State Dependent Utility Functions

185 1

This implies U  (x) = x ρ−1 for x > 0 and (U  )−1 (y) = y ρ−1 for y > 0. Note that U  (x) = (ρ − 1)x ρ−2 < 0. ρ−2  (x) = − (ρ−1)x = 1−ρ ARA(x) = − UU  (x) x > 0. x ρ−1 x RT (x) = 1−ρ >0 

(x) RRA(x) = − xU U  (x) = 1 − ρ > 0.

(exponential) U (x) = −e−αx < 0 for x > 0 and α > 0.

This implies U  (x) = αe−αx for x > 0 and (U  )−1 (y) = − α1 log αy for y > 0. Note that U  (x) = −α 2 e−αx < 0.  (x) 2 e−αx = − −α = α > 0. ARA(x) = − UU  (x) αe−αx RT (x) =

1 α

>0



(x) RRA(x) = − xU U  (x) = αx > 0. This completes the example.

Note that all of these utility functions’ risk tolerances are linear in x. This leads to a wider class of state independent utility functions with linear risk tolerance. Example 13 (State Independent Linear Risk Tolerance Utility Functions) Any state independent utility function U with RT (x) = d0 + d1 x for d0 , d1 ∈ R is said to have linear risk tolerance. It can be shown that this class of utility functions satisfy   1 dx U (x) = c1 + c2 e d0 +d1 x for c1 , c2 ∈ R. Proof Define Let U (x) dln(U (x)) dx

−U  (x) U  (x) = A(x).  (x) = U  (x). Then, −UU(x)

= A(x).

= A(x). Computing the indefinite integral yields ln(U (x)) = c0+ A(x)dx for c0 ∈ R. Or, U  (x) = ec0 + A(x)dx = c2 e A(x)dx for c2 ∈ R. Computing the indefinite integral again gives   U (x) = c1 + c2 e A(x)dx for c1 ∈ R. 1 Since A(x) = RT1(x) = d0 +d for linear risk tolerance, substitution completes 1x the proof. This completes the example.

9.5 State Dependent Utility Functions For the subsequent chapters, we return to the original probability space (Ω, F , P) excluding independent gambles and we assume that the trader’s preference relation ρ

# has a state dependent EU representation with respect to P. We will always invoke the following assumption.

186

9 Utility Functions

Assumption (State Dependent Utility Function) The state dependent utility function U : (0, ∞) × Ω → R satisfies the following: for all ω ∈ Ω a.s. P, (i) (ii) (iii) (iv)

U (x, ω) is B(0, ∞) ⊗ FT measurable, U (x, ω) is continuous and differentiable in x ∈ (0, ∞), U (x, ω) is strictly increasing and strictly concave in x ∈ (0, ∞), and (Inada Conditions) lim U  (x, ω) = ∞ and lim U  (x, ω) = 0. x→∞

x↓0

Condition (i) is a necessary measurability condition. Condition (ii) allows the use ρ

of calculus to find an optimal solution. Condition (iii) implies that preferences # are strictly monotone (see Lemma 20). Condition (iii) also states that the utility function is strictly concave, which implies risk aversion for independent gambles, which is ρ

weaker than assuming the investor’s preferences # are risk averse (see Theorem 45). The Inada conditions (iv) guarantee an interior solution to the subsequently defined utility maximization problem (in Chaps. 10 and 12). The state dependent utility function U : (0, ∞) × Ω → R will be defined over terminal wealth (or consumption in Chap. 12) denominated in units of the mma. The fact that this utility function is state dependent implies that this representation of a trader’s preferences is very general (see Kreps [134, Chapter 3]). It includes, as special cases, recursive utility functions and habit formation (see Back [5, Chapter 21] and Skiadas [179, Chapter 6]). Examples of utility functions that satisfy the above assumption follow. Example 14 (State Independent Utility Functions) (logarithmic) U (x) = ln(x) for x > 0. This utility function satisfies the assumption. Indeed, from Example 12 above, we have that properties (i)–(iii) are satisfied. To show property (iv), note that U  (x) = 1   x > 0 for x > 0 implies lim U (x, ω) = ∞ and lim U (x, ω) = 0. x→∞

x↓0

ρ

(power) U (x) = xρ > 0 for x > 0 and ρ < 1, ρ = 0. This utility function satisfies the assumption. Indeed, from Example 12 above, we have that properties (i)–(iii) are satisfied. To show property (iv), note that U  (x) = x ρ − 1 for x > 0 implies lim U  (x, ω) = ∞ and lim U  (x, ω) = 0. x↓0

x→∞

(exponential) U (x) = −e−αx < 0 for x > 0 and α > 0. This utility function violates the assumption. From Example 12 above, we have that properties (i)–(iii) are satisfied. But, property (iv) is violated. Note that U  (x) = αe−αx for x > 0. This gives lim U  (x, ω) = 0 but lim U  (x, ω) = α < ∞. x→∞

x↓0

This completes the example. We will need the following properties of the utility function’s derivative in x ∈ (0, ∞) for a given ω ∈ Ω. Lemma 24 (Properties of U  ) For every ω ∈ Ω a.s. P, (i) U  (x, ω) : (0, ∞) × Ω → R is strictly decreasing in x ∈ (0, ∞), and

9.6 Conjugate Duality

187

−1 (ii) the inverse function, I (y, ω) = U  (y, ω) : (0, ∞) × Ω → R, exists, it is B(0, ∞) ⊗ FT measurable, and it is strictly decreasing in y ∈ (0, ∞) with I (0, ω) = ∞ and I (∞, ω) = 0. Proof (i) Strictly decreasing is implied by strict concavity. (ii) These properties follow directly from the properties of U  . This completes the proof. We note that this lemma implies (suppressing the ω ∈ Ω notation), by the definition of an inverse function, that U  (I (y)) = y for 0 < y < ∞, I (U  (x)) = x for 0 < x < ∞.

and

(9.6)

Remark 63 (Money Market Account Numeraire) In this utility function formulation, wealth (or consumption in Chap. 12) is normalized by the mma’s value BT . This is without loss of generality. Indeed, suppose instead that one postulates the utility function is defined over wealth denominated in units of a single perishable consumption good, U c (WT , ω), where WT (ω) denotes time T wealth in units of the consumption good. Let ψT (ω) > 0 be dollars per consumption good at time T . Then, nominal wealth is XT (ω) = WT (ω)ψT (ω). The above formulation considers time T dollar wealth in units of the mma, T (ω) T defined by XT = BTX(ω) = WBT Tψ(ω) . Then, because the utility function is state dependent, we can define an equivalent utility function over units of the mma   BT (ω) c (W , ω). The utility function U (·, ω) , ω = U by U (XT , ω) := U c XT ψ T T (ω) BT (ω) defined over wealth in units of the mma embedds the ratio ψ in the definition T (ω) of U . This utility function U (·, ω), after the appropriate transformation, will yield the same optimal trading strategy and terminal wealth, in units of the consumption good, as the utility function U c defined over wealth using the consumption good as the numeraire. This completes the remark.

9.6 Conjugate Duality To solve the trader’s optimization problem in Chaps. 10–12 using convex optimization, we need to define the convex conjugate of the utility function U . Definition 39 (Convex Conjugate) The convex conjugate of U is defined by U˜ (y, ω) = sup [U (x, ω) − xy], x>0

y > 0.

(9.7)

188

9 Utility Functions

We state without proof the following properties of this convex conjugate. This lemma can be found in Mostovyi [151, p. 139]. Lemma 25 (Properties of the Convex Conjugate) For all ω ∈ Ω a.s. P, (i) U˜ (y, ω) is B(0, ∞) ⊗ FT measurable, differentiable, decreasing, and strictly convex in y ∈ (0, ∞) with U˜ (0, ω) = U (∞, ω), (ii) the derivative is U˜  (y, ω) = −(U  )−1 (y, ω) = −I (y, ω),

y > 0,

(iii) the supremum in expression (9.7) is attained at x = I (y, ω), i.e. U˜ (y, ω) = U (I (y, ω), ω) − yI (y, ω),

y > 0,

(iv) the biconjugate relation holds U (x, ω) = inf

  U˜ (y, ω) + xy ,

x>0

(9.8)

y>0

with the infimum attained at y = U  (x, ω), i.e. U (x, ω) = U˜ (U  (x, ω), ω) + xU  (x, ω),

x > 0.

Example 15 (Convex Conjugate Functions) This example uses the state independent utility functions in Example 14. (logarithmic) U (x) = ln(x) for x > 0, gives U˜ (y) = −lny − 1. Proof U˜ (y) = sup [ln(x) − xy]. Setting the first derivative equal to zero yields x>0

− y = 0. This implies x = y1 . This is the unique maximum since ln(x) − xy is strictly concave for all y > 0 (the second derivative is strictly negative). Substitution yields U˜ (y) = ln y1 − 1. Algebra completes the proof. 1 x

(power) U (x) =

xρ ρ

for x > 0 and ρ < 1, ρ = 0, gives 

y U˜ (y) = 

ρ 1−ρ

ρ 1−ρ



.

ρ Proof U˜ (y) = sup [ xρ − xy]. Setting the first derivative equal to zero yields

x>0

x ρ−1

− y = 0. This implies x = y



1 ρ−1



. This is the unique maximum since

9.7 Reasonable Asymptotic Elasticity xρ ρ

− xy is strictly concave for all y > 0 (the second derivative is strictly negative). 

Substitution yields U˜ (y) = 

ρy

189

ρ ρ−1

ρ



y

ρ ρ−1

ρ





−y

1 ρ−1





y. Algebra gives U˜ (y) =

y

ρ ρ−1

ρ



−

. Simplification completes the proof.

9.7 Reasonable Asymptotic Elasticity The next assumption is needed as a sufficient condition to guarantee the existence of a solution to the investor’s portfolio problem as defined in Chaps. 10–12 below. It is ”almost” necessary in that, if it is not satisfied, examples can be found where the investor’s portfolio optimization problem has no solution (see Kramkov and Schachermayer [132]). Assumption (Reasonable Asymptotic Elasticity (AE(U) < 1)) Let U be a state dependent utility function. We assume that for all ω ∈ Ω a.s. P, U satisfies AE(U, ω) = limsup x→∞

x U  (x, ω) < 1. U (x, ω)

(9.9)

 For a given ω ∈ Ω, this assumption relates the ; marginal utility U (x) to the U (x)  average utility x for large x via the ratio U (x) (U (x) /x ) . The condition that AE(U,ω) < 1 states that for large enough wealth x, the marginal utility must be less than the average utility. If AE(U) < 1 is violated, then as wealth gets large, a trader’s marginal utility exceeds or equals their average utility, implying more wealth is desired and can be attained. Intuitively, an optimal wealth may not exist for such a utility function. An alternative characterization of the reasonable asymptotic elasticity assumption further clarifies its meaning. This characterization is based on the utility func (x,ω) tion’s asymptotic relative risk aversion, defined as ARRA(U, ω) = − lim xU U  (x,ω) , x→∞

recall that RRA(x, ω) was discussed in Sect. 9.4 above. Theorem 48 (Characterization of AE(U) < 1) Suppose U  (x, ω) is differentiable on x ∈ (0, ∞) and ARRA(U, ω) exists a.s. P. Then, AE(U, ω) exists a.s. P and ARRA(U, ω) > 0 if and only if AE(U, ω) < 1. Proof This proof is from Schachermayer [171]. Fix a ω ∈ Ω. We suppress the ω argument in the following expressions. (Step 1) Suppose U (∞) < ∞. Then, for x0 > 0, 0 ≤ xU  (x) = (x − x0 )U  (x) + x0 U  (x). Hence, 0 ≤ lim sup xU  (x) = lim sup (x − x0 )U  (x) + lim sup x0 U  (x). x→∞

x→∞

x→∞

190

9 Utility Functions

But, lim U  (x) = 0. And, by the mean value theorem, x→∞

U (x) − U (x0 ) = U  (z)(x − x0 ) for some z ∈ [x0 , x]. Since U  is decreasing, this implies U  (x)(x − x0 ) ≤ U(x) − U (x0 ). 

Hence, lim sup xU  (x) ≤ lim sup (U (x) − U (x0 )) = lim sup U (x) −U (x0 ). x→∞

x→∞

x→∞

This holds for all x0 → ∞, which implies that lim sup xU  (x) = 0. x→∞

(Case a) If U (∞) = 0, then applying L’Hospital’s gives  rule  xU  (x) xU  (x) U  (x)+xU  (x) lim U (x) = lim − = 1 − lim   U (x) U (x) . x→∞ x→∞ x→∞ This completes the proof for Case (a).  (x) (Case b) If U (∞) = 0, then lim xU U (x) = 0. In this case, define x→∞

U˜ (x) = U (x) − U (∞). Then, U˜ (∞)  = 0. This is Case (a), thus x U˜  (x) x U˜  (x) . Substitution gives = 1 − lim − ˜  lim U (x) x→∞ U˜ (x) x→∞     (x) xU (x) xU (x) − lim xU lim U (x)−U = 1 − lim  (∞) U (∞) = 0, hence U (x) . But, x→∞ x→∞ x→∞    (x) xU  (x) − = 1 − lim lim xU  U (x) . x→∞ U (x) x→∞ This completes the proof for Case (b). (Step 2) Suppose U (∞) = ∞. For x ≥ 1, x   dz = U  (1) + x (U  (z) + zU  (z))dz xU  (x) = U  (1) + 1 (zU  (z)) 1  x = U  (1) + U (x) − U (1) + 1 zU  (z)dz. So, xU  (x) U (x)



(1) = UU (x) +1− Taking limits

U (1) U (x)

+

x 1

zU  (z)dz . U (x)

x zU  (z)dz U  (1) U (1) + 1 − lim U + lim 1 U (x) U (x) (x) x→∞ x→∞  x x→∞  1 zU (z)dz = 1 + lim . U (x) x→∞ x  (Casea) If 1 zU (z)dz = ∞, then applying L’Hospital’ rules x  (x) zU  (z)dz = lim xU . lim 1 U (x)  x→∞ x→∞ U (x) xU  (x) x→∞ U (x)

lim

= lim

gives

   (x) xU  (x) − = 1 − lim Hence, lim xU  U (x) . x→∞ U (x) x→∞ This completes the proof of Case (a). x ∞ zU  (z)dz (Case b) If 1 zU  (z)dz < ∞, then lim U1 (x)−U (1) = lim

x zU  (z)dz 1 x x→∞ 1 U  (z)dz

= 0.

x→∞ x   (ξ )(x−1) 1 zU (z)dz  By the mean value theorem, x U  (z)dz = ξUU (δ)(x−1) for some δ, ξ ∈ [1, x]. 1 x  (z)dz  zU (ξ ) Let η = ξδ . Then, 1 x U  (z)dz = ξUU (ηξ ) . Taking the limit as ξ → ∞ for fixed η, 1  x x   zU (z)dz zU (z)dz we get lim 1 x U  (z)dz = lim 1 x U  (z)dz and x→∞ 1 ξ →∞ 1 ξ U  (ξ )  ξ →∞ U (ηξ )

lim

=

ξ U  (ξ ) .  ξ →∞ U (ξ )

lim

completes the proof of Case (b). This completes the proof.

ξ U  (ξ )  ξ →∞ U (ξ )

Combined these give lim

= 0, which

9.8 Differential Beliefs

191

In terms of relative risk aversion, the reasonable asymptotic elasticity assumption states that as wealth becomes infinitely large, the investor remains risk averse. Example 16 (AE(U ) < 1 and AE(U ) = 1) Applying this definition to the state independent utility functions in Example 14 shows that they satisfy the reasonable asymptotic elasticity assumption. (logarithmic) U (x) =ln(x)for x > 0.   x· x1 1 AE(U ) = limsup ln(x) = limsup ln(x) = 0. This utility function satisfies x→∞

AE(U ) < 1. (power) U (x) =

x→∞

xρ

for x > 0 and ρ < 1, ρ = 0.   ρ−1 = limsup ρ = ρ < 1. This utility function satisfies AE(U ) = limsup x·xx ρ ρ

x→∞

x→∞

ρ

AE(U ) < 1. Here is an example that violates the reasonable asymptotic elasticity condition. x (violates AE(U ) < 1) U (x) = lnx for x > 0.   1 1 x   x ln(x) − (lnx)2 x 1 AE(U ) = limsup = 1. = limsup 1 − lnx x x→∞

ln(x)

x→∞

This completes the example.

9.8 Differential Beliefs This section discusses the changes to the utility function framework needed when we introduce differential beliefs in subsequent chapters, where the trader’s beliefs differ from the statistical probability measure P. Suppose that P represents the statistical probability measure and Pi represents the trader’s beliefs where Pi and i P are equivalent probability measures. Thus dP dP > 0 is a FT measurable random   i variable and E dP dP = 1. It will be convenient to consider the modified utility function U (x, ω) =

dPi U (x, ω). dP

(9.10)

i Because dP dP > 0, U inherits all of the assumed properties of U and those given in Lemma 24. For example, the first derivative is

U  (x, ω) =

dPi  U (x, ω) > 0. dP

(9.11)

The convex conjugate of the modified utility function is   dPi ˜ dP ˜ . U (y, ω) = sup [U (x, ω) − xy] = U y dP dPi x>0

(9.12)

192

9 Utility Functions

Proof U˜ (y, ω) = sup [U (x, ω) − xy] x>0   i U (x, ω) − xy = sup dP dP x>0   dPi dP i U (x, ω) − xy = sup dP dP dP dPi x>0     dPi dP dP i ˜ = dP sup U (x, ω) − xy dPi = dP dP U y dPi . x>0

This completes the proof.

9.9 Notes The characterization of an trader’s preferences is a well studied topic in both economics and statistical decision theory. A classical book is Fishburn [61]. For excellent presentations in microeconomic texts see Kreps [134] and Mas-Colell, Whinston, and Green [142]; in finance texts see Back [5], Follmer and Schied [63], and Skiadas [179]; and in statistical decision theory texts see Berger [11] and DeGroot [43].

Chapter 10

Complete Markets (Utility Over Terminal Wealth)

This chapter studies an individual’s portfolio optimization problem. In this optimization, the solution differs depending on whether the market is complete or incomplete. This chapter investigates the optimization problem in a complete markets setting, and the next chapter analyzes incomplete markets.

10.1 The Set-Up Given is a normalized market (S, F, P) where the value of the money market account Bt = 1 for all t ≥ 0. We assume that there are no arbitrage opportunities in the market, i.e. Assumption (NFLVR)

Ml = ∅ where Ml = {Q ∼ P : S is a Q local martingale}.

This assumption is imposed for two reasons. One, an optimal solution will not exist to the trader’s portfolio problem if NFLVR is violated (see the proof of Theorem 63 in Chap. 13). Two, this assumption enables a characterization of the trader’s budget constraint that facilitates solving the problem using convex optimization. As mentioned in the introduction to this chapter, we assume that the market is complete. Assumption (Complete Markets) Choose a Q ∈ Ml . The market (S, F, P) is complete with respect to Q. By the Second Fundamental Theorem 14 in Chap. 2, this implies that the set of equivalent local martingale measures is a singleton, i.e. Ml = {Q} . Remark 64 (Stochastic Processes in a Complete Market) The assumption of a complete market with respect to Q restricts the class of semimartingale processes © Springer Nature Switzerland AG 2021 R. A. Jarrow, Continuous-Time Asset Pricing Theory, Springer Finance, https://doi.org/10.1007/978-3-030-74410-6_10

193

194

10 Complete Markets (Utility Over Terminal Wealth)

S possible for the risky assets in the market. By the definition of a complete market (see Chap. 2), the stochastic process for S must be such that synthetic construction of a arbitrary Q integrable random time T payoff is possible where the resulting value process of the admissible s.f.t.s. that generates the payoff must be a Q martingale. Not all stochastic processes for S satisfy this restriction. For example, in the finite dimensional Brownian motion market of Sect. 2.8 in Chap. 2, the volatility matrix of the risky asset price processes must have a rank equal to the number of Brownian motions underlying the risk asset price process. When S has a discontinuous sample path process, this restriction implies that the probability density of the jump amplitude must be discrete with the number of possible jump magnitudes less than or equal to the number of traded risky assets (see Cont and Tankov [34, Chapter 9.2]). This completes the remark. Denote the probability density function of the equivalent local martingale measure by YT = dQ dP ∈ Ml ⊂ Dl . YT is the unique local martingale deflator Ml . Recall that  Dl = Y ∈ L+0 : Y0 = 1, XY is a P local martingale,  X = 1 + α · dS, (α0 , α) ∈ A (1) Dl = {YT ∈ L0+ : ∃Z ∈ Dl , YT = ZT }   dQ Ml = Y ∈ Dl : ∃Q ∼ P, YT = dP Ml = {YT ∈ L0+ : ∃Z ∈ Ml , YT = ZT } (see Chap. 2 for a discussion of local martingale deflators). Note that because YT = dQ dP , E[YT ] = 1 and Yt = E [YT |Ft ] ∈ Ml is a P martingale.

10.2 Problem Statement This section defines the investor’s wealth optimization problem. We assume that the ρ

investor’s preferences # have a state dependent EU representation with respect to P where the utility function is defined over terminal wealth U (XT , ω) and satisfies the assumption given in Chap. 9, repeated here for convenience. Assumption (Utility Function) The utility function U : (0, ∞) × Ω → R is such that for all ω ∈ Ω a.s. P, (i) (ii) (iii) (iv)

U (x, ω) is B(0, ∞) ⊗ FT measurable, U (x, ω) is continuous and differentiable in x ∈ (0, ∞), U (x, ω) is strictly increasing and strictly concave in x ∈ (0, ∞), and (Inada Conditions) lim U  (x, ω) = ∞ and lim U  (x, ω) = 0. x↓0

x→∞

10.2 Problem Statement

195

We do not consider utility over intermediate consumption in this chapter. This simplification is imposed for pedagogical reasons. Indeed, if one understands the solution to the portfolio optimization problem for utility over terminal wealth, then the solution to the portfolio optimization problem with intermediate consumption is easier to understand and to implement. The solution to the portfolio optimization problem with utility defined over both intermediate consumption and terminal wealth is discussed in Chap. 12. Because the complete market setting is a special case of the solution given in Chap. 12, there will not be a separate chapter studying complete markets with utility defined over intermediate consumption and terminal wealth. Recall that the set of admissible s.f.t.s. is denoted A (x) = {(α0 , α) ∈ (O, L (S)) : Xt = α0 (t) + αt · St , ∃c ≤ 0, t Xt = x + 0 αu · dSu ≥ c, ∀t ∈ [0, T ] . For the investor’s optimization problem, we restrict consideration to admissible s.f.t.s. where the lower bound c = 0, i.e. admissible s.f.t.s.’s where the wealth process is always nonnegative. This is the set of nonnegative wealth s.f.t.s. with initial wealth x ≥ 0, denoted N (x) = {(α0 , α) ∈ (O, L (S)) : Xt = α0 (t) +  αt · St , t Xt = x + 0 αu · dSu ≥ 0, ∀t ∈ [0, T ] . Note that N (x) ⊂ A (x) where x ≥ 0. To facilitate solving the investor’s optimization problem, we need to define two new sets. The first is the set of wealth processes generated by the nonnegative wealth s.f.t.s. with x ≥ 0 denoted    t X (x) = X ∈ L+0 : ∃(α0 , α) ∈ N (x), Xt = x + αu · dSu , ∀t ∈ [0, T ] . 0

The second is the set of time T terminal wealths (random variables) generated by these wealth processes with x ≥ 0, i.e. T C (x) = {XT ∈ L0+ : ∃(α0 , α) ∈ N (x), x + 0 αt · dSt = XT } = {XT ∈ L0+ : ∃Z ∈ X (x), XT = ZT }. We use these two sets below. Formally, the trader’s portfolio optimization problem can now be stated. For simplicity of notation, we suppress the dependence of the state dependent utility function on the state ω ∈ Ω writing U (x) = U (x, ω) everywhere below.

196

10 Complete Markets (Utility Over Terminal Wealth)

Problem 1 (Choose the Optimal Trading Strategy (α0 , α) for x ≥ 0) v(x) =

sup

where

E [U (XT )]

(α0 ,α)∈N (x)

N (x) = {(α0 , α) ∈ (O, L (S)) : Xt = α0 (t) +  αt · St , t Xt = x + 0 αu · dSu ≥ 0, ∀t ∈ [0, T ] . In the classical literature this problem was solved using dynamic programming, which requires the additional assumption that the stock price process follows a Markov diffusion process. Although often a reasonable assumption, this problem can alternatively be solved using martingale methods and convex duality without adding this assumption. This alternative approach has the advantage that it builds on the insights obtained from the earlier chapters on derivative pricing and hedging. Consequently, we follow the second approach. The second approach solves this optimization problem in two steps. The two steps are: 1. to solve a static optimization problem which chooses the optimal time T wealth (the optimal “derivative”), and then 2. to determine the trading strategy that attains this “derivative.” Step 2 is equivalent to the construction of a synthetic derivative. This problem was solved in Sect. 2.7 in Chap. 2. Hence, the only new challenge in solving the investor’s optimization problem is step 1. We now focus on solving step 1. Problem 2 (Choose the Optimal Derivative XT for x ≥ 0) v(x) =

sup E [U (XT )]

XT ∈C (x)

where 

C (x) = {XT ∈ L0+ : ∃(α0 , α) ∈ N (x), x +

T

αt · dSt = XT }.

0

To solve this problem, we give an alternative characterization of the constraint set C (x), which represents those payoffs XT ∈ L0+ that can be generated by a nonnegative wealth s.f.t.s. (α0 , α) ∈ N (x) with initial wealth x ≥ 0. This characterization requires the assumption that the market satisfies NFLVR. The intuition behind this alternative characterization uses risk neutral valuation from Chap. 2. Consider the unique local martingale measure Q ∈ Ml , which exists by the NFLVR assumption and market completeness. When using risk neutral valuation in a complete market to compute present values (see Theorem 18 in Chap. 2), the constraint set C (x) should be the same as the set of all payoffs whose present values E Q [XT ] are affordable at time 0, i.e. less than or equal to x. This is indeed the case, as the next theorem shows.

10.2 Problem Statement

197

Theorem 49 (Budget Constraint Equivalence)   C (x) = XT ∈ L0+ : E Q [XT ] ≤ x   Proof Define C1 (x) = XT ∈ L0+ : E Q [XT ] ≤ x . Step 1: Show C (x) ⊂ C1 (x). T Take XT ∈ C (x), then ∃(α0 , α) ∈ N (x), x + 0 αt · dSt = XT . t The stochastic process x + 0 αu · dSu ≥ 0 is a local martingale under Q that is bounded below by 0.Hence, by Lemma 4 in Chap. 1, it is a supermartingale. Thus,  T Q E x + 0 αt · dSt ≤ x. This implies E Q [XT ] ≤ x. Hence, XT ∈ C1 (x). Step 2: Show C1 (x) ⊂ C (x). Take XT ∈ C1 (x). Then, E Q [XT ] < ∞. Since the market is complete with respect to Q, t ∃(α0 , α) ∈ N (x) such that x + 0 αu · dSu = E Q [XT |Ft ] for all t ∈ [0, T ]. T Hence, x + 0 αt · dSt = XT . This implies XT ∈ C (x), which completes the proof. Using this new budget constraint, we can rewrite the optimization problem using Q the local martingale deflator YT = dQ dP ∈ Ml , which satisfies E [XT ] = E [YT XT ]. Problem 3 (Choose the Optimal Derivative XT for x ≥ 0) v(x) =

sup E [U (XT )]

XT ∈C (x)

where

  C (x) = XT ∈ L0+ : E [YT XT ] ≤ x for YT =

dQ dP

∈ Ml .

To solve this problem, we define the Lagrangian L (XT , y) = E [U (XT )] + y (x − E [XT YT ]) .

(10.1)

Using the Lagrangian transforms a constrained optimization problem to an unconstrained optimization, where the unconstrained optimization’s first order conditions can be used to characterize the solution. With this observation in mind, we first show that the original constrained optimization problem is equivalent to solving the primal problem using the Lagrangian, i.e. "

! primal :

sup XT ∈L0+

inf L (XT , y) = y>0

sup E [U (XT )] = v(x).

XT ∈C (x)

198

10 Complete Markets (Utility Over Terminal Wealth)

Proof Note that  inf {y (x − E [XT YT ])} = y>0

0 if x − E [YT XT ] ≥ 0 −∞ otherwise.

Hence,  inf {E [U (XT )] + y (x − E [XT YT ])} = y>0

 inf L (XT , y) = y>0

So, sup XT ∈L0+

=



inf L (XT , y) = sup

XT ∈L0+

y>0

E [YT XT ] ≤ x otherwise.

E [U (XT )] if E [YT XT ] ≤ x −∞ otherwise.

"

!

E [U (XT )] if −∞

E [U (XT )] if E [YT XT ] ≤ x −∞ otherwise.

sup E [U (XT )] = v(x).

XT ∈C (x)

This completes the proof. The dual problem using the Lagrangian is ⎞

⎛ dual :

inf ⎝ sup L (XT , y)⎠ . y>0

XT ∈L0+

For optimization problems we know that the dual problem provides an upper bound on the value function of the primal problem, i.e. ⎞

⎛

"

!

inf ⎝ sup L (XT , y)⎠ ≥ sup

inf L (XT , y) = v(x).

y>0

y>0

XT ∈L0+

XT ∈L0+

Proof We have L (XT , y) = E [U (XT )] + y (x − E [XT YT ]). Note that sup L (XT , y) ≥ L (XT , y) ≥ inf L (XT , y) for all (XT , y). XT ∈L0+

y>0

Taking the infimum over y on the left side and the supremum over XT on the right side ⎞ ⎛ gives " ! inf ⎝ sup L (XT , y)⎠ ≥ sup

inf L (XT , y) . This completes the proof.

y>0

y>0

XT ∈L0+

XT ∈L0+

10.3 Existence of a Solution

199

The primal and dual problems’ value functions are equal if there is no duality gap. Often, solving the dual problem is easier than solving the primal problem directly. For our problem, this turns out to be the case. Hence, to solve the investor’s optimization problem we will show that there is no duality gap and we solve the dual problem. This is the task to which we now turn.

10.3 Existence of a Solution We need to show that there is no duality gap and that the optimum is attained, equivalently, the Lagrangian has a saddle point. To prove the existence of a saddle point, we need two assumptions in addition to the previous assumption that S satisfies NFLVR. Assumption (Reasonable Asymptotic Elasticity (AE(U) 0, which is an uninteresting optimization problem. Given these assumptions, we can now prove the following theorem. Theorem 50 (Existence of a Unique Saddle Point and Characterization of v(x) for x ≥ 0) Given the above assumptions, there exists a unique saddle point, i.e. there exists a unique (Xˆ T , y) ˆ such that ⎛

⎞

"

!

ˆ = sup inf ⎝ sup L (XT , y)⎠ = L (Xˆ T , y)

inf L (XT , y) = v(x).

y>0

y>0

XT ∈L0+

XT ∈L0+

Define v(y) ˜ = E[U˜ (yYT )] where U˜ (y, ω) = sup [U (x, ω) − xy], Then, v and v˜ are in conjugate duality, i.e.

(10.3) y > 0.

x>0

˜ + xy) , ∀x > 0, v(x) = inf (v(y)

and

(10.4)

y>0

v(y) ˜ = sup (v(x) − xy) , ∀y > 0. x>0

(10.5)

200

10 Complete Markets (Utility Over Terminal Wealth)

In addition, (i) v˜ is strictly convex, decreasing, differentiable on int (dom(v)) ˜ = ∅, (ii) v is strictly concave, increasing, and differentiable on (0, ∞), and (iii) defining yˆ to be where the infimum is attained in expression (10.4) we have v(x) = v( ˜ y) ˆ + x y. ˆ

(10.6)

Proof This theorem is a special case of Theorem 53 in Chap. 11. This completes the proof.

10.4 Characterization of the Solution This section characterizes the solution to the portfolio optimization problem.

10.4.1 The Characterization To characterize the solution, we focus on the dual problem, i.e. ⎛

⎞

v(x) = inf ⎝ sup L (XT , y)⎠ y>0

XT ∈L0+

⎛

⎞

= inf ⎝ sup E [U (XT ) − yXT YT ] + xy ⎠ . y>0

XT ∈L0+

To solve, we first exchange the sup and expectation operator. ⎡

⎤

v(x) = inf E ⎣ sup (U (XT ) − yXT YT ) + xy ⎦ y>0

(10.7)

XT ∈L0+

The justification for this step is proven in the appendix to this chapter. We now solve this problem, working from the inside out. Step 1. (Solve for XT ∈ L0+ ) Fix y and solve for the optimal Xˆ T ∈ L0+ , i.e. sup (U (XT ) − yXT YT ) .

XT ∈L0+

10.4 Characterization of the Solution

201

To solve this problem, fix a ω ∈ Ω, and consider the related problem sup

(U (XT (ω), ω) − yXT (ω)YT (ω)) .

XT (ω)∈R+

This is a simple optimization problem on the real line. The first order condition for an optimum gives U  (XT (ω), ω) − yYT (ω) = 0, Xˆ T (ω) = I (yYT (ω), ω)

or

with I (·, ω) = (U  (·, ω))−1 .

Given the properties of I (·, ω), Xˆ T (ω) is FT -measurable, hence Xˆ T (ω) ∈ L0+ . Note that Xˆ T depends on y. Step 2. (Solve for y > 0) Given the optimal Xˆ T , next we solve for the optimal Lagrangian multiplier y. ˆ Using conjugate duality (see Lemma 25 in Chap. 9), we have U˜ (yYT ) = sup [U (XT ) − yXT YT ] = [U (Xˆ T ) − y Xˆ T YT ]. XT ∈L0+

This transforms the problem to ⎡

⎤

inf E ⎣ sup (U (XT ) − yXT YT ) + xy ⎦ = inf y>0

XT ∈L0+

  E[U˜ (yYT )] + xy .

y>0

Taking the derivative of the right side with respect to y yields E[U˜  (yYT )YT ] + x = 0.

(10.8)

This step requires taking the derivative underneath the expectation operator. The justification for this step is proven in the appendix to this chapter. Noting from Lemma 25 in Chap. 9 that U˜  (y) = −I (y), we get that the optimal yˆ satisfies   E I (yY ˆ T )YT = E[Xˆ T YT ] = E Q [Xˆ T ] = x,

(10.9)

which is that the optimal terminal wealth Xˆ T satisfies the budget constraint with an equality. Solving this equation gives the optimal y. ˆ Step 3. (Characterization of Xˆ ∈ X (x) for x ≥ 0) The optimal time t wealth is denoted Xˆ t . Given the market is complete with respect to Q, we know that there exists a nonnegative wealth s.f.t.s. process

202

10 Complete Markets (Utility Over Terminal Wealth)

generating Xˆ T at time T where x +

t 0

αu · dSu = Xˆ t is a Q martingale, i.e.

  Xˆ t = E Q Xˆ T |Ft

(10.10)

for all t ∈ [0, T ]. This implies that the optimal wealth process Xˆ ∈ X (x) when multiplied by the local martingale deflator process, Xˆ t Yt , is a P martingale. Indeed, for all t ∈ [0, T ],

      YT

Q ˆ ˆ ˆ |F F Y X Yt = Xˆ t Yt . = E E XT YT |Ft = E XT t t T t Yt Step 4. (Nonnegativity of Xˆ ∈ X (x) ) In the optimization problem, time T wealth must be nonnegative. This is captured by the restriction that Xˆ T ∈ L0+ , i.e. Xˆ T ≥ 0. From expression (10.10), this implies that the optimal wealth process Xˆ t must also be nonnegative for all t ∈ [0, T ], i.e.   E Q Xˆ T |Ft = Xˆ t ≥ 0. Remark 65 (Random Variables Versus Stochastic Processes) The optimization problem determines the random variable Xˆ T ∈ L0+ , which is the optimal time T wealth. But, since the market is complete with respect to Q, there exists an  t Q ˆ (α0 , α) ∈ N (x) with x ≥ 0 such that Xt = x + 0 αu · dSu = E XT |Ft for all t ∈ [0, T ]. This characterizes the optimal wealth (stochastic) process X ∈ X (x). From this optimal wealth process, the nonnegative wealth s.f.t.s. generating it can be determined using the methods in Chap. 2. We note that if the risky assets are redundant, as defined in Chap. 2, then the optimal trading strategy will not be unique. However, if the risky assets are nonredundant, then it is unique (see Sect. 2.4 in Chap. 2 for further discussion). A sufficient condition on the risky asset price process S such that the risky assets are non-redundant is given in Theorem 10 of Chap. 2. The NFLVR and complete market assumptions, by the Second Fundamental Theorem 14 of asset pricing in Chap. 2, also identify the unique equivalent local martingale probability measure Q ∈ Ml , which in turn identifies the unique local martingale deflator YT = dQ dP ∈ Ml ⊂ Dl . This local martingale deflator uniquely determines the local martingale deflator process, Y ∈ Ml ⊂ Dl via the P martingale condition Yt = E [YT |Ft ] for all t ∈ [0, T ]. These observations will prove useful, for comparison, when we study an incomplete market in Chap. 11 below. This completes the remark.

10.5 The Shadow Price

203

10.4.2 Summary For easy reference we summarize the previous characterization results. The optimal value function v(x) =

sup E [U (XT )]

XT ∈C (x)

for x ≥ 0 has the solution for terminal wealth given by ˆ T) Xˆ T = I (yY

with I = (U  )−1 = −U˜ 

where YT = dQ dP ∈ Ml ⊂ Dl is the local martingale deflator associated with the unique local martingale measure Q ∈ Ml . This generates the local martingale deflator process Y ∈ Ml ⊂ Dl via the expression Yt = E [YT |Ft ] for all t ∈ [0, T ]. The Lagrangian multiplier yˆ is the solution to the budget constraint   ˆ T ) = x, E YT I (yY where the optimal wealth Xˆ T satisfies the budget constraint with an equality   E YT Xˆ T = x. The optimal wealth process Xˆ ∈ X (x) for x ≥ 0 exists because by market completeness with respect to Q, there exists a nonnegative wealth s.f.t.s. (α0 ,α) ∈  t Q ˆ ˆ ˆ N (x) with time T payoff XT where Xt = x + 0 αu · dSu = E XT |Ft ≥ 0 for all t. This optimal nonnegative wealth s.f.t.s. is unique if and only if the risky assets in the market are non-redundant. The optimal wealth process Xˆ ∈ X (x), when multiplied by the local martingale deflator process, Xˆ t Yt ≥ 0 is a P martingale.

10.5 The Shadow Price Using expression (10.6), we can obtain the shadow price of the budget constraint. The shadow price is the benefit, in terms of expected utility, of increasing the initial wealth x by 1 unit (of the mma). The shadow price equals the Lagrangian multiplier.

204

10 Complete Markets (Utility Over Terminal Wealth)

Theorem 51 (Shadow Price of the Budget Constraint) yˆ = v  (x) = E[U  (Xˆ T )]

(10.11)

Proof The first equality is obtained by taking the derivative of v(x) = v( ˜ y) ˆ + x y. ˆ The second equality follows from the first order condition for an optimum ˆ T . Taking expectations gives E[U  (Xˆ T )] = E[yY ˆ T ] = yˆ < ∞ U  (Xˆ T ) = yY because E[YT ] = 1. This completes the proof.

10.6 The Local Martingale Deflator A key insight of the individual optimization problem is the characterization of the trader’s local martingale deflator, in this case the probability density YT = dQ dP ∈ Ml of an equivalent local martingale measure Q ∈ Ml , in terms of the individual’s utility function and initial wealth. To obtain this characterization, using Theorem 51 above, we rewrite the first order condition for the optimal wealth Xˆ T as YT =

U  (Xˆ T ) U  (Xˆ T ) = . yˆ E[U  (Xˆ T )]

(10.12)

As seen, the local martingale deflator equals the investor’s marginal utility of wealth, normalized by the expected marginal utility of terminal wealth. Given investors trade in a competitive market, prices are taken as exogenous. And since the market is complete with respect to Q, prices uniquely determine the equivalent local martingale measure Q. At an optimum, the individual equates her marginal utility (normalized) to the market determined probability density YT = dQ dP ∈ Ml . Of course, in equilibrium (Part III of this book), the local martingale deflator is endogenously determined by all of the investors’ collective trades and the market clearing conditions. Since the local martingale deflator YT ∈ Ml is a probability density with respect to P, the state price density process is given by the P martingale relation Yt = E [YT |Ft ] =

E[U  (Xˆ T ) |Ft ] E[U  (Xˆ T )]

(10.13)

for all t ∈ [0, T ]. Remark 66 (Asset Price Bubbles) In this individual optimization problem, prices and hence the local martingale measure Q are taken as exogenous. The above analysis does not require that Q is a martingale measure. Hence, given Chap. 3, because Q can be a strict local martingale measure, this optimization problem applies to a market with asset price bubbles. That is, the portfolio optimization

10.6 The Local Martingale Deflator

205

problem’s solution and the optimal trading strategy apply to a market with price bubbles, if Q is a strict local martingale measure. This completes the remark. Remark 67 (Systematic Risk) In this remark, we use the non-normalized market ((B, S), F, P) representation to characterize systematic risk. Recall that the state price density is the key input to the systematic risk return relation in Chap. 4, Theorem 38. To apply this theorem, we assume that the local martingale deflator YT ∈ Ml is a martingale deflator, i.e. YT ∈ M. Then, the risky asset returns over the time interval [t, t + Δ] satisfy

'

E[U  (Xˆ T ) |Ft+Δ ] Bt

E [Ri (t) |Ft ] = r0 (t) − cov Ri (t), (1 + r0 (t)) Ft 

ˆ B t+Δ E[U (XT ) |Ft ] &

(10.14) 1 i (t) , r0 (t) = p(t,t+Δ) −1 where the ith risky asset’s return is Ri (t) = Si (t+Δ)−S Si (t) is the default-free spot rate of interest where p(t, t + Δ) is the time t price of a default-free zero-coupon bond maturing at time t + Δ, and the state price density is

Ht =

Yt Bt

=

E[YT |Ft ] Bt

=

E[U  (Xˆ T )|Ft ] . E[U  (Xˆ T )]Bt

When the time step is very small, i.e. Δ ≈ dt, the expression and this relation simplifies to

Bt Bt+Δ (1+r0 (t))

' E[U  (Xˆ T ) |Ft+Δ ]

E [Ri (t) |Ft ] ≈ r0 (t) − cov Ri (t),

Ft . E[U  (Xˆ T ) |Ft ]

≈ 1,

&

(10.15)

This last expression clarifies the meaning of systematic risk. Recalling the discussion in Chap. 4, systematic risk is characterized by the right side of expres(1 + r0 (t)) with the ratio sion (10.15) where we have replaced the term HHt+Δ t   E[U  (Xˆ T )|Ft+Δ ] . The intuition for this ratio is as follows. E[U  (Xˆ T )|Ft ]    ˆ • When E[U (XTˆ )|Ft+Δ ] is large, systematic risk is small (note the negative sign). E[U (XT )|Ft ] This occurs when the marginal utility of wealth is large, hence wealth is scarce. An asset whose return is large when wealth is scarce is valuable. Such an asset is “anti-risky.”   

ˆ

• When E[U (XTˆ )|Ft+Δ ] is small, systematic risk is large. This occurs when the E[U (XT )|Ft ] marginal utility of wealth is small, hence wealth is plentiful. An asset whose return is large when wealth is plentiful is less valuable then when wealth is scarce. Such an asset is “risky.”

This completes the remark.

206

10 Complete Markets (Utility Over Terminal Wealth)

10.7 The Optimal Trading Strategy This section solves step 2 of the optimization problem, which is to determine the optimal trading strategy (α0 , α) ∈ N (x) for x ≥ 0. This is done using the standard techniques developed for the synthetic construction of derivatives in Chap. 2. Since the market is complete with respect to Q, we know there exists a nonnegative wealth s.f.t.s. (α0 , α) ∈ N (x) such that Xˆ t = x +



t

αu · dSu = E Q [Xˆ T |Ft ]

(10.16)

0

for all t ∈ [0, T ] where x = Xˆ0

and

Xˆ t = α0 (t) + αt · St .

(10.17)

We need to characterize this nonnegative wealth s.f.t.s. (α0 (t), αt ) given the initial wealth x ≥ 0. To illustrate this determination, suppose that S is a Markov diffusion process. Due to the diffusion assumption, S is a continuous process. Given this assumption, we can write the optimal wealth process as a deterministic function of time t and the risky asset prices St , i.e. E Q [Xˆ T |Ft ] = E Q [Xˆ T |St ] = Xˆ t (St ). Using Ito’s formula (assuming the appropriate differentiability conditions hold), we have that Xˆ t (Su )=x +

 0

t

1 ∂ Xˆ u du + ∂u 2

 t ˆ  t n  n ) ∂ 2 Xˆ u ( ∂ Xu d S (u), S (u) + · dSu i j 2 (u) ∂S 0 i=1 j =1 0 ∂Su ij (10.18)

ˆ

ˆ

ˆ

∂X ∂X  where ∂∂SX = ( ∂S , . . . , ∂S ) ∈ Rn . Equating the integrands of dSt in the two n 1 stochastic integral equations immediately above, the holdings in the risky assets are easily determined to be

∂ Xˆ u (Su ) = αu . ∂Su This is the standard “delta” used to construct a “synthetic derivative,” in this case the optimal terminal wealth, XˆT . Finally, given the holdings in the risky assets αt , the holdings in the mma are obtained from expression (10.17). If the risky assets are non-redundant, then this nonnegative wealth s.f.t.s. is unique. Otherwise, it is not (see Sect. 2.4 in Chap. 2 for additional discussion). Finally, if the underlying asset price process is not a Markov diffusion process, then for a large class of processes

10.8 An Example

207

one can use Malliavin calculus to determine the trading strategy (see Detemple et al. [50] and Nunno et al. [51]).

10.8 An Example This section presents an example to illustrate the abstract expressions of the previous sections. This example is from Pham [156, p. 196].

10.8.1 The Market Suppose the normalized market (r = 0) consists of a single risky asset and a mma where the risky asset price process follows a geometric Brownian motion, i.e. dSt = St (bdt + σ dWt ) or 1 2 St = S0 ebt− 2 σ t+σ Wt

(10.19)

for all t ∈ [0, T ] where b, σ are strictly positive constants and Wt is a standard Brownian motion with W0 = 0 that generates the filtration F. Because there is only one risk asset trading, the risky asset market is non-redundant. In Chap. 5 it was shown that for geometric Brownian motion, the market is complete with respect to Q ∈ Ml . Hence, by the Second Fundamental Theorem 14 of asset pricing in Chap. 2, the equivalent local martingale measure is unique. In that Chapter it was also shown that Q is an equivalent martingale measure with probability density YT =

1 2 dQ = e−θ·WT − 2 θ T > 0 dP

(10.20)

for θ = σb . Thus, the evolution of the risky asset’s price is a martingale under Q, and it is given by dSt = St σ dWtθ or 1 2 θ St = S0 e− 2 σ t+σ Wt

(10.21)

where dW θ (t) = dW (t) + θ (t)dt is a Brownian motion under Q. This implies that under Q, using dW (t) = dW θ (t) − θ (t)dt, we have that the martingale deflator is YT = e−θ·WT +θ θ

2T − 1 θ 2T 2

1 2 T

= e−θ·WT + 2 θ θ

> 0.

(10.22)

208

10 Complete Markets (Utility Over Terminal Wealth)

10.8.2 The Utility Function Let preferences be represented by a state independent power utility function U (x) =

xρ ρ

for x > 0 and ρ < 1, ρ = 0. Recall that 1

I (y) = y ρ−1 for y > 0.

10.8.3 The Optimal Wealth Process Using the solution as previously determined, the optimal wealth process is ˆ T ) = (yY ˆ T )−κ Xˆ T = I (yY where κ =

1 1−ρ

> 0. And, for an arbitrary time t,     ˆ T ) |Ft = E Q (yY ˆ T )−κ |Ft . Xˆ t = E Q I (yY

Algebra and substitution gives     θ 1 2 = yˆ −κ E Q YT −κ |Ft = yˆ −κ E Q eθκ·WT − 2 θ κT |Ft 1 2 κT

= yˆ −κ e− 2 θ

θ + 1 (θκ)2 (T −t) 2

eθκ·Wt

.

Recall that yˆ is the solution to the budget constraint   ˆ T ) = x. Xˆ 0 = E Q I (yY Substitution for time 0 gives 1 2 κT + 12 (θκ)2 T

x = yˆ −κ e− 2 θ

.

Combined we get that the optimal wealth process is θ 2 Xˆ t = xeθκ·Wt − 2 (θκ) t . 1

(10.23)

10.8 An Example

209

10.8.4 The Optimal Trading Strategy To get the optimal trading strategy we apply Ito’s formula to expression (10.23), which yields d Xˆ t = Xˆ t θ κdWtθ . Proof Dropping all the subscripts and superscripts, 1 ∂2X ∂X θ dX = ∂X ∂t dt + 2 ∂W 2 dt + ∂W dW . 1 1 2 2 = −X 2 (θ κ) dt + 2 X(θ κ) dt + Xθ κdW θ . This completes the proof. Since the market is complete with respect to Q, we know that there exists a nonnegative wealth s.f.t.s. (αˆ 0 , α) ˆ ∈ N (x) with x ≥ 0 such that d Xˆ t = αˆ t dSt = αˆ t St σ dWtθ . Equating the coefficients of the Brownian motion terms in the above two expressions for d Xˆ t yields αˆ t St σ = Xˆ t θ κ or αˆ t =

Xˆ t θκ St σ .

(10.24)

Note that the holdings in the mma are determined by the expression αˆ 0 (t) = Xˆ t − αˆ t St for all t. This optimal nonnegative s.f.t.s. is unique. Expressing these holdings as a proportion of wealth, we get (using θ = σb ) πˆ t =

b αˆ t St θκ = 2 . = σ σ (1 − ρ) Xˆ t

(10.25)

This shows that for power utility functions in a geometric Brownian motion market, the optimal portfolio weight in the risky asset is a constant across time and independent of the level of wealth.

10.8.5 The Value Function Last, to obtain the value note that  function,   1 2 v(x) = E U (Xˆ T ) = E U (xeθκ·WT − 2 (θκ) T )   θκ·WT − 12 (θκ)2 T ρ   1 ρ 2 (xe ) = xρ e− 2 ρ(θκ) T E eθκρ·WT =E ρ  1  1 1 ρ ρ 2 2 2 2 = xρ e− 2 ρ(θκ) T e 2 (θκ) ρ T = xρ e− 2 ρ(1−ρ)(θκ) T .

210

10

Substituting in κ = v(x) =

xρ ρ e

1 1−ρ

Complete Markets (Utility Over Terminal Wealth)

> 0 and θ =

1 2 b2 − 12 ρ(1−ρ)( 1−ρ ) 2T σ

b σ

gives

, or

v(x) =

ρ b2 x ρ − 12 1−ρ T σ2 . e ρ

(10.26)

This completes the example.

10.9 Notes Optimization in a complete market is often studied as a special case of an incomplete market. Excellent references for solving the investor’s optimization problem include Dana and Jeanblanc [42], Duffie [52], Karatzas and Shreve [124], Merton [147], and Pham [156].

Appendix Proof of Expression (10.7) To use this result in an incomplete market, Chap. 11, we note that the following proof holds for any YT ∈ Ds as well. Proof (Exchange of sup and E[·] operator) It is trivial that sup E [U (XT ) − yXT YT ] ≤ E[ sup (U (XT ) − yXT YT )].

XT ∈L0+

XT ∈L0+

We want to prove the opposite inequality. Since U is strictly concave, there exists a unique solution XT∗ to   sup [U (XT ) − yXT YT ] = U (XT∗ ) − yXT∗ YT . XT ∈L0+

But,   sup E [U (XT ) − yXT YT ] ≥ E U (XT∗ ) − yXT∗ YT

XT ∈L0+

= E[ sup (U (XT ) − yXT YT )] XT ∈L0+

which completes the proof.

Appendix

211

Proof of Expression (10.8) To use this result in an incomplete market, Chap. 11, we note that the following proof holds for any YT ∈ Ds as well. Proof (Exchange of E[·] and Derivative) for δ > 0. Since the derivative exists, we use the left derivative. ˜ ˜ (yYT ) ˜ ˜ T )−U lim U ((y+Δ)YΔT )−U (yYT ) = lim U ((y+Δ)Y · lim (y+Δ)YΔT −yYT (y+Δ)YT −yYT ΔYT →0 U˜ ((y+Δ)YT )−U˜ (yYT ) lim · YT = lim U˜  (ξ YT )YT (y+Δ)YT −yYT ΔYT →0 Δ→0

Δ→0

=

Δ→0

a.s. P where Δ < 0.

The last equality follows from the mean value theorem (Guler [66], p. 3), i.e. there exists ξ ∈ [y + Δ, y] such that U˜ ((y + Δ)YT ) − U˜ (yYT ) = U˜  (ξ YT ) [(y + Δ)Y  T − yYT ]. Thus,

∂E[U˜ (yYT )] ∂y

= lim

E U˜ ((y+Δ)YT )−U˜ (yYT ) Δ

Δ→0

= lim E[U˜  (ξ YT )YT ]. Δ→0

Now E[U˜ (yYT )] < ∞ because v(y) ˜ < ∞ and YT is the supermartingale deflator such that v(y) ˜ = E[U˜ (yYT )]. By Kramkov and Schachermayer [132, Lemma 6.3 (iv) and (iii), p. 944], AE (U ) < 1 implies there exists a constant C and z0 > 0 such that −U˜  (z)z < C U˜ (z) for 0 < z ≤ z0 , and U˜ (μz) < K(μ)U˜ (z) for 0 < μ < 1 and 0 < z ≤ z0 where K(μ) is a constant depending upon μ. Combined, −U˜  (μz)μz < C U˜ (μz) < CK(μ)U˜ (z) implies that there exists a z0 > 0 such that ¯ ¯ −U˜  (μz)μz < K(μ) U˜ (z) for 0 < z ≤ z0 where K(μ) is a constant depending upon μ for 0 < μ < 1. Letting z = yYT and μ = yξ < 1, because Δ < 0 so that ξ < y. Then, ¯ ξ )U˜ (yYT ). Since the right side is P integrable, using the −U˜  (ξ YT )YT < ξ1 K( y dominated convergence theorem, lim E[U˜  (ξ YT )YT ] = E[ lim U˜  (ξ YT )YT ] = E[U˜  (yYT )YT ]. Δ→0

Δ→0

The last equality follows from the continuity of U˜  (·). The continuity of U˜  (·) follows because U˜ (·) is strictly convex, hence U˜  (·) is a strictly increasing function, which is therefore differentiable a.s. P (see Royden [167, Theorem 2, p. 96]), and hence continuous. This completes the proof.

Chapter 11

Incomplete Markets (Utility Over Terminal Wealth)

This chapter studies the investor’s portfolio optimization problem in an incomplete market. The solution in this chapter parallels the solution for the complete market setting in Chap. 10.

11.1 The Set-Up Given is a normalized market (S, F, P) where the value of a money market account Bt = 1 for all t ≥ 0. As before, we assume that there are no arbitrage opportunities in the market, i.e. Assumption (NFLVR)

Ml = ∅ where Ml = {Q ∼ P : S is a Q local martingale}.

If NFLVR is violated, a solution to the trader’s portfolio optimization problem will not exist (see the proof of Theorem 63 in Chap. 13). In addition, this assumption enables a representation of the trader’s budget constraint that facilitates the use of convex optimization. In this chapter the market may be incomplete. In an incomplete market, given an equivalent local martingale measure Q ∈ Ml , there exist integrable derivatives that cannot be synthetically constructed using a value process that is a Q martingale. If the set of equivalent local martingale measures is not a singleton, then the cardinality of |Ml | = ∞, i.e. the set contains an infinite number of elements. In this case, derivative prices are not uniquely determined because the equivalent local martingale measure in not unique. Here, the investor’s portfolio optimization problem provides an important method to identify the relevant local martingale measure for the trader to price derivatives.

© Springer Nature Switzerland AG 2021 R. A. Jarrow, Continuous-Time Asset Pricing Theory, Springer Finance, https://doi.org/10.1007/978-3-030-74410-6_11

213

214

11 Incomplete Markets (Utility Over Terminal Wealth)

11.2 Problem Statement This section presents the investor’s portfolio optimization problem. We assume that ρ

the investor’s preferences # have a state dependent EU representation with respect to P where the utility function if defined over terminal wealth U (XT , ω) and satisfies the assumption given in Chap. 9, repeated here for convenience. Assumption (Utility Function) The utility function U : (0, ∞) × Ω → R is such that for all ω ∈ Ω a.s. P, (i) (ii) (iii) (iv)

U (x, ω) is B(0, ∞) ⊗ FT measurable, U (x, ω) is continuous and differentiable in x ∈ (0, ∞), U (x, ω) is strictly increasing and strictly concave in x ∈ (0, ∞), and (Inada Conditions) lim U  (x, ω) = ∞ and lim U  (x, ω) = 0. x↓0

x→∞

We do not consider utility over intermediate consumption in this chapter. The solution to the portfolio optimization problem with intermediate consumption is given in Chap. 12. Recall that the set of nonnegative wealth s.f.t.s. for x ≥ 0 is N (x) = {(α0 , α) ∈ (O, L (S)) : Xt = α0 (t) +  αt · St , t Xt = x + 0 αu · dSu ≥ 0, ∀t ∈ [0, T ] . The set of value processes generated by the nonnegative wealth s.f.t.s. for x ≥ 0 is    t αu · dSu , ∀t ∈ [0, T ] , X e (x) = X ∈ L+0 : ∃(α0 , α) ∈ N (x), Xt = x + 0

and the set of time T terminal wealths generated by these value processes for x ≥ 0 is T C e (x) = {XT ∈ L0+ : ∃(α0 , α) ∈ N (x), x + 0 αt · dSt = XT } = {XT ∈ L0+ : ∃Z ∈ X e (x), XT = ZT }. where the superscript “e” stands for equality in the constraint set. The trader’s problem is to choose a nonnegative wealth s.f.t.s. to maximize their expected utility of terminal wealth. For simplicity of notation, we suppress the dependence of the utility function on the state ω ∈ Ω writing U (x) = U (x, ω) everywhere below.

11.2 Problem Statement

215

Problem 4 (Choose the Optimal Trading Strategy (α0 , α) for x ≥ 0) v(x) =

sup

where

E [U (XT )]

(α0 ,α)∈N (x)

N (x) = {(α0 , α) ∈ (O, L (S)) : Xt = α0 (t) +  αt · St , t Xt = x + 0 αu · dSu ≥ 0, ∀t ∈ [0, T ] . Similar to a complete market, we first solve the static problem to choose the optimal time T wealth (a “derivative”). Then, after solving the static problem, we use the methods for the synthetic construction of a derivative (see Sect. 2.7.2 in Chap. 2) to determine the trading strategy that generates the optimal wealth. Problem 5 (Choose the Optimal Derivative XT for x ≥ 0) v(x) =

sup

XT ∈C e (x)

E [U (XT )]

where 

C e (x) = {XT ∈ L0+ : ∃(α0 , α) ∈ N (x), x +

T

(11.1)

αt · dSt = XT }.

0

To facilitate the determination of the solution, we solve an equivalent problem, where we allow some of the time T wealth to be discarded, called free disposal. Define    t αu · dSu ≥ Xt , ∀t ∈ [0, T ] . X (x) = X ∈ L+0 : ∃(α0 , α) ∈ N (x), x + 0

The set of time T wealths generated by this set of stochastic processes is  C (x) = {XT ∈ L0+ : ∃(α0 , α) ∈ N (x), x +

T

αt · dSt ≥ XT }.

0

Note that the superscript “e” is omitted from these sets. The equivalent optimization problem with free disposal is v(x) =

sup E [U (XT )]

XT ∈C (x)

where 

C (x) = {XT ∈ L0+ : ∃(α0 , α) ∈ N (x), x +

T

(11.2)

αt · dSt ≥ XT }.

0

Given more wealth is preferred to less (U is increasing in x), it is intuitive that the solution to these two problems should be the same, as the next lemma proves.

216

11 Incomplete Markets (Utility Over Terminal Wealth)

Lemma 26 (Free Disposal) XT is an optimal solution to problem (11.1) if and only if XT is an optimal solution to problem (11.2). Proof (Step 1) Since C e (x) ⊂ C (x), we have sup E [U (XT )] ≤ sup E [U (XT )]. XT ∈C e (x) XT ∈C (x) T (Step 2) Suppose ∃(α0 , α) ∈ N (x), x + 0 αt · dSt > Xˆ T where Xˆ T is an optimal solution of sup E [U (XT )]. XT ∈C (x)    T = Let ZT = x + 0 αt · dSt . Then, E [U (ZT )] > E U Xˆ T sup E [U (XT )] since U (·) is strictly increasing. XT ∈C (x)

But,

sup

XT ∈C e (x)

E [U (XT )] ≥ E [U (ZT )] because ZT ∈ C e (x).

This implies that

sup

XT ∈C e (x)

E [U (XT )] >

sup E [U (XT )]. This contradicts

XT ∈C (x)

(Step 1). Thus, the solution Xˆ T to T sup E [U (XT )] must satisfy x + 0 αt ·dSt = Xˆ T for some (α0 , α) ∈ N (x). XT ∈C (x)

This completes the proof. Remark 68 (Self-Financing Trading Strategy with Cash Flows) For comparison with Chap. 12 when intermediate consumption is included, consider a cash flow process At (ω) : Ω × [0, T ] → [0, ∞), which is an adapted, right continuous, and nondecreasing process with A0 = 0. Such a cash flow process was considered in Chap. 8 when discussing super- and sub-replication in an incomplete market. Let V+ be the set of adapted, right continuous, and nondecreasing processes with initial values equal to 0. A s.f.t.s. with positive cash flows is a nonnegative wealth s.f.t.s. (α0 , α) ∈ N (x) and a cash flow process At ∈ V+ such that the value process Xt = α0 (t) + αt · St t satisfies Xt + At = x + 0 αu · dSu for all t ∈ [0, T ]. As seen, the s.f.t.s. “finances” both the value process Xt and the cash flow process At . Note that this implies that the nonnegative wealth s.f.t.s. (α0 , α) ∈ N (x) t satisfies x + 0 αu · dSu ≥ Xt for all t, i.e. XT ∈ C (x). Using this new notation, it can easily be seen that  C (x) = {XT ∈ L0+ : ∃(α0 , α) ∈ N (x), ∃A ∈ V+ , x +

T

αt · dSt = XT + AT }.

0

This completes the remark. To facilitate the determination of the optimal solution, we seek an alternative characterization of the constraint set that is easier to use. Recall that we have the

11.2 Problem Statement

217

following sets of local martingale deflators and their relationships (see Chap. 2, Sect. 2.5.3).  Dl = Y ∈ L+0 : Y0 = 1, XY is a P local martingale,  X = 1 + α · dS, (α0 , α) ∈ N (1) Dl = {YT ∈ L0+ : ∃Z ∈ Dl , YT = ZT }   dQ Ml = Y ∈ Dl : ∃Q ∼ P, YT = dP Ml = {YT ∈ L0+ : ∃Z ∈ Ml , YT = ZT } M = {Y ∈ L+0 : YT =

dQ , Q ∈ M}. dP

M = {Y ∈ L0+ : ∃Z ∈ M , YT = ZT }. M ⊂ Ml ⊂ Dl ⊂ L+0

stochastic processes random variables

M ⊂ Ml ⊂ Dl ⊂ L0+

We use these sets of local martingale deflators below. The constraint set C (x) represents those payoffs XT ∈ L0+ that can be dominated (because of free disposal) by a nonnegative wealth s.f.t.s. (α0 , α) ∈ N (x) with initial wealth x. Consider the set of local martingale deflators YT ∈ Ml . Using the insights from super- and sub-replication in Chap. 8, Corollary 9, each of these deflators generates a possible present value for the payoff XT , E [XT YT ]. Intuitively, since it is unknown which local martingale deflator should be used to compute the present value of the payoffs, C (x) should be the same as the set of all payoffs whose “worst case” present values are affordable at time 0, i.e. less than or equal to x. This is indeed the case, as the next theorem shows. Theorem 52 (Budget Constraint Equivalence 1) C (x) = {XT ∈ L0+ : sup E [XT YT ] ≤ x}. YT ∈Ml

Proof Define C1 (x) = {XT ∈ L0+ : sup E [XT YT ] ≤ x}. YT ∈Ml

(Step 1) Show C (x) ⊂ C1 (x). T Take XT ∈ C (x), then ∃(α0 , α) ∈ N (x) such that x + 0 αt · dSt ≥ XT . t Let Wt = x + 0 αu · dSu . Choose YT ∈ Ml . Then, there exists Z ∈ Ml such that YT = ZT .

218

11 Incomplete Markets (Utility Over Terminal Wealth)

By definition of Ml , Wt Zt is a P local martingale bounded below by zero. Hence, by Lemma 4 in Chap. 1, Wt Zt is a P supermartingale. Thus, E [ZT WT ] ≤ Z0 W0 = x, or equivalently E [YT WT ] ≤ x. Since this is true for all YT ∈ Ml , XT ∈ C1 (x). (Step 2) Show C1 (x) ⊂ C (x). Take ZT ∈ C1 (x). Define Zt = ess sup E[ YYTt ZT |Ft ] = ess sup E Q [ZT |Ft ] ≥ 0. YT ∈Ml

Q∈Ml

Part 1 of step 2 is to note that Zt is a supermartingale for all Q ∈ Ml , this follows by Theorem 6 in Chap. 1. Part 2 of step 2 is to use the optional decomposition Theorem 7 in Chap. 1, which implies that ∃α ∈ L (S) and an adapted  t right continuous nondecreasing process At with A0 = 0 such that Zt = Z0 + 0 αu · dSu − At for all t. Let αt be the position in the risky assets. Define the position in the mma α0 (t) by the expression Zt = α0 (t) + αt · St for all t. By the definition of Zt we get that t Z0 + 0 αu · dSu = α0 (t) + αt · St + At . Hence, the trading strategy is self-financing with cash flow At≥ 0 (free disposal) and initial value x = Z0 . t We have Z0 + 0 αu · dSu = Zt + At ≥ Zt ≥ 0 since At ≥ 0 and Zt ≥ 0. Hence, t Z0 + 0 αu · dSu is bounded below by zero, i.e. (α0 , α) ∈ N (x) with x = Z0 . Thus, ZT ∈ C (x). This completes the proof. Remark 69 (Alternative Version of the Budget Constraint) Notice that in this budget constraint characterization there are an infinite number of constraints, one for each YT ∈ Ml . Indeed, we can rewrite the constraint as E [XT YT ] ≤ x

for all YT ∈ Ml .

To prove this, note that this constraint is implied by sup E [XT YT ] ≤ x. YT ∈Ml

Conversely, taking the supremum of the left side of the previous expression across all YT ∈ Ml gives back the supremum constraint. This completes the remark. Given this remark, we can write the optimization problem as Problem 6 (Choose the Optimal Derivative XT for x ≥ 0) v(x) = sup E [U (XT )]

where

XT ∈L0+

E [XT YT ] ≤ x for all YT ∈ Ml . As in Chap. 10 when solving the investor’s optimization problem in a complete market, we are going to solve this optimization problem (the primal problem) by solving its dual problem and showing that there is no duality gap. Unfortunately, it can be shown that, in general, there does not exist an element YT ∈ Ml such that the solution to the dual problem is attained (see Pham [156, p. 186] or Kramkov and

11.2 Problem Statement

219

Schachermayer [132]). To guarantee the existence of an optimal solution to the dual problem, we need to “fill-in the interior” of the set of local martingale probability measures Ml . This is the task to which we now turn. Define the set Ds = {YT ∈ L0+ : Y0 = 1, ∃(Z(T )n )n≥1 ∈ Ml , YT ≤ lim Zn (T ) a.s.}. n→∞

(11.3)

Ds is the smallest convex, solid, closed subset of L0+ (in the P−convergence topology) that contains Ml (see Pham [156, p. 186]). Solid means that if 0 ≤ YˆT ≤ YT and YT ∈ Ds , then YˆT ∈ Ds . We note that the random variable YT ∈ Ds need not be the density of a probability measure with respect to P since E(YT ) < 1 is possible. Indeed, E[YT ] ≤ E[ lim Zn (T )] ≤ lim E[Zn (T )] = 1, where the n→∞

n→∞

second inequality is due to Fatou’s lemma. It can be shown that Ds is equal to the set of supermartingale deflators, i.e. Ds = {YT ∈ L0+ : ∃Z ∈ Ds , YT = ZT } where  Ds = Y ∈ L+0 : Y  0 = 1, XY is a P supermartingale,  X = 1 + α · dS, (α0 , α) ∈ N (1) . Proof (Also See Schachermayer [171]) This follows because by the bipolar theorem in Brannath and Schachermayer [21], Ds is the smallest convex, solid, closed subset of L0+ (in the P convergence topology) that contains Ml , which also is expression (11.3). This completes the proof. In an incomplete market, the supermartingale deflators take the role that the set of martingale deflators play in a complete market. Consequently, we need to understand their properties, especially in relation to the set of martingale deflators. First, we note that, in general, the set of supermartingale deflator processes depend on the probability measure P. Indeed, by Girsanov’s Theorem (see Theorem 5 in Chap. 1), an equivalent change of measure can change a supermartingale to a martingale or a submartingale. Second, the supermartingale deflator Y ∈ Ds is itself a P supermartingale. This follows by considering the nonnegative wealth s.f.t.s. (α0 , α) = (1, 0) ∈ N (1), which represents a buy and hold trading strategy in only the mma. In this case Xt = 1 for all t, and the statement follows from the definition of the set Ds . Since Y0 = 1, this implies E[YT ] ≤ 1. Similar to a martingale deflator, Y ∈ Ds need not be a probability density with respect to P (see Chap. 2, Sect. 2.5.2). If Y is a probability density with respect to P with E [YT ] = 1, then Yt = E [YT |Ft ] is a P martingale (see Lemma 3 in Chap. 1). We will use this insight below when characterizing the state price density.

220

11 Incomplete Markets (Utility Over Terminal Wealth)

The set of local martingale deflators Dl is a strict subset of the set of supermartingale deflator processes Ds . Indeed, this follows because a nonnegative local martingale is a supermartingale by Lemma 4 in Chap. 1. Hence, we have the follow relationships: stochastic processes

Ml ⊂ Dl ⊂ Ds ⊂ L+0 Ml ⊂ Dl ⊂ Ds ⊂ L0+ .

random variables

We can now prove the following equivalence. Lemma 27 (Budget Constraint Equivalence 2) C (x) = {XT ∈ L0+ : E [XT YT ] ≤ x for all YT ∈ Ds } Proof Let C1 (x) = {XT ∈ L0+ : E [XT YT ] ≤ x for all YT ∈ Ds }. (Step 1) Show C1 (x) ⊂ C (x). If XT ∈ C1 (x), then E [XT YT ] ≤ x for all YT ∈ Ds . Since Ml ⊂ Ds , E [XT YT ] ≤ x for all YT ∈ Ml . Hence XT ∈ C (x). (Step 2) Show C (x) ⊂ C1 (x). If XT ∈ C (x), then E[XT Zn (T )] ≤ x for all Zn (T ) ∈ Ml . Choose a YT ∈ Ds . Note that there exists a sequence Zn (T ) ∈ Ml such that YT ≤ lim Zn (T ) a.s. n→∞

Thus, XT YT ≤ lim XT Zn a.s. since XT ≥ 0. n→∞

Taking expectations, E[XT YT ] ≤ E[ lim XT Zn (T )] ≤ lim E[XT Zn (T )] n→∞

n→∞

where the second inequality is due to Fatou’s lemma. But, E[XT Zn (T )] ≤ sup E [XT YT ] for all n. Y ∈Ml

Hence, lim E[XT Zn (T )] ≤ sup E [XT YT ]. n→∞

YT ∈Ml

This implies that E[XT YT ] ≤ sup E [XT YT ] ≤ x. YT ∈Ml

This is true for an arbitrary YT ∈ Ds , hence E[XT YT ] ≤ x for all YT ∈ Ds . This shows that XT ∈ C1 (x). This completes the proof. We can finally rewrite the optimization problem in a form where a solution to its dual problem can be proven to exist. Problem 7 (Choose the Optimal Derivative XT for x ≥ 0) v(x) =

sup E [U (XT )]

XT ∈C (x)

11.2 Problem Statement

221

where C (x) = {XT ∈ L0+ : E [XT YT ] ≤ x for all YT ∈ Ds }. To solve this problem, define the Lagrangian function L (XT , y, YT ) = E [U (XT )] + y (x − E [XT YT ]) .

(11.4)

The primal problem can be written as "

! primal :

inf

sup XT ∈L0+

y>0,YT ∈Ds

L (XT , y, YT ) =

sup E [U (XT )] = v(x).

XT ∈C (x)

Proof Note that E [XT YT ] ≤ x for all YT ∈ Ds if and only if sup E [XT YT ] ≤ x. Hence, YT ∈Ds

sup (E [XT YT ] − x) ≤ 0. But,

YT ∈Ds

sup (E [XT YT ] − x) = − inf (x − E [XT YT ]). Substitution yields

YT ∈Ds

inf (x − E [XT YT ]) ≥ 0.

YT ∈Ds

YT ∈Ds

Hence, inf

7

< inf (x − E [XT YT ]) =

y

YT ∈Ds

y>0

Or, inf

y>0,YT ∈Ds

{y (x − E [XT YT ])} =

⎧ ⎨ 0

sup = sup

XT ∈L0+

inf

y>0,YT ∈Ds

⎧ ⎨ 0

if

⎩ −∞

otherwise. sup E [XT YT ] ≤ x

YT ∈Ds

otherwise.

sup E [XT YT ] ≤ x

YT ∈Ds

L (XT , y, YT )

⎧ ⎨ E [U (XT )] if ⎩

YT ∈Ds

⎩ −∞

This implies {E [U (XT )] + y (x − E [XT YT ])} inf y>0,Y ∈D s T ⎧ ⎨ E [U (XT )] if sup E [XT YT ] ≤ x YT ∈Ds = ⎩ −∞ otherwise. Substitution yields ⎧ ⎨ E [U (XT )] if inf L (XT , y, YT ) = ⎩ y>0,YT ∈Ds −∞ So, " ! XT ∈L0+

sup E [XT YT ] ≤ x

if

−∞

sup E [XT YT ] ≤ x

YT ∈Ds

otherwise

otherwise.

222

=

11 Incomplete Markets (Utility Over Terminal Wealth)

sup E [U (XT )] = v(x).

XT ∈C (x)

This completes the proof. The dual problem is ⎞

⎛ dual :

inf

y>0,YT ∈Ds

⎝ sup L (XT , y, YT )⎠ . XT ∈L0+

For optimization problems we know that ⎞

⎛ inf

y>0,YT ∈Ds

⎝ sup L (XT , y, YT )⎠ ≥ sup XT ∈L0+

"

! XT ∈L0+

L (XT , y, YT ) = v(x).

inf

y>0,YT ∈Ds

Proof We have L (XT , y) = E [U (XT )] + y (x − E [XT YT ]). Note that sup L (XT , y, YT ) ≥ L (XT , y, YT ) ≥ inf L (XT , y, YT ) XT ∈L0+

y>0

for all (XT , y, YT ). Taking the infimum over (y, YT ) on the left side and the supremum over gives ⎛ XT on the right side ⎞ " ! ⎠ ⎝ inf sup L (XT , y, YT ) ≥ sup inf L (XT , y, YT ) . This y>0,YT ∈Ds

XT ∈L0+

XT ∈L0+

y>0,YT ∈Ds

completes the proof. The solutions to the primal and dual problems are equal if there is no duality gap. We solve the primal problem by solving the dual problem and showing that there is no duality gap.

11.3 Existence of a Solution We need to show that there is no duality gap and that the optimum is attained, equivalently, a saddle point exists for the Lagrangian. To prove the existence of a saddle point, we need two assumptions in addition to the previous assumption that S satisfies NFLVR. These are the same two assumptions used in Chap. 10 in a complete market to guarantee the existence of a solution to the trader’s optimization problem. Assumption (Reasonable Asymptotic Elasticity) AE(U, ω) = limsup x→∞

xU  (x, ω) 0 such that v(x) < ∞. The motivation for the second assumption is obvious. If it is violated, then v(x) = ∞ for all x > 0, which results in an uninteresting optimization problem. We can now prove that the investor’s optimization problem exists and has a unique solution. Theorem 53 (Existence of a Unique Saddle Point and Characterization of v(x) for x ≥ 0) Given the above assumptions, there exists a unique saddle point, i.e. there exists a unique (Xˆ T , YˆT , y) ˆ such that ⎞

⎛ inf

YT ∈Ds , y>0

⎝ sup L (XT , YT , y)⎠ = L (Xˆ T , YˆT , y) ˆ XT ∈L0+

"

! = sup

XT ∈L0+

inf

YT ∈Ds , y>0

L (XT , YT , y) = v(x).

(11.5)

  Define v(y) ˜ = inf E U˜ (yYT ) where YT ∈Ds

U˜ (y, ω) = sup [U (x, ω) − xy],

y > 0.

x>0

Then, v and v˜ are in conjugate duality, i.e. ˜ + xy) , ∀x > 0 v(x) = inf (v(y)

(11.6)

y>0

v(y) ˜ = sup (v(x) − xy) , ∀y > 0

(11.7)

x>0

In addition, (i) v˜ is strictly convex, decreasing, differentiable on (0, ∞), (ii) v is strictly concave, increasing, and differentiable on (0, ∞), (iii) defining yˆ to be where the infimum is attained in expression (11.6), v(x) = v( ˜ y) ˆ + x y, ˆ

(11.8)

(iv) v(y) ˜ = inf E[U˜ (yYT )] = inf E[U˜ (yYT )]. YT ∈Ds

YT ∈Ml

Proof To prove this theorem, we show conditions 1 and 2 of the Lemma 28 in the appendix to this chapter hold. Under the above assumptions, these conditions follow from Zitkovic [187, Theorem 4.2]. To apply this theorem note the following facts. First, choosing the stochastic clock in Zitkovic [187] as in Example 2.6 (2), p. 757 gives the objective as the utility of terminal wealth. Second, choose the random

224

11 Incomplete Markets (Utility Over Terminal Wealth)

endowment to be identically zero. Third, our hypotheses on the utility function defined above satisfy Zitkovic [187, Definition 2.3, p. 755] except for condition (b). But, for utility of terminal wealth satisfying the reasonable asymptotic utility assumption (AE(U ) < 1), these conditions are automatically satisfied, see Karatzas and Zitkovic [125, Example 3.2, p. 1838]. Although Zitkovic [187] requires that S is locally bounded, this condition is unnecessary for utility of terminal wealth as shown in Karatzas and Zitkovic [125]. This completes the proof. Remark 70 (Alternative Sufficient Conditions for Existence) The sufficient conditions (i) there exists an x > 0 such that v(x) < ∞ and (ii) AE(U ) < 1 can be replaced by the alternative sufficient conditions v(y) ˜ < ∞ for all y > 0 and v(x) > −∞ for all x > 0. The same proof as given for Theorem 53 works but with Mostovyi [151] Theorem 2.3 replacing Zitkovic [187, Theorem 4.2]. In this circumstance, the utility function does not need to satisfy condition (b) of Zitkovic [187, Definition 2.3]. Here, Mostovyi [151] extends the domain of U to include x = 0 and the range of U to include −∞ at x = 0. In addition, the assumption that the set of equivalent local martingale measures Ml = ∅ can be replaced by the set of local martingale deflator processes Dl = ∅. A sketch of the proof is as follows. The existence of a buy and hold nonnegative wealth s.f.t.s. strategy implies that the constraint set C e (x) for v is nonempty, and Dl = ∅ implies that the constraint set for v˜ is nonempty because Dl ⊂ Ds = ∅. Given the appropriate topologies for C e (x) and Dl , assuming that v(y) ˜ < ∞ for all y > 0 and v(x) > −∞ for all x > 0 together imply, because of the strict convexity of v(y) ˜ for all y > 0 and the strict concavity of v(x) for all x > 0, that a unique solution exists to each of these problems. Then, the conditions of Lemma 28 in the appendix of Chap. 11 apply to obtain the remaining results. This completes the remark.

11.4 Characterization of the Solution This section characterizes the solution to the investor’s optimization problem.

11.4.1 The Characterization To characterize the solution, we focus on the solution to the dual problem, i.e. v(x) =

inf

⎛ =

inf

y>0,YT ∈Ds

( sup L (XT , y, YT ))

y>0,YT ∈Ds XT ∈L0 +

⎞

⎝ sup E [U (XT ) − yXT YT ] + xy ⎠ . XT ∈L0+

11.4 Characterization of the Solution

225

To solve, we first exchange the sup and expectation operator. ⎡ v(x) =

inf

y>0,YT ∈Ds

⎤

E ⎣ sup (U (XT ) − yXT YT ) + xy ⎦

(11.9)

XT ∈L0+

The proof of this step is identical to that given in the appendix to Chap. 10. We now solve this problem, working from the inside out. Step 1. (Solve for XT ∈ L0+ ) Fix y, YT and solve for the optimal wealth Xˆ T ∈ L0+ : sup [U (XT ) − yXT YT ] .

XT ∈L0+

To solve this problem, fix a ω ∈ Ω, and consider the related problem (U (XT (ω), ω) − yXT (ω)YT (ω)) .

sup XT (ω)∈R+

This is a simple optimization problem on the real line. The first order condition for an optimum gives U  (XT (ω), ω) − yYT (ω) = 0, Xˆ T (ω) = I (yYT (ω), ω)

or

with I (·, ω) = (U  (·, ω))−1 .

Given the properties of I (·, ω), Xˆ T (ω) is FT −measurable, hence Xˆ T (ω) ∈ L0+ . Note that Xˆ T depends on y, YT . Step 2. (Solve for YT ∈ Ds ) Given the optimal wealth Xˆ T and y, solve for the optimal supermartingale deflator YˆT ∈ Ds . Using conjugate duality (see Lemma 25 in Chap. 9), we have U˜ (yYT ) = sup [U (XT ) − XT yYT ] = [U (Xˆ T ) − Xˆ T yYT ]. XT ∈L0+

This transforms the problem to ⎡ inf

y>0,YT ∈Ds

inf

y>0,YT ∈Ds

⎤

E ⎣ sup (U (XT ) − yXT YT ) + xy ⎦ = XT ∈L0+

(E[U˜ (yYT )] + xy) = inf ({ inf E[U˜ (yYT )]} + xy). y>0

YT ∈Ds

226

11 Incomplete Markets (Utility Over Terminal Wealth)

We solve the inner most infimum problem first. Defining v(y) ˜ as the value function for this infimum, let YˆT be the solution such that v(y) ˜ = E[U˜ (y YˆT )] = inf E[U˜ (yYT )]. YT ∈Ds

Note that YˆT depends on y. Step 3. (Solve for y > 0) Given the optimal wealth Xˆ T and supermartingale deflator YˆT , solve for the optimal Lagrangian multiplier y > 0. ˜ + xy) inf (v(y) y>0

Taking the derivative of the right side with respect to y, one gets v˜  (y) + x = 0. Substituting the definition for v˜ and taking the derivative yields E[U˜  (y YˆT )YˆT ] + x = 0. This step requires taking the derivative underneath the expectation operator. The same proof as in the appendix to Chap. 10 applies. Noting that U˜  (y) = −I (y) from Lemma 25 in Chap. 9, we get   E I (yˆ YˆT )YˆT = E[Xˆ T YˆT ] = x,

(11.10)

which is that the optimal terminal wealth Xˆ T satisfies the budget constraint with an equality. Solving this equation gives the optimal y. ˆ Step 4. (Characterization of Xˆ ∈ X (x) for x ≥ 0 ) Step 1 solves for the optimal wealth Xˆ T ∈ L0+ , a random variable. Since Xˆ T ∈ t C (x), there exists a stochastic process Z ∈ X (x) such that Zt = x + 0 αu · dSu ∈ L+0 with Xˆ T = ZT for some (α0 , α) ∈ N (x). For convenience, let us label this process Z as Xˆ ∈ L+0 . This gives the first characterization of the optimal wealth process Xˆ using a nonnegative wealth s.f.t.s. (α0 , α) ∈ N (x). We can also obtain a second characterization. By the definition of the supermartingale deflator YˆT ∈ Ds , there exists a stochastic process Y ∈ Ds with YˆT = YT . Label this process Y ∈ Ds as Yˆ . Then, by the definition of Ds , Yˆt Xˆ t is a supermartingale under P. Hence,   E Xˆ T YˆT |Ft ≤ Xˆ t Yˆt

11.4 Characterization of the Solution

227

for all t ∈ [0, T ]. But by expression (11.10) we have that E[Xˆ T YˆT ] = x = Xˆ 0 Yˆ0 . By Lemma 3 in Chap. 1, Xˆ t Yˆt is a P martingale. Hence,

' & YˆT

ˆ ˆ Xt = E XT

Ft Yˆt

(11.11)

for all t ∈ [0, T ]. Step 5. (Nonnegativity of Xˆ ∈ X (x)) As noted in Step 4 above, the wealth process Xˆ ≥ 0 because Xˆ T ∈ C (x), which is the set of random variables dominated by nonnegative wealth s.f.t.s’s in the set N (x). Remark 71 (Random Variables Versus Stochastic Processes) The optimization problem determines the random variable Xˆ T ∈ L0+ , which is the optimal time T wealth. But, since Xˆ T ∈ C (x), there exists a nonnegative stochastic process t Z ∈ X (x) such that ZT = XT and Zt = x + 0 αu · dSu for some nonnegative wealth s.f.t.s. (α0 , α) ∈ N (x). Note that the equality in this statement is justified by the free disposal Lemma 26. We note that if the risky assets are redundant, as defined in Chap. 2, then the optimal trading strategy will not be unique. However, if the risky assets are non-redundant, then it is (see Sect. 2.4 in Chap. 2 for further discussion). A sufficient condition on the risky asset price process S such that the risky assets are non-redundant is given in Theorem 10 in Chap. 2. This characterization of the optimal wealth (stochastic) process X ∈ X (x) can be used to determine the nonnegative wealth s.f.t.s. generating it using the methods for constructing a synthetic derivative in Chap. 2. The optimization problem also determines the random variable YˆT ∈ Ds , which is the trader’s optimal supermartingale deflator. By the definition of Ds there exists a stochastic process Y ∈ Ds with YˆT = YT . Label this process Y ∈ Ds as Yˆ . Yˆ is the optimal supermartingale deflator process. Unlike the situation in a complete market, in an incomplete market the supermartingale deflator process can exhibit many different properties. In order of increasing restrictiveness, these properties are listed below. Yˆ ∈ Ds is a supermartingale deflator process. process and YˆT = dQ Yˆ ∈ Ds where Yˆ is a supermartingale deflator dP is a  

ˆ ˆ probability density for P. Here, Yt = E YT Ft and X ∈ X (x) is a Q supermartingale. 3. Yˆ ∈ Dl ⊂ Ds where Yˆ is a local martingale deflator process, i.e. given a X ∈ X (x), XYˆ is a P local martingale.

1. 2.

228

11 Incomplete Markets (Utility Over Terminal Wealth)

4. Yˆ ∈ Ml ⊂ Dl where Yˆ is a local martingale deflator process, YˆT = dQ dP is a  

probability density with respect to P. Here, Yˆt = E YˆT Ft and X ∈ X (x) is a Q local martingale. 5. Yˆ ∈ M ⊂ Ml where Yˆ is a martingale deflator process, YˆT = dQ dP is a  

probability density with respect to P. Here, Yˆt = E YˆT Ft and X ∈ X (x) is a Q martingale. We note that for cases (2,4 and 5) where YˆT ∈ Ds is a probability density for P, Yˆ is a P martingale. Necessary and sufficient conditions for case (5), where the supermartingale deflator is both a martingale deflator and a probability density with respect to P, i.e. YˆT ∈ M, are contained in Kramkov and Weston [133]. This completes the remark.

11.4.2 Summary For easy reference we summarize the previous characterization results. The optimal value function v(x) =

sup E [U (XT )]

XT ∈C (x)

for x ≥ 0 has the solution for terminal wealth given by Xˆ T = I (yˆ YˆT )

with I = (U  )−1 = −U˜ 

where YˆT ∈ Ds is the optimal supermartingale deflator. The optimal supermartingale deflator is the solution to v(y) ˜ = inf E[U˜ (yYT )]. YT ∈Ds

By the definition of the set Ds , there is a supermartingale deflator process Yˆ ∈ Ds ⊂ L0+ such that its time t value is YˆT . The Lagrangian multiplier yˆ is the solution to the budget constraint   E YˆT I (yˆ YˆT ) = x. The optimal wealth Xˆ T satisfies the budget constraint with an equality   E YˆT Xˆ T = x.

11.6 The Supermartingale Deflator

229

The optimal wealth process Xˆ ∈ X (x) exists since Xˆ T ∈ C (x), i.e. there exists ˆ a nonnegative wealth  t s.f.t.s. (α0 , α) ∈ N (x) with time T payoff equal to that XT where Xˆ t = x + 0 αu · dSu for all t. This optimal nonnegative wealth s.f.t.s. is unique if and only if the risky assets in the market are non-redundant. The optimal portfolio wealth process Xˆ ∈ X (x) when multiplied by the supermartingale deflator, Xˆ t Yˆt ≥ 0, is a P martingale. This implies that for an arbitrary time t ∈ [0, T ], we have &

YˆT Xˆ t = E Xˆ T Yˆt

'

Ft ≥ 0.

It is important to note, for subsequent use, that the solution YˆT ∈ Ds depends on the utility function U .

11.5 The Shadow Price Using the characterization of the value function, we can obtain the shadow price of the budget constraint. The shadow price is the benefit, in terms of expected utility, of increasing the initial wealth by 1 unit (of the mma). The shadow price equals the Lagrangian multiplier. Theorem 54 (The Shadow Price of the Constraint) yˆ = v  (x) ≥ E[U  (Xˆ T )]

(11.12)

with equality if and only if the supermartingale deflator YˆT ∈ Ds is a probability density with respect to P, i.e. E[YˆT ] = 1. Proof The first equality is obtained by taking the derivative of v(x) = v( ˜ y) ˆ + x y. ˆ The second inequality follows from the first order condition for an optimum U  (Xˆ T ) = yˆ YˆT . Taking expectations gives E[U  (Xˆ T )] = E[yˆ YˆT ] = yE[ ˆ YˆT ] ≤ yˆ because E[YˆT ] ≤ 1. Equality occurs if and only if E[YˆT ] = 1. This completes the proof.

11.6 The Supermartingale Deflator The key insight of the individual optimization problem is the characterization of the trader’s supermartingale deflator process Y ∈ Ds in terms of the trader’s utility function and initial wealth. To obtain this characterization, using Theorem 54 above,

230

11 Incomplete Markets (Utility Over Terminal Wealth)

we rewrite the first order condition for Xˆ T : U  (Xˆ T ) U  (Xˆ T ) U  (Xˆ T ) =  ≤ YˆT = yˆ v (x) E[U  (Xˆ T )]

(11.13)

with equality if and only if the supermartingale deflator YˆT ∈ Ds is a probability density with respect to P, i.e. E[YˆT ] = 1. This is similar to the result relating the local martingale measure to preferences in the solution to the investor’s optimization problem in Chap. 10. Recall that in a complete market, the supermartingale deflator is a local martingale deflator, i.e. YˆT ∈ Ml ⊂ Dl ⊂ Ds . The difference in an incomplete market is that the supermartingale deflator YˆT ∈ Ds need not be a local martingale deflator nor a probability density with respect to P, i.e. YˆT ∈ / Ml is possible. Assume now that YˆT ∈ Ml , i.e. YˆT is a local  martingale deflator and a probability density with respect to P where E YˆT = 1. However, YˆT need not be the probability density of a martingale measure. In this case, the individual’s state price density is given by the P martingale,  E[U  (Xˆ ) |F ]  T t Yˆt = E YˆT |Ft =  ˆ E[U (XT )]

(11.14)

for all t ∈ [0, T ]. Here, the supermartingale deflator equals the investor’s marginal utility of wealth, normalized by the first derivative of the optimal value function. Given investors trade in a competitive market, prices are taken as exogenous. Because the market is incomplete, prices need not uniquely determine the equivalent local martingale measure Q ∈ Ml . The individual’s optimization problem determines a unique local martingale measure from among all YT ∈ Ml that the investor can utilize to price derivatives. Different investors may have different local martingale deflators YT ∈ Ml . Remark 72 (Asset Price Bubbles) In this individual’s optimization problem, prices and hence the local martingale measure Q are taken as exogenous. The above analysis does not require the condition that Q is a martingale measure. In addition, even if the individual’s supermartingale deflator is a state price density, i.e. YˆT ∈ Ml , it need not be a martingale measure (it could be a strict local martingale measure). Hence, the solution to the trader’s optimization problem applies to a market with asset price bubbles (see Chap. 3). This completes the remark. Remark 73 (Systematic Risk) In this remark, we use the non-normalized market ((B, S), F, P) representation to characterize systematic risk. Recall that the state price density is the key input to the systematic risk return relation in Chap. 4, Theorem 38. To apply this theorem, we assume that the supermartingale deflator YˆT ∈ Ds is a martingale deflator, i.e. YˆT ∈ M. Then, the risky asset returns over the

11.7 The Optimal Trading Strategy

231

time interval [t, t + Δ] satisfy

'

E[U  (Xˆ T ) |Ft+Δ ] Bt

E [Ri (t) |Ft ] = r0 (t) − cov Ri (t), (1 + r0 (t)) Ft

E[U  (Xˆ T ) |Ft ] Bt+Δ &

(11.15) 1 i (t) , r0 (t) = p(t,t+Δ) −1 where the ith risky asset’s return is Ri (t) = Si (t+Δ)−S Si (t) is the default-free spot rate of interest where p(t, t + Δ) is the time t price of a default-free zero-coupon bond maturing at time t + Δ, and the state price density is  ˆ T |Ft ] Ht = BYtt = E[YB = E[U ( XˆT )|Ft ] . t E[U (XT )]Bt

When the time step is very small, i.e. Δ ≈ dt, the expression and this relation simplifies to

Bt Bt+Δ (1

' E[U  (Xˆ T ) |Ft+Δ ]

E [Ri (t) |Ft ] ≈ r0 (t) − cov Ri (t),

Ft . E[U  (Xˆ T ) |Ft ]

+ r0 (t)) ≈ 1,

&

(11.16)

This is the same expression that we obtained in Chap. 10 in a complete market. The same discussion regarding the interpretation of systematic risk therefore applies. This completes the remark.

11.7 The Optimal Trading Strategy This section solves step 2 of the optimization problem which is to determine the optimal trading strategy (α0 , α) ∈ N (x) for x ≥ 0. This is done using the standard techniques developed for the synthetic construction of derivatives in Chap. 2. Since the optimal wealth Xˆ T ∈ C (x), we know there exists a nonnegative wealth s.f.t.s. (α0 , α) ∈ N (x) such that Xˆ T = x +



T

αt · dSt ,

0

and an optimal wealth process Xˆ ∈ X (x) such that Xˆ t = x +



t

αu · dSu

0

for all t ∈ [0, T ]. This uses the free disposal Lemma 26. We need to characterize this nonnegative wealth s.f.t.s. (α0 , α) ∈ N (x) given the initial wealth x ≥ 0. To illustrate this determination, suppose that S is a Markov diffusion process. Due to the diffusion assumption, S is a continuous process. Given this assumption, we can write the optimal wealth process as a deterministic function

232

11 Incomplete Markets (Utility Over Terminal Wealth)

of time t and the risky asset prices St , i.e.

'

' & YˆT

YˆT

ˆ ˆ E XT

Ft = E XT

St = Xˆ t (St ). Yˆt Yˆt &

Using Ito’s formula (assuming the appropriate differentiability conditions), we have Xˆ T (ST ) = x +



T

0

1 ∂ Xˆ t dt + ∂t 2



T 0

 T ˆ n  n  ) ∂ 2 Xˆ t ( ∂ Xt · dSt d S (t), S (t) + i j 2 ∂S ∂Sij 0 i=1 j =1 (11.17)

ˆ

ˆ

ˆ

∂X ∂X  where ∂∂SX = ( ∂S , . . . , ∂S ) ∈ Rn . Equating the coefficients of dSt in the two n 1 preceding stochastic integral equations implies that the trading strategy is

∂ Xˆ t (St ) = αt . ∂St This is the standard “delta” used to construct the synthetic derivative XˆT as in Chap. 2. Finally, α0 is determined given α by the value process expression Xt = α0 + α · St . If the risky assets are non-redundant, then this nonnegative wealth s.f.t.s. is unique. Otherwise, it is not (see Sect. 2.4 in Chap. 2 for additional discussion). If the underlying asset price process is not Markov, then for a large class of processes one can use Malliavin calculus to determine the trading strategy (see Detemple et al. [50] and Nunno et al. [51]).

11.8 An Example This section presents an example to illustrate the abstract expressions of the previous sections. This example is obtained from Pham [156, p. 197].

11.8.1 The Market We consider the normalized Brownian motion market given in Sect. 2.8 of Chap. 2. Suppose the normalized market (rt = 0 for all t ≥ 0) consists of n risky assets and

11.8 An Example

233

a mma where the risky asset price process evolves as ! dSi (t) = Si (t) bi (t)dt +

D 

" (11.18)

σid (t)dWd (t)

d=1

for all t ∈ [0, T ] and i = 1, . . . , n where Wt = W (t) = (W1 (t), . . . , WD (t)) ∈ RD are independent Brownian motions with Wd (0) = 0 for all d = 1, . . . , D, and ⎡ ⎢ ⎢ σt = σ (t) = ⎢ ⎣

σ11 (t) · · · σ1D (t)

⎤

⎥ ⎥ .. .. ⎥. . . ⎦ σn1 (t) · · · σnD (t)

(11.19)

n×D

In vector notation, we can write the evolution of the stock price process as dSt = diag(St )bt dt + diag(St )σt dWt

(11.20)

where dSt = (dS1 (t), . . . , dSn (t)) ∈ Rn and diag(St ) equals the n × n diagonal matrix with elements (S1 (t), . . . , Sn (t)) in the diagonal. We assume that (1) rank (σt ) = n for all t a.s. P, and T . −1 . .  .2 bt . dy < ∞. (2) 0 .σt σt σt Since rank (σt ) = min{D, n}, this implies that D ≥ n, and it implies that the risky assets are non-redundant (see Sect. 2.4 of Chap. 2). By Theorems 19 and 20 in Chap. 2, the market satisfies NFLVR and by the First Fundamental Theorem 13 of asset pricing in Chap. 2, Ml = ∅. It was shown in Theorem 21 in Chap. 2 that the set of local martingale measures is Ml = {Qν :

T

dQν dP

T

− 0 (θt +νt )·dWt − 2 0 θt +νt  = eν  E dQ dP = 1, ν ∈ K(σ )} 1

2 dt

> 0,

where  −1 θt = σt σt σt bt and  K (σ ) = {ν ∈ L (W ) : 0

T

νt 2 dt < ∞, σt νt = 0

for all t}.

234

11 Incomplete Markets (Utility Over Terminal Wealth)

We want the market to be incomplete. Using Theorem 22 in Chap. 2, we assume that D > n. For later use, we need the following sets. Dl = {YT ∈ L0+ : YT = e−

T 0

(θt +νt )·dWt − 12

T 0

θt +νt 2 dt

> 0, ν ∈ K(σ )}

is the set of local martingale deflators, and Ml = {YT ∈ L0+ : YT = e−

T 0

(θt +νt )·dWt − 12

T 0

θt +νt 2 dt

> 0, E [YT ] = 1, ν ∈ K(σ )}

is the set of local martingale deflators that are probability densities with respect to P, hence Yt = E [YT |Ft ] for YT ∈ Ml .

11.8.2 The Utility Function Let preferences be represented by a state independent logarithmic utility function U (x) = ln(x)

for x > 0.

Recall that I (y) =

1 y

and

U˜ (y) = −lny − 1 for y > 0.

11.8.3 The Optimal Supermartingale Deflator Using Theorem 53, the optimal supermartingale deflator YˆT is the solution to   v(y) ˜ = inf E U˜ (yYT ) . YT ∈Ds

From Theorem 53 again, we have that   v(y) ˜ = inf E U˜ (yYT ) YT ∈Dl

11.8 An Example

235

  because Ml ⊂ Dl and v(y) ˜ = inf E U˜ (yYT ) . Karatzas et al. [122] (see also YT ∈Ml

Pham [156, p. 197]) show that for this Brownian motion market the solution is attained in the set Dl . We solve for the optimal element in this set. In this regard, note that E U˜ (yYT ) = −E [ln (yYT )] − 1     T 1 T 2 = −E ln ye− 0 (θt +νt )·dWt − 2 0 θt +νt  dt − 1   T T = −E ln (y) − 0 (θt + νt ) · dWt − 12 0 θt + νt 2 dt − 1.     T If E 0 θt + νt 2 dt = ∞, then E U˜ (yYT ) = ∞. Hence, we can restrict   T consideration to those ν such that E 0 θt + νt 2 dt < ∞. In this case,   T T E 0 (θt + νt ) · dWt = 0 since 0 (θt + νt ) · dWt is a martingale (see Protter [158, p. 171]).     T Hence, E U˜ (yYT ) = −ln (y) + 12 E 0 θt + νt 2 dt − 1. Since θt + νt 2 = θt 2 + νt 2 , inspection shows that the infimum is obtained at νt = 0. Thus, the optimal supermartingale deflator is YˆT = e−

T 0

θt ·dWt − 12

T 0

θt 2 dt

.

We note that this solution is independent of y.

11.8.4 The Optimal Wealth Process The optimal wealth process is Xˆ T = I (yˆ YˆT ) =

 1 T 1 1 T 2 = e 0 θt ·dWt + 2 0 θt  dt . yˆ yˆ YˆT

And, for an arbitrary time t, & Xˆ t = E

' 1 YˆT

Ft . yˆ YˆT Yˆt

Algebra gives  1 t 1 1 t 2 Xˆ t = = e 0 θs ·dWs + 2 0 θs  ds . yˆ yˆ Yˆt

236

11 Incomplete Markets (Utility Over Terminal Wealth)

Recall that yˆ is the solution to the budget constraint &

YˆT Xˆ 0 = E I (yˆ YˆT ) Yˆt

' = x.

Substitution for time 0 yields x=

1 . yˆ

Combined we get that the optimal wealth process satisfies Xˆ t = xe

t

 1 t 2 0 θs ·dWs + 2 0 θs  ds

.

(11.21)

11.8.5 The Optimal Trading Strategy To get the optimal trading strategy we apply Ito’s formula to expression (11.21), which yields d Xˆ t = Xˆ t θt  2 dt + Xˆ t θt · dWt . Proof Dropping all the subscripts and superscripts, 1 ∂2X ∂X dX = ∂X ∂t dt + 2 ∂W 2 dt + ∂W dW . = X 12 θ  2 dt + 12 X θ  2 dt + Xθ · dW . This completes the proof. Since the optimal wealth Xˆ T ∈ C (x), there exists a nonnegative wealth s.f.t.s. (α0 , α) ˆ ∈ N (x) with x ≥ 0 such that d Xˆ t = αt · dSt = diag (St ) αt · bt dt + diag (St ) αt · σt dWt . Equating the coefficients of the Brownian motion terms yields (diag (St ) αt ) σt = θt Xˆ t where the holdings in the mma are determined by the expression αˆ 0 (t) = Xˆ t − αˆ t · St . Because the risky assets are non-redundant, this nonnegative wealth s.f.t.s. is unique.  −1 bt . The algebra is: We solve this expression for αt noting that θt = σt σt σt σt (αt diag (St )) = θt Xˆ t

11.9 Differential Beliefs

237

 −1 σt (αt diag (St )) = σt σt σt bt Xt  −1    σ bt Xt  t σt (α  t diag (St )) = σt σt σt σt σt σt (αt diag (St )) = bt Xt . Since σt is of full rank which equals n, by Theil [183, p. 11], σt σt also has rank equal to n and is invertible. Hence, expressing the holdings in the risky assets as a proportion of wealth, we get πˆ t =

−1 (αt diag (St ))  = σt σt bt . Xt

(11.22)

11.8.6 The Value Function Last, to obtain the value function, note that     t  1 t 2 v(x) = E U (Xˆ T ) = E U (xe 0 θs ·dWs + 2 0 θs  ds )  = ln (x) + E 0

T

1 θs · dWs + 2



T

0



1 θs  ds = ln (x) + E 2



2

T

 θs  ds . 2

0

 −1 Substituting θt = σt σt σt bt gives the optimal value function 1 v(x) = ln (x) + E 2



T 0

 .  −1 . .  .2 bt . ds . .σt σt σt

(11.23)

This completes the example.

11.9 Differential Beliefs This section discusses the changes to the investor’s optimization problem that are needed when we introduce differential beliefs in subsequent chapters, where the trader’s beliefs differ from the statistical probability measure P. Suppose, therefore, that P represents the statistical probability measure and Pi represents trader i’s i beliefs where Pi and P are equivalent probability measures. Thus dP dP > 0 is a   dPi FT measurable random variable and E dP = 1. The investor’s objective function is  Ei [U (x, ω)] = E

 dPi U (x, ω) . dP

(11.24)

238

11 Incomplete Markets (Utility Over Terminal Wealth)

Recall from Sect. 9.8 in Chap. 9 that given the modified utility function its convex conjugate is  sup x>0

dPi U (x) − yx dP

  dPi ˜ dP . = U y dP dPi



dPi dP U (x, ω),

(11.25)

The reason for this change is that  Ds = Y ∈ L+0 : Y0 = 1, XY is a P supermartingale,  X = 1 + α · dS, (α0 , α) ∈ N (1) depends on the probability measure P, and we want to keep Ds the same across all investors. The investor’s optimal value function is given by 

dPi U (XT ) vi (x) = sup E dP XT ∈C (x) ⎡ =

inf

y>0,YT ∈Ds

E ⎣ sup XT ∈L0+

!7 = inf y>0



 (11.26)

dPi U (XT ) − yXT YT dP



⎤ + xy ⎦

" <  dP dPi ˜ inf E + xy . U yYT dP dPi YT ∈Ds 

The optimal solution Xˆ T solves  sup XT ∈L0+

dPi U (XT ) − yX ˆ T YˆT dP

 ,

i.e. dPi  ˆ U (XT ) − yˆ YˆT = 0, dP

(11.27)

or   dP ˆ ˆ XT = I yˆ YT dPi

with I = (U  )−1 = −U˜  .

Furthermore, YˆT ∈ Ds solves  ˆ = inf E v˜i (y) YT ∈Ds

  dP dPi ˜ . U yY ˆ T dP dPi

(11.28)

11.10 Notes

239

As before, the solution yˆ solves inf (v˜i (y) + xy) y>0

or v˜i (y) ˆ + x = 0. Computing the derivative yields     dP ˆ  ˆ ˜ E U yˆ YT YT + x = 0. dPi Using I = (U  )−1 = −U˜  and the optimal Xˆ T shows that yˆ satisfies the budget constraint, i.e.       dP ˆ ˆ E I yˆ YT YT = E Xˆ T YˆT = x. dPi Finally, using the fact that v(x) = v( ˜ y) ˆ + x yˆ yields the shadow price of the constraint v  (x) = y. ˆ The optimal supermartingale deflator YˆT =

dPi  ˆ dP U (XT )

yˆ

=

dPi  ˆ dP U (XT ) . v  (x)

(11.29)

This completes the modifications to the investor’s optimization problem for differential beliefs.

11.10 Notes Excellent references for solving the investor’s optimization problem in an incomplete market are Dana and Jeanblanc [42], Duffie [52], Karatzas and Shreve [124], Merton [147], and Pham [156].

240

11

Incomplete Markets (Utility Over Terminal Wealth)

Appendix Lemma 28 (Existence of a Saddle Point) Assume that (1) a solution Xˆ T exists to v(x) =

sup

E [U (XT )] and

XT ∈C e (x)

˜ = inf E[U˜ (yYT )]. (2) a solution YˆT exists to v(y) YT ∈Ds

Then, v and v˜ are in conjugate duality, i.e. v(x) = inf (v(y) ˜ + xy) , ∀x > 0

(11.30)

y>0

v(y) ˜ = sup (v(x) − xy) , ∀y > 0

(11.31)

x>0

In addition, (i) v˜ is strictly convex, decreasing, differentiable on (0, ∞), (ii) v is strictly concave, increasing, and differentiable on (0, ∞), (iii) defining yˆ to be where the infimum is attained in expression (11.30), v(x) = v( ˜ y) ˆ + x y. ˆ And, (Xˆ T , YˆT , y) ˆ is a saddle point of L (Xˆ T , YˆT , y). ˆ Proof (Part 1) If a solution Xˆ T exists to v(x) =

sup

XT ∈C e (x)

E [U (XT )], then by the

free disposal Lemma 26, it has the identical solution and value function as v(x) = sup E [U (XT )].

XT ∈C (x)

Given v(x) define v ∗ (y) = sup (v(x) − xy) , ∀y > 0. x>0

First, because v(x) < ∞ for some x > 0, v is proper. v is increasing and strictly concave because v(x) = sup E [U (XT )] and U is increasing and strictly XT ∈C (x)

concave. This implies v(x) < ∞ for all x > 0 (see Pham [156, p. 181]). v(x) is strictly concave on (0, ∞), hence continuous on (0, ∞), and therefore upper semicontinuous. By Pham [156, Theorem B.2.3, p. 219], v and v ∗ are in conjugate duality, where v(x) = inf (v ∗ (y) + xy). y>0

By Pham [156, Proposition B.2.4, p. 219], we get that v ∗ (y) is differentiable on int (dom(v)). ˜ By Pham [156, Proposition B.3.5, p. 219], since v ∗ is strictly convex, we get v is differential on (0, ∞). To complete the proof, we need to show that v ∗ (y) = v(y). ˜

Appendix

241

(Step 1) By the definition of U˜ we have U (x) ≤ U˜ (y) + xy for all x > 0 and y > 0. Thus, U (XT ) ≤ U˜ (yYT ) + XT yYT for XT ∈ C (x). Taking expectations yields E [U (XT )] ≤ E[U˜ (yYT )]+yE [XT YT ] ≤ E[U˜ (yYT )]+yx. The last inequality uses XT ∈ C (x). Taking the supremum on the left side, the infimum on the right side, and using the definition of v(y) ˜ = inf E[U˜ (yYT )], we get: YT ∈Ds

v(x) ≤ v(y) ˜ + xy for all y > 0, or v(x) ≤ inf (v(y) ˜ + xy). y>0

(Step 2) We have U˜ (y) = U (I (y)) − yI (y) for all y > 0. ˜ Hence, U˜ (y YˆT ) = U (I (y YˆT )) − yYT I (y YˆT ) where YˆT attains v(y). Take expectations to get     v(y) ˜ = E U (I (y YˆT )) − yE YˆT I (y YˆT ) . ˜ + xy = Choose y such that XT = I (y YˆT ) ∈ C (x). Then, this equals v(y) E [U (XT )]. Taking the supremum on the right side yields v(y) ˜ + xy ≤ v(x). The infimum of the left side gives inf (v(y) ˜ + xy) ≤ v(x). y>0

Combined steps 1 and 2 show inf (v(y) ˜ + xy) = v(x). y>0

Because v(x) ˜ < ∞ for some x > 0, v˜ is proper. Because v(y) ˜ = inf E[U˜ (yYT )], we see that v˜ is strictly convex, hence YT ∈Ds

continuous on int (dom(v)) ˜ . By Pham [132, Theorem B.2.3, p. 219], v(y) ˜ = sup (v(x) − xy) , ∀y > 0. x>0

˜ + xy) = v(x). It exists Define yˆ to be where the infimum is attained in inf (v(y) y>0

because v(x) < ∞ and v˜ is strictly convex. Then, v(x) = v( ˜ y) ˆ + x y. ˆ ˆ is a saddle point if and only if the (Part 2) By Guler [66, p.278], L (Xˆ T , YˆT , y) solution to the primal problem equals the solution to the dual ⎛ problem, i.e. ⎞ " ! ⎝ sup L (XT , YT , y)⎠. v(x) = sup inf inf L (XT , YT , y) = YT ∈Ds , y>0

XT ∈L0+

YT ∈Ds , y>0

XT ∈L0+

We show this later condition. First, v(x) = inf (v(y) ˜ + xy). Using the definitions, this is equivalent to " !y>0 v(x) = inf inf E[U˜ (yYT )] + xy y>0

⎛

YT ∈Ds

⎡

⎤

⎞

= inf ⎝ inf E ⎣ sup U (XT ) − yXT YT ⎦ + xy ⎠ . y>0

YT ∈Ds

XT ∈L0+

242

11

Incomplete Markets (Utility Over Terminal Wealth)

Exchange the sup and E[·] operator, which is justified by the same proof as in the appendix to⎛Chap. 10. ⎛ ⎞ ⎞ v(x) = inf ⎝ inf ⎝ sup E [U (XT ) − yXT YT ]⎠ + xy ⎠ y>0

⎛

YT ∈Ds

⎛

XT ∈L0+

⎞⎞

= inf ⎝ inf ⎝ sup E [U (XT )] − y (E [XT YT ] − x)⎠⎠ y>0

⎛

YT ∈Ds

XT ∈L0+

⎛

⎞⎞

= inf ⎝ inf ⎝ sup L (XT , YT , y)⎠⎠. y>0

YT ∈Ds

XT ∈L0+

This completes the proof. Remark 74 (Extension to Chapter 12) This same proof works with XT ∈ C (x) ⊂ L0+ replaced by (c, XT ) ∈ C (x) ⊂ L+0 × L0+ and  Y ∈ Ds ⊂ T 0 sup E 0 U1 (ct )dt + U2 (XT ) and v(y) L+ where v(x) = ˜ = (c,XT )∈C (x)   T inf E 0 U˜ 1 (yYt )dt + U˜ 2 (yYT ) . This completes the remark.

Y ∈Ds

Chapter 12

Incomplete Markets (Utility Over Intermediate Consumption and Terminal Wealth)

This chapter studies the investor’s optimization problem in an incomplete market where the investor has a utility function defined over both terminal wealth and intermediate consumption. The presentation parallels the portfolio optimization problem studied in Chap. 11, and it is based on Jarrow [94].

12.1 The Set-Up We need to alter the previous set-up to include intermediate consumption. Given is a normalized market (S, F, P) where the value of a money market account (mma) Bt = 1 for all t ≥ 0. Here, the utility function is defined over terminal wealth and intermediate consumption. Both terminal wealth and consumption are denominated in units of the mma. As discussed in Chap. 9, this is without loss of generality. In addition to choosing a trading strategy, now an investor also chooses a consumption plan c : ×[0, T ] → [0, ∞), which is a nonnegative optional process. Thus, we consider a triplet, consisting of a trading strategy and consumption plan (α0 , α, c) ∈ (O, L (S), O). The value of the trading strategy at time t is equal to Xt = α0 (t) + αt · St .

(12.1)

This is the same wealth process as in the previous portfolio optimization problems studied in Chaps. 10 and 11 without consumption. Next, define the cumulative time t consumption as 

t

Ct =

cu du

(12.2)

0

© Springer Nature Switzerland AG 2021 R. A. Jarrow, Continuous-Time Asset Pricing Theory, Springer Finance, https://doi.org/10.1007/978-3-030-74410-6_12

243

244

12 Incomplete Markets (Utility Over Intermediate Consumption and Terminal Wealth)

t where ct ≥ 0 all t a.s. P. Note that 0 cs ds is well defined for all t ∈ [0, T ], although it could be +∞. If cumulative consumption is finite a.s. P, then C ∈ L+0 (adapted and cadlag) is continuous and of finite variation (see Lemma 1 in Chap. 1). We next introduce the extension of a self-financing trading strategy when there is consumption. Definition 40 (Self-Financing with Consumption) A trading strategy (α0 , α, c) ∈ (O, L (S), O) is said to be a self-financing trading strategy (s.f.t.s) with consumption if  t αu · dSu (12.3) Xt + Ct = x + 0

for all t ∈ [0, T ] a.s. P. Here, the change in value of the trading strategy “finances” both the value process Xt and the cumulative consumption Ct . We note that this trading strategy is not “truly” self-financing because there is a cash outflow for consumption. Note the analogy to the s.f.t.s. with cash flows as defined when discussing super- and sub-replication in Chap. 8 and Remark 68 in Chap. 11. We define the set of nonnegative wealth s.f.t.s.with consumption for x ≥ 0 as N (x) = {(α0 , α, c) ∈ (O, L (S), O) : Xt = α0 (t) +  αt · St , t x + 0 αu · dSu = Xt + Ct ≥ 0, ∀t ∈ [0, T ] .

(12.4)

t We note that for the mathematics, Xt +Ct = x+ 0 αu ·dSu becomes the fundamental quantity for defining the relevant sets of deflators and deflator processes. It represents the sum of the value process plus cumulative consumption funded by the mma and risky asset trading strategy across time. The set of wealth processes plus cumulative consumption generated by the nonnegative wealth s.f.t.s. with consumption is denoted    t X e (x) = (X + C) ∈ L+0 : ∃(α0 , α, c) ∈ N (x), Xt + Ct = x + αu · dSu , ∀t ∈ [0, T ] 0

and the set of time T terminal wealths and consumption generated by these wealth processes is T C e (x) = {(XT + CT ) ∈ L0+ : ∃(α0 , α, c) ∈ N (x), x + 0 αt · dSt = XT + CT } = {(XT + CT ) ∈ L0+ : ∃Z ∈ X e (x), XT + CT = ZT }. As before, the “e” superscript denotes that the trading strategy equals the value process and cumulative consumption.

12.1 The Set-Up

245

The sets of relevant local martingale and supermartingale deflators, analogous to those in Chap. 11, are given by  Ds = Y ∈ L+0 : Y0 = 1, (X + C)Y is a P supermartingale,  X + C = 1 + α · dS, (α0 , α, c) ∈ N (1) Ds = {YT ∈ L0+ : Y0 = 1, ∃(Z(T )n )n≥1 ∈ Ml , YT ≤ lim Zn (T ) a.s.} n→∞

= {YT ∈ L0+ : ∃Z ∈ Ds , YT = ZT }  Dl = Y ∈ L+0 : Y0 = 1, (X + C)Y is a P local martingale,  X + C = 1 + α · dS, (α0 , α, c) ∈ N (1) Dl = {YT ∈ L0+ : ∃Z ∈ Dl , YT = ZT } 

dQ Ml = Y ∈ Dl : ∃Q ∼ P, YT = dP



Ml = {YT ∈ L0+ : ∃Z ∈ Ml , YT = ZT } M = {Y ∈ L+0 : YT =

dQ , Q ∈ M} dP

M = {Y ∈ L0+ : ∃Z ∈ M , YT = ZT } We have the following relationships among the various sets. stochastic processes random variables

M ⊂ Ml ⊂ Dl ⊂ Ds ⊂ L+0 M ⊂ Ml ⊂ Dl ⊂ Ds ⊂ L0+

Note that since the s.f.t.s. with consumption plan α = 0, c = 0 for all t ≥ 0 with initial wealth x = 1 is a feasible nonnegative wealth s.f.t.s. with consumption, Y ∈ Dl implies that Y is a nonnegative P local martingale and a P supermartingale by Lemma 4 in Chap. 1. By the Doob Meyer decomposition (Protter [158, Theorem 13, p. 115]) for supermartingales, Y is also a semi-martingale. We note that Y need not be a probability density with respect to P (and therefore not a P martingale). As in Chap. 11, we assume there are no arbitrage opportunities in the market, i.e. Assumption (NFLVR) Ml = ∅ where Ml = {Q ∼ P : S is a Q local martingale}.

246

12 Incomplete Markets (Utility Over Intermediate Consumption and Terminal Wealth)

If NFLVR is violated, a solution to the trader’s optimization problem will not exist (see the Proof of Theorem 63 in Chap. 13). In addition, this assumption enables a representation of the trader’s budget constraint that facilitates the use of convex optimization. In this chapter the market may be incomplete. In an incomplete market, given an equivalent local martingale measure Q ∈ Ml , there exist integrable derivatives that cannot be synthetically constructed using a value process that is a Q martingale. If the set of equivalent local martingale measures is not a singleton, then the cardinality of |Ml | = ∞, i.e. the set contains an infinite number of elements. In this case, derivative prices are not uniquely determined. Here, the investor’s portfolio optimization problem provides a method to uniquely identify the relevant local martingale measure for the investor to price derivatives.

12.2 Problem Statement This section presents the investor’s optimization problem. We assume that the ρ

investor’s preferences # have a state dependent EU representation with respect to P given by 

T

E

 U1 (ct , t, ω)dt + U2 (XT , ω)

(12.5)

0

where U1 and U2 are utility functions with the properties assumed in Chap. 9, repeated here for convenience. Assumption (Utility Function) The utility function Ui : (0, ∞) × Ω → R for i = 1, 2 is such that for all ω ∈ Ω a.s. P, (i) (ii) (iii) (iv)

Ui (x, ω) is B(0, ∞) ⊗ FT measurable, Ui (x, ω) is continuous and differentiable in x ∈ (0, ∞), Ui (x, ω) is strictly increasing and strictly concave in x ∈ (0, ∞), and (Inada Conditions) lim Ui (x, ω) = ∞ and lim Ui (x, ω) = 0. x↓0

x→∞

And, with the following modifications: 1. U1 (c, t, ω) is an optional process in (t, ω) ∈ [0, T ] × Ω for every c > 0, and T 2. 0 |U1 (c, t, ω)| dt < ∞ a.s. P for every c > 0. The investor’s optimization problem is to choose a nonnegative wealth s.f.t.s. with consumption to maximize their expected utility of consumption and terminal wealth. The entire wealth and consumption process is constrained to be nonnegative. For simplicity of notation, we suppress the dependence of the utility functions on the state ω ∈ Ω writing U1 (c, t) = U1 (c, t, ω) and U2 (x) = U2 (x, ω) everywhere below. Formally, the optimization problem is

12.2 Problem Statement

247

Problem 8 (Choose the Optimal (α0 , α, c) for x ≥ 0)  v(x) =

sup

T

E

 U1 (ct , t)dt + U2 (XT )

where

0

(α0 ,α,c)∈N (x)

N (x) = {(α0 , α, c) ∈ (O, L (S), O) : Xt = α0 (t) +  αt · St , t x + 0 αu · dSu = Xt + Ct ≥ 0, ∀t ∈ [0, T ] . To solve this problem, as before, we can break the problem up into two steps. First, we solve for the optimal “derivative” (c, XT ) ∈ O × L0+ where a derivative is now a wealth process with intermediate cash flows. Of course, this is more consistent with actual markets because many derivatives have intermediate cash flows. Second, we find the trading strategy that generates the optimal “derivative.” As before, the solution to the second step uses the insights from Chap. 2 on the synthetic construction of a derivative’s payoffs. Problem 9 (Choose the Optimal Derivative (c, XT ) for x ≥ 0)  v(x) =

sup

T

E

(c,XT ) ∈C e (x)

 U1 (ct , t)dt + U2 (XT )

where

0

C e (x) = {(c, XT ) ∈ O × L0+ : ∃(α0 , α, c) ∈ N (x), x +

T 0

αt · dSt =XT + CT }.

As in Chap. 11, we solve an equivalent problem where we add free disposal. In this regard we define the set of dominated value plus cumulative consumption processes, i.e.    t X (x) = (X + C) ∈ L+0 : ∃(α0 , α, c) ∈ N (x), x + αu · dSu ≥ Xt + Ct , ∀t ∈ [0, T ] . 0

Lemma 26 in Chap. 11 justifies the equivalence (the same proof applies with consumption). Problem 10 (Choose the Optimal Derivative (c, XT ) for x ≥ 0)  v(x) =

sup

E

(c,XT ) ∈C (x)

T

 U1 (ct , t)dt + U2 (XT )

where

0

C (x)= {(c, XT ) ∈ O × L0+ : ∃(α0 , α, c) ∈ N (x), x +

T 0

αt · dSt ≥ XT + CT }.

Note that the “e” superscript no longer appears on the constraint set.

248

12 Incomplete Markets (Utility Over Intermediate Consumption and Terminal Wealth)

To facilitate the solution, we give an alternative characterization of the budget constraint. Note that C (x) represents those payoffs XT + CT ∈ L0+ that are dominated by a nonnegative wealth with consumption process (α0 , α, c) ∈ N (x) with initial wealth x. Consider the set of local martingale deflator processes Y ∈ Ml . Using the intuition underlying risk neutral valuation in Chap. 2, each of these local martingale deflator processes generates a possible present value for the payoff XT + CT , given by E [(XT + CT ) YT ]. Since it unknown which local martingale deflator process should be used, the constraint set C (x) should be the same as the set of all payoffs whose “worst case” present values are affordable at time 0, i.e. less than or equal to x. This is indeed the case, as the next theorem shows. Theorem 55 (Budget Constraint Equivalence) 7


0,Y ∈Ds

(c,XT )∈O×L0+

=

L (XT , y, YT )

Θ(c, XT ) where

sup (c,XT )∈O×L0+

Θ(c, XT ) =

=

sup

⎧      ⎨ E T U1 (ct , t)dt + U2 (XT ) if sup E XT YT + T Yu cu du ≤ x 0 0 ⎩

(c,XT )∈C (x)

E

Y ∈Ds

 T 0

−∞



otherwise

U1 (ct , t)dt + U2 (XT ) = v(x).

This completes the proof. The dual problem is ⎞

⎛ dual :

inf

y>0,Y ∈Ds

⎝

sup (c,XT )∈O×L0+

L (c, XT , y, Y )⎠ .

252

12 Incomplete Markets (Utility Over Intermediate Consumption and Terminal Wealth)

For optimization problems we know that ⎞

⎛ inf

y>0,Y ∈Ds

⎝

L (c, XT , y, Y )⎠ ≥

sup

(c,XT )∈O×L0+

"

! sup (c,XT )∈O×L0+

inf

y>0,Y ∈Ds

L (c, XT , y, Y ) = v(x).

Proof We have      T T L (c, XT , y, Y ) = E 0 U (ct , t)dt + U (XT ) +y x−E 0 ct Yt dt + XT YT . Note that sup L (c, XT , y, Y ) ≥ L (c, XT , y, Y ) ≥ inf L (c, XT , y, Y ) y>0,Y ∈Ds

(c,XT )∈O×L0+

for all (c, XT , y, Y ). Taking the infimum over (y, Y ) on the left side and the supremum over (c, XT ) on the right ⎞ ⎛ side gives inf

y>0,Y ∈Ds

⎝

sup

L (c, XT , y, Y )⎠ ≥

(c,XT )∈O×L0+

sup (c,XT )∈O×L0+

"

!

inf

y>0,Y ∈Ds

L (c, XT , y, Y ) .

This completes the proof. The solutions to the primal and dual problems are equal if there is no duality gap. We will solve the primal problem by solving the dual problem and showing that there is no duality gap.

12.3 Existence of a Solution We need to show that there is no duality gap and that the optimum is attained, or equivalently, a saddle point exists. To prove this existence, we need three assumptions in addition to the previous assumption that S satisfies NFLVR. The first and third assumptions are identical to those used in Chap. 11 to guarantee the existence of a solution when the utility function is only defined over terminal wealth. The second assumption is needed to guarantee existence because the utility function now includes intermediate consumption.

12.3 Existence of a Solution

253

Assumption (Reasonable Asymptotic Elasticity) !

cU1 (c, t, ω) AE(U1 , ω) = limsup ess sup c→∞ t∈[0,T ] U1 (c, t, ω) AE(U2 , ω) = limsup x→∞

"

xU2 (x, ω) 0, which results in an uninteresting optimization problem. We can now prove that the investor’s optimization problem exists and has a unique solution. Theorem 58 (Existence of a Unique Saddle Point and Characterization of v(x)) Given the above assumptions, there exists a unique saddle point, i.e. there exists a unique (c, ˆ Xˆ T , YˆT , y) ˆ such that inf

(

sup

sup

(

Y ∈Ds , y>0 (c,XT )∈O×L0+

=

L (c, XT , Y, y)) = L (c, ˆ Xˆ T , Yˆ , y) ˆ

inf

(c,XT )∈O×L0+ Y ∈Ds , y>0

Define v(y) ˜ = inf E Y ∈Ds

 T 0

L (c, XT , Y, y)) = v(x)

(12.6)

 U˜ 1 (yYt , t)dt + U˜ 2 (yYT ) where

U˜ i (y, ω) = sup [Ui (x, ω) − xy],

y > 0 for i = 1, 2.

x>0

Then, v and v˜ are in conjugate duality, i.e. ˜ + xy) , ∀x > 0 v(x) = inf (v(y) y>0

(12.7)

254

12 Incomplete Markets (Utility Over Intermediate Consumption and Terminal Wealth)

v(y) ˜ = sup (v(x) − xy) , ∀y > 0

(12.8)

x>0

In addition, (i) v˜ is strictly convex, decreasing, differentiable on (0, ∞), (ii) v is strictly concave, increasing, and differentiable on (0, ∞), (iii) defining yˆ to be where the infimum is attained in expression (12.7), v(x) = v( ˜ y) ˆ + x y, ˆ (iv) v(y) ˜ = inf E Y ∈Ds

= inf E Y ∈Ml

 T 0

 T 0

U˜ 1 (yYt , t)dt + U˜ 2 (yYT )

(12.9)



 U˜ 1 (yYt , t)dt + U˜ 2 (yYT ) .

Proof To prove this theorem, we show conditions 1 and 2 of the Lemma 28 in the appendix of Chap. 11 hold, as modified by adding consumption. Under the above assumptions, these conditions follow from Zitkovic [187, Theorem 4.2]. To apply this theorem, first we need to choose the random endowment to be identically zero. For utility of consumption U1 , the above assumptions are the hypotheses of Zitkovic [187, Theorem 4.2]. For the utility of terminal wealth U2 , choose the stochastic clock in Zitkovic [187] as in Example 2.6 (2), p. 757. Second, for utility of terminal wealth satisfying the reasonable asymptotic utility assumption (AE(U ) < 1), the boundedness of utility assumption (above) is automatically satisfied, see Karatzas and Zitkovic [125, Example 3.2, p. 1838]. Although Zitkovic [187] requires that S is locally bounded, as shown in Karatzas and Zitkovic [125] this condition is not necessary for the choice of the stochastic clock given above, which is deterministic. This completes the proof. Remark 75 (Alternative Sufficient Condition) The sufficient conditions (i) there exists an x > 0 such that v(x) < ∞, (ii) AE(U ) < 1, and (iii) the boundedness of utility assumptions can be replaced by the alternative sufficient conditions v(y) ˜ < ∞ for all y > 0 and v(x) > −∞ for all x > 0. The same proof as given for Theorem 58 works but with Mostovyi [151], Theorem 2.3 replacing Zitkovic [187] Theorem 4.2. Here, Mostovyi [151] extends the domain of U1 to include c = 0 and the range of U1 to include −∞ at c = 0. In addition, the assumption that the set of local martingale measures Ml = ∅ can be replaced by the set of local martingale deflator processes Dl = ∅. A sketch of the proof is as follows. The existence of a buy and hold nonnegative wealth s.f.t.s. strategy implies that the constraint set C (x) for v is nonempty, and Dl = ∅ implies that the constraint set for v˜ is nonempty because Dl ⊂ Ds = ∅. ˜ < ∞ for all Given the appropriate topologies for C (x) and Dl , assuming that v(y) y > 0 and v(x) > −∞ for all x > 0 together imply, because of the strict convexity of v(y) ˜ for all y > 0 and the strict concavity of v(x) for all x > 0, that a unique solution exists to each of these problems. Then, the conditions of Lemma 28 in

12.4 Characterization of the Solution

255

the appendix of Chap. 11 apply to obtain the remaining results. This completes the remark. Remark 76 (No Terminal Wealth) A special case of this theorem is the case where the utility  of terminal wealth U2 (x) = 0 for all x and the trader’s objective function T is E 0 U1 (ct , t)dt . This completes the remark.

12.4 Characterization of the Solution To solve this general problem, we will divide the solution procedure into three steps: (1) we characterize the optimal consumption problem, assuming that utility of terminal wealth is zero (U2 (x) = 0 for all x), (2) we characterize the optimal wealth problem, assuming utility of consumption is zero (U1 (c, t) = 0 for all c, t), and then (3) we combine the two characterizations.

12.4.1 Utility of Consumption (U2 ≡ 0) The simplified problem is given by Problem 12 (Choose the Optimal Derivative c)  v1 (x1 ) =



T

sup E

U1 (ct , t)dt

c ∈C1 (x1 )

where

0

  C1 (x1 ) = c ∈ O : E

T





ct Y1 (t)dt ≤ x1 for all Y ∈ Ds .

0

To solve this problem, we define the Lagrangian function  L (c, y1 , Y1 ) = E

T

   U1 (ct , t)dt+ + y1 x − E

0



T

ct Y1 (t)dt

.

0

We focus on solving the dual problem, i.e.  dual :

12.4.1.1

v1 (x1 ) =

inf

y1 >0,Y1 ∈Ds

 sup L (c, y1 , Y1 ) . c∈O

The Solution

To solve this problem, we work from the inside of the right side of this expression to the outside.

256

12 Incomplete Markets (Utility Over Intermediate Consumption and Terminal Wealth)

Step 1. (Solve for c ∈ O) Fix y1 , Y1 and solve for the optimal consumption c ∈ O.  sup L (c, y1 , Y1 ) = sup E c∈O



 (U1 (ct , t) − y1 ct Y1 (t)) dt + y1 x1 .

0

c∈O

The sup and the E

T

 (·) operator be be interchanged.

Proof First, it is always true that 



T



T

(U1 (ct , t) − y1 ct Y1 (t)) dt ≤ E

E 0

   sup (U1 (ct , t) − y1 ct Y1 (t)) dt

0

c∈O

for all possible c ∈ O. Taking the sup of the left side gives 

T

sup E





(U1 (ct , t) − y1 ct Y1 (t)) dt ≤ E

0

c∈O

T

   sup (U1 (ct , t) − y1 ct Y1 (t)) dt .

0

c∈O

Next, let c∗ be such that

U1 (ct∗ , t) − y1 ct∗ Y1 (t) = sup (U1 (ct , t) − y1 ct Y1 (t)) . c∈O

Since c∗ is a feasible, 

 

U1 (ct∗ , t) − y1 ct∗ Y1 (t) dt ≤ sup E

T

E 0

i.e.  E

T 0

(U1 (ct , t) − y1 ct Y1 (t)) dt

0

c∈O





T

   sup (U1 (ct , t) − y1 ct Y1 (t)) dt ≤ sup E c∈O

T

 (U1 (ct , t) − y1 ct Y1 (t)) dt .

0

c∈O

This completes the proof. This implies that  sup E c∈O

T

  (U1 (ct , t) − y1 ct Y1 (t)) dt = E

0

0

T



  sup (U1 (ct , t) − y1 ct Y1 (t)) dt . c∈O

Consider the optimization problem within the integral, sup (U1 (ct , t) − y1 ct Y1 (t)) . c∈O

12.4 Characterization of the Solution

257

To solve this problem, fix a ω ∈ Ω, and consider the related problem sup

(U1 (ct (ω), t, ω) − y1 ct (ω)Y1 (t, ω))

ct (ω)∈R+

for all t ∈ [0, T ]. This is a simple optimization problem on the real line. The first order condition for an optimum gives U1 (ct (ω), t, ω) − y1 Y1 (t, ω) = 0,

or

with I1 (·, t, ω) = (U1 (·, t, ω))−1

cˆt (ω) = I1 (y1 Y1 (t, ω), t, ω)

for all t ∈ [0, T ]. Given the properties of I1 (·, t, ω), cˆt (ω) ∈ O. Note that we have U˜ 1 (y1 Y1 (t), t) = sup [U1 (ct , t) − y1 ct Y1 (t)] = [U1 (cˆt , t) − y1 cˆt Y1 (t)]. c∈O

This yields 

   sup (U1 (ct , t) − y1 ct Y1 (t)) dt + x1 y1

T

sup L (c, y1 , Y1 ) = E 0

c∈O

 =E

T

c∈O

 ˜ U1 (y1 Y1 (t), t)dt + x1 y1 .

0

Hence, inf

y1 >0,Y1 ∈Ds

  sup L (c, y1 , Y1 ) = c∈O

7 = inf y1 >0

inf

y1 >0,Y1 ∈Ds

  E inf

Y1 ∈Ds

T

  E

T

  ˜ U1 (y1 Y1 (t), t)dt + x1 y1

0



U˜1 (y1 Y1 (t), t)dt + x1 y1

< .

0

Step 2. (Solve for Y1 ∈ Ds ) Given the optimal consumption cˆ and y1 , solve for the optimal supermartingale deflator Y1 ∈ Ds . This is the solving   E inf

Y1 ∈Ds

0

T

U˜1 (y1 Y1 (t), t)dt

 .

258

12 Incomplete Markets (Utility Over Intermediate Consumption and Terminal Wealth)

Defining v˜1 (y1 ) as the value function for this infimum, let Yˆ1 be the solution such that  T   T  ˜ ˆ ˜ v˜1 (y1 ) = E U1 (y1 Y1 (t), t)dt = inf E U1 (y1 Y1 (t), t)dt . Y1 ∈Ds

0

0

Note that Yˆ1 depends on y1 . Step 3. (Solve for y1 > 0) Given the optimal consumption process cˆ and the optimal supermartingale deflator Yˆ1 , solve for the optimal Lagrangian multiplier y1 > 0, inf (v˜1 (y1 ) + x1 y1 ). y1 >0

Taking the derivative of the right side with respect to y1 , one gets v˜1  (y1 ) + x1 = 0. Substituting the definition for v˜1 and taking the derivative yields 

 U˜ 1 (y1 Yˆ1 (t), t)Yˆ1 (t)dt + x1 = 0.

T

E 0

This step requires taking the derivative underneath the expectation operator. The same proof as in the appendix to Chap. 10 applies. From Lemma 25 in Chap. 9, we  have U˜1 (y1 , t) = −I1 (y1 , t). Thus, 

T

E

 I1 (yˆ 1 Yˆ1 (t), t)Yˆ1 (t)dt = x1 .

0

Solving this equation gives the optimal yˆ1 . Using the solutions from the first step, we write this as  E

T

 ˆ cˆt Y1 (t)dt = x1 ,

0

which is that the optimal consumption process cˆ satisfies the budget constraint with an equality. ˆ ∈ X (x) for x ≥ 0 ) Step 4. (Characterization of (Xˆ (1) + C) Over the time horizon [0, T ], the trader invests in assets to obtain their optimal consumption. This implies that the wealth process Xˆ (1) terminates with Xˆ T(1) = 0. We now implicitly characterize this wealth process.

12.4 Characterization of the Solution

259

Step 1 solves for the optimal consumption process cˆ ∈ O. Since cˆ ∈ C1 (x1 ), (1) by the definition of C1 (x1 ), there exists a nonnegative wealth s.f.t.s (α0 , α (1) , c) ˆ ∈  T (1) ˆ N (x) with consumption such that x1 + 0 αu · dSu = CT . The equality uses the free disposal Lemma 26 in Chap. 11. The time t value of the nonnegative wealth  t (1) (1) s.f.t.s. with consumption is x1 + 0 αu ·dSu = Xˆ t + Cˆ t .   t (1) Since Yˆ1 ∈ Ds , we have that Yˆ1 (t) x1 + 0 αu · dSu is a supermartingale t under P. This implies that Yˆ1 (t)(Xˆ t(1) + Cˆ t ) = Yˆ1 (t)Xˆ t(1) + 0 cˆu Yˆ1 (u)du is a P supermartingale.     (1) (1) Hence, E Yˆ1 (T )Cˆ T = E Yˆ1 (T )(Xˆ T + Cˆ T ) ≤ Y0 X0 + Y0 C0 = x1 , (1)

since Y0 = 1, X0 = x1, and C0 = 0. But,  from the budget constraint T ˆ ˆ ˆ E Y1 (T )CT = E 0 cˆu Y1 (u)du = x1 . The first equality was given in the proof of Theorem 57. (1) By Lemma 3 in Chap. 1, Yˆ1 (t)(Xˆ t + Cˆ t ) is a martingale under P, i.e.   E Yˆ1 (T )(Xˆ T(1) + Cˆ T ) |Ft = Yˆ1 (t)(Xˆ t(1) + Cˆ t )

(12.10)

ˆ for all t ∈ [0, T ]. This completes the characterization of Xˆ (1) + C. Step 5. (Nonnegativity of Cˆ t ) As noted in Step 4 above, the cumulative consumption process Cˆ ≥ 0 because cˆ ∈ C1 (x1 ), which are the set of consumption processes generated by s.f.t.s’s in the set N (x) of nonnegative wealth plus cumulative consumption processes with x ≥ 0. Remark 77 (Random Variables Versus Stochastic Processes) The optimization problem determines the stochastic process cˆ ∈ O, which is the optimal consumption process. Since cˆ ∈ C1 (x), there exists a nonnegative stochastic process Z ∈ X (x) t (1) (1) such that ZT = Cˆ T + Xˆ T with Xˆ T = 0 and Zt = x + 0 αu · dSu for all t for some (α0 , α, c) ∈ N (x) with x ≥ 0. Note that the equality in this statement is justified by the free disposal Lemma 26 in Chap. 11. We denote Z = Xˆ (1) + Cˆ ∈ X (x). These expressions implicitly characterize the optimal wealth (stochastic) process Xˆ (1) and it can be used to determine the nonnegative wealth s.f.t.s. (α0 , α, c) ∈ N (x) with consumption generating it. As noted earlier, a second characterization of the optimal wealth process is given in expression (12.10). This characterization depends on the optimal supermartingale deflator (stochastic) process Yˆ1 ∈ Ds . Unlike the situation in a complete market, the supermartingale deflator can exhibit many different properties. In order of increasing restrictiveness these are properties 1–5 below. 1. Yˆ1 ∈ Ds is a supermartingale deflator process. 2. Yˆ1 ∈ Ds where Yˆ1 is a supermartingale deflator process and Yˆ1 (T ) = dQ dP is a   (1) probability density for P. Here, Yˆ1 (t) = E Yˆ1 (T ) |Ft and (Xˆ + C) ∈ X (x) is a Q supermartingale.

260

12 Incomplete Markets (Utility Over Intermediate Consumption and Terminal Wealth)

3. Yˆ1 ∈ Dl ⊂ Ds where Yˆ1 is a local martingale deflator process and given a (Xˆ (1) + C) ∈ X (x), (Xˆ (1) + C)Yˆ1 is a P local martingale. 4. Yˆ1 ∈ Ml ⊂ Dl where Yˆ1 is a local martingale deflator process and Yˆ1 (T ) = dQ dP   is a probability density for P. Here, Yˆ1 (t) = E Yˆ1 (T ) |Ft and (Xˆ (1) + C) ∈ X (x) is a Q local martingale. 5. Yˆ1 ∈ M ⊂ Ml where Yˆ1 is a martingale deflator process and Yˆ1 (T ) = dQ dP is a   (1) probability density for P. Here, Yˆ1 (t) = E Yˆ1 (T ) |Ft and (Xˆ + C) ∈ X (x) is a Q martingale. We note that for cases (2, 4 and 5) where Yˆ1 ∈ Ds is a probability density for P, Yˆ1 is a P martingale. Necessary and sufficient conditions for case (5), where the supermartingale deflator is both a martingale deflator and a probability density with respect to P, i.e. Yˆ1 ∈ M , are contained in Kramkov and Weston [133]. This completes the remark.

12.4.1.2

The Shadow Price

Using the characterization of the value function, we can obtain the shadow price of the budget constraint. The shadow price is the benefit, in terms of expected utility, of increasing the initial wealth by 1 unit (of the mma). The shadow price equals the Lagrangian multiplier. Theorem 59 (Shadow Price of the Budget Constraint) yˆ1 = v1 (x1 ) ≥ E[U1 (cˆt , t)] for all t ∈ [0, T ] with equality if and only if the supermartingale deflator deflator Yˆ1 (T ) ∈ Ds is a probability density with respect to P, i.e. E[Yˆ1 (T )] = 1. Proof The first equality is obtained by taking the derivative of v1 (x1 ) = v˜1 (yˆ1 ) + x1 yˆ1 . From the first order conditions for an optimal consumption we have U1 (cˆt , t) − ˆ yˆ1 Y1 (t) = 0. Taking expectations gives E[U1 (cˆt , t)] = E[yˆ1 Yˆ1 (t)] ≤ yˆ1 since E[Yˆ1 (T )] ≤ 1. Equality occurs if and only if E[Yˆ1 (T )] = 1. This completes the proof. Remark 78 (The implies that when E[Yˆ1 (T )] = 1,   Shadow  Price)   This theorem   v1 (x1 ) = E U1 (ct , t) = E U1 (cT , T ) for all t ∈ [0, T ]. This equality will prove useful below in the characterization of systematic risk. This completes the remark.

12.4 Characterization of the Solution

12.4.1.3

261

The Supermartingale Deflator Process

To obtain the characterization of the supermartingale deflator process Yˆ1 ∈ Ds , we rewrite the first order condition for consumption as follows Yˆ1 (t) =

U1 (cˆt , t) U  (cˆt , t) U1 (cˆt , t) = 1 ≤ yˆ1 v1 (x1 ) E[U1 (cˆt , t)]

for all t ∈ [0, T ] with equality if and only if the supermartingale deflator Yˆ1 (T ) ∈ Ds is a probability density with respect to P, i.e. E[Yˆ1 (T )] = 1. Assume now that Yˆ1 (T ) ∈ Ml , i.e. Yˆ1 (T ) is a local martingale deflator and a probability density with respect to P. In this case, Yˆ1 ∈ Ml is a P martingale, i.e.   Yˆ1 (t) = E Yˆ1 (T ) |Ft . Here, the local martingale deflator process satisfies Yˆ1 (t) =

U1 (cˆt , t) =E E[U1 (cˆt , t)]



 U1 (cˆT , T )

Ft E[U1 (cˆT , T )]

(12.11)

for all t ∈ [0, T ]. Remark 79 (Asset Price Bubbles) In this individual’s optimization problem, prices and hence the local martingale measure Q are taken as exogenous. The above results never impose the condition that Q is a martingale measure. If the individual’s supermartingale deflator is a local martingale deflator, i.e. Yˆ1 (T ) ∈ Ml , and a probability density with respect to P, it need not be a martingale measure (it could generate a strict local martingale measure). Hence, this analysis applies to a market with asset price bubbles (see Chap. 3), and the above solution applies to an optimal consumption process in the presence of price bubbles. This completes the remark. Remark 80 (Systematic Risk) In this remark, we use the non-normalized market ((B, S), F, P) representation to characterize systematic risk. Recall that the state price density is the key input to the systematic risk return relation in Chap. 4, Theorem 38. To apply this theorem, we assume that the supermartingale deflator Yˆ1 (T ) ∈ Ds is a martingale deflator, i.e. Yˆ1 (T ) ∈ M. Then, the risky asset returns over the time interval [t, t + Δ] satisfy

 

U  (cˆt+Δ , t + Δ) Bt E [Ri (t) |Ft ] = r0 (t) − cov Ri (t), 1  (1 + r0 (t))

Ft U1 (cˆt , t) Bt+Δ (12.12) 1 i (t) , r0 (t) = p(t,t+Δ) −1 where the ith risky asset’s return is Ri (t) = Si (t+Δ)−S Si (t) is the default-free spot rate of interest where p(t, t + Δ) is the time t price of a default-free zero-coupon bond maturing at time t + Δ, and the state price density is

262

Ht =

12 Incomplete Markets (Utility Over Intermediate Consumption and Terminal Wealth) Yˆ1 (t) Bt

U1 (cˆt ,t) . In the simplification of HHt+Δ E[U1 (cˆt ,t)]Bt t that E[U1 (cˆt , t)] = E[U1 (cˆt+Δ , t + Δ)].

=

in the above expression we

use the fact When the time step is very small, i.e. Δ ≈ dt, the expression and this relation simplifies to

Bt Bt+Δ (1

 U1 (cˆt+Δ , t + Δ)

E [Ri (t) |Ft ] ≈ r0 (t) − cov Ri (t),

Ft . U1 (cˆt , t)

+ r0 (t)) ≈ 1,



(12.13)

This is similar to the expression obtained in Chap. 11, except that the conditional expectation of the marginal utility terminal wealth is replaced by the marginal utility of current consumption. This completes the remark.

12.4.1.4

The Optimal Trading Strategy

We can now solve for the optimal trading strategy (α0(1) , α (1) , c) ˆ ∈ N (x) with consumption using the standard techniques developed for the synthetic construction of derivatives in Chap. 2. We are given an optimal wealth and consumption process t (1) (1) (1) Xˆ t + Cˆ t = Xˆ t + 0 cˆu du where cˆ ∈ C1 (x1 ) and Xˆ T = 0. By the definition of the budget constraint set C1 (x1 ), there exists a nonnegative wealth s.f.t.s with T consumption (α0(1) , α (1) , c) ∈ N (x) such that x1 + 0 αu(1) · dSu = Cˆ T and where t x1 + 0 αu(1) · dSu = Cˆ t + Xˆ t(1) for all t ∈ [0, T ]. To illustrate the procedure, assume that S is a Markov diffusion process. The diffusion assumption implies that S is a continuous process. Then, we can write the conditional expectation as a deterministic function g(t, St ) of the risky asset price vector St and time t, i.e.

'

' & &

Yˆ1 (T ) ˆ (1) Yˆ1 (T ) ˆ (1)

(1) ˆ ˆ ˆ ˆ (XT + CT ) Ft =E (XT + CT ) St =g(t, St ). Xt + Ct =E

Yˆ1 (t) Yˆ1 (t) As shown in Chaps. 10 and 11, one can use Ito’s formula on g(t, St ) to determine the trading strategy α (1) by matching the integrands of the stochastic integral with respect to St . Given the holdings in the risky asset α (1) , the value process expression (1) (1) (1) (1) Xt + Ct = α0 (t) + αt · St determines the holdings in the mma α0 . If the risky assets are non-redundant, then this s.f.t.s. with consumption is unique (see Sect. 2.4 of Chap. 2). This completes the identification.

12.4 Characterization of the Solution

12.4.1.5

263

Summary

For easy reference we summarize the previous characterization results. The optimal value function is  T  v1 (x1 ) = sup E U1 (ct , t)dt where c ∈C1 (x1 )

0

7



T

C1 (x1 ) = c ∈ O : sup E Y1 ∈Ds




0:

E

 T 0

(1) Yˆ1 (t)(Xˆ t + Cˆ t )

• yˆ1 = v1 (x1 ), and U  (cˆ ,t) • Yˆ1 (t) = 1 t = yˆ1

U1 (cˆt ,t) v1 (x1 )

is a P martingale,

for all t ∈ [0, T ]. In particular, Yˆ1 (T ) =

U1 (cˆT ,T ) . v1 (x1 )

12.4.2 Utility of Terminal Wealth (U1 ≡ 0) This optimization problem was solved in Chap. 11. We summarize the solution here with the modified notation for easy reference. The optimal value function is v2 (x2 ) =

sup

XT ∈C2 (x2 )

E [U2 (XT )]

where

C2 (x2 ) = {XT ∈ L0+ : E [XT Y2 (T )] ≤ x2 for all Y2 (T ) ∈ Ds }.

264

12 Incomplete Markets (Utility Over Intermediate Consumption and Terminal Wealth)

The optimal solution is (2) (2) • terminal wealth Xˆ T ∈ L0+ : Xˆ T = I2 (y2 Y2 (T )) with I2 = (U2 )−1 , (2) (2) (2) • trading strategy (α0 , α (2) ) is determined such that α0 (t) + αt · St = x1 +  t (2) ˆ (2) 0 αu · dSu = Xt for all t ∈ [0, T ]. It is unique if and only if the risky assets in the market are non-redundant. • supermartingale deflator Yˆ2 (T ) ∈ Ds :

v˜2 (y2 ) = E[U˜ 2 (y2 Yˆ2 (T ))] =

inf

E[U˜ 2 (y2 Y2 (T ))],

Y2 (T )∈Ds

  E I2 (yˆ2 Yˆ2 (T ))Yˆ2 (T ) = x2 ,

• shadow price yˆ2 > 0:

• wealth characterization Xˆ (2) ∈ X (x): • yˆ2 = v2 (x2 ), and • Yˆ2 (T ) =

U2 (Xˆ T ) yˆ2 (2)

=

Xˆ t(2) Yˆ2 (t) is a P martingale,

U  (Xˆ T ) v2 (x2 ) (2)

• the supermartingale deflator process Yˆ2 ∈ Ds is that process with time T value Yˆ2 (T ).

12.4.3 Utility of Consumption and Terminal Wealth This section combines the two previous solutions to characterize the combined problem including both intermediate consumption and terminal wealth. This approach to solving the general problem is based on Karatzas and Shreve [124]. Recall that the combined problem is to solve  v(x) =

sup

E

(c,XT ) ∈C (x)

C (x) = {(c, XT ) ∈ O

T

× L0+

 U1 (ct , t)dt + U2 (XT )

where

0

  : E XT YT +

T

 ct Y (t)dt ≤ x for all Y ∈ Ds }.

0

ˆ α, Denote the optimal solutions as c, ˆ X, ˆ Yˆ , y. ˆ We claim that the solution to this combined problem can be obtained by solving a simpler problem, given the solutions to the previous two. The simpler problem is to find that proportion of the initial wealth x ≥ 0 to allocate to the consumption problem and the remaining proportion of initial wealth to allocate to the terminal wealth problem such that these initial wealth allocations optimize the sum of both the utility of consumption and the utility of terminal wealth problems, respectively. This insight is formalized in the next theorem.

12.4 Characterization of the Solution

265

Theorem 60 (Equivalence of Solutions) Let the initial wealth proportion allocated to consumption, x1∗ ≥ 0, be the solution to V (x) = sup (v1 (x1 ) + v2 (x − x1 )) . x1 ∈[0,x]

Then, this wealth allocation obtains the value function for the original optimization problem, i.e.

V (x) = v1 (x1∗ ) + v2 (x − x1∗ ) = v(x). Proof First, the optimal solution x1∗ to V (x) satisfies v1 (x1∗ ) − v2 (x − x1∗ ) = 0. Or, v1 (x1∗ ) = v2 (x − x1∗ ). There are two remaining steps. (Step 1) Show V (x) ≤ v(x). Note that X(1) + X(2) = X and c for x1∗ and x2 = x − x1∗ are a feasible solution for the combined problem. Since v(x) is the supremum across all possible solutions, we get the stated result. (Step 2) Show V (x) ≥ v(x). ˆ cˆ are the optimal solutions to the We give a proof by contradiction. Suppose X, combined problem where ∗ v(x) > v1 (x1∗ ) + v2 (x − x1 ).     T Here v(x) = E 0 U1 (cˆt , t)dt + E U2 Xˆ T and     T E 0 cˆt Yˆ (t)dt + E Xˆ T Yˆ (T ) = x where Yˆ is the optimal supermartingale deflator for the combined problem. Note that this is an equality at the optimum.   T Define x1 by E 0 cˆt Yˆ (t)dt = x1 . Then, E Xˆ T Yˆ (T ) = x −   T E 0 cˆt Yˆ (t)dt = x − x1 . We see that cˆt is a feasible solution for C1(x1 ). T This implies v1 (x1 ) ≥ E 0 U1 (cˆt , t)dt . C2 (x − x1 ). Similarly, Xˆ T is a feasible solution   for  This implies that v2 (x2 ) ≥ E U2 Xˆ T . Combined, we get v(x) > v1 (x1∗ )+v2 (x−x1∗ ) = sup (v1 (x1 ) + v2 (x − x1 )) ≥ E x1 ∈[0,x]    ˆ E U2 XT = v(x).



T 0

This contradiction completes the proof. Given this theorem, we can now determine the combined solution.

 U1 (cˆt , t)dt +

266

12 Incomplete Markets (Utility Over Intermediate Consumption and Terminal Wealth)

Corollary 11 (General Solution) For simplicity of notation, the optimal solutions for the two separate problems are denoted without “hats.” (1)

(2)

αˆ t = αt + αt for all t ∈ [0, T ], (1) (2) ˆ Xt = Xt + Xt for all t ∈ [0, T ), since XT(1) = 0, Xˆ T = XT(2) cˆt = ct for all t ∈ [0, T ], yˆ = y1 = y2 , cˆt ,t) ˆ for all t ∈ [0, T ), Y (t) = Y1 (t) = Uv1 ( (x)

Yˆ (T ) = Y1 (T ) = Y2 (T ) =

U1 (cˆT ,T ) v  (x)

=

U2 (Xˆ T ) v  (x) .

Proof We have v1 (x1∗ ) + v2 (x − x1∗ ) = v(x). (Step 1) Taking derivatives v2 (x − x1∗ ) = v  (x). But, v2 (x − x1∗ ) = v1 (x1∗ ). Since y1 = v1 (x1∗ ), y2 = v2 (x − x1∗ ), yˆ = v  (x), we get y1 = y2 = y. ˆ (Step 2) The value functions imply         T T (2) E 0 U1 (ct , t)dt + E U2 (XT ) = E 0 U1 (cˆt , t)dt + E U2 (Xˆ T ) By uniqueness of the solutions to the three problems we get (1) (2) αt + αt = αˆ t , (1) (2) where Xt + Xt = Xˆ t (2) ˆ since XT(1) = 0, and XT = XT ct = cˆt . The only qualification here is that if the risky assets are redundant, then there will always exist a set of holdings in the mma and risky assets where the first equality (1) (2) holds. If the risky assets are non-redundant, then αt and αt are unique in the summation. (Step 3) Last, U  (c ,t) U  (cˆ ,t) Y1 (t) = 1 t and Yˆ (t) = 1  t for t ∈ [0, T ). But, cˆt = ct , which implies y1

v (x)

y

y

Yˆ (t) = Y1 (t) for t ∈ [0, T ). In the combined problem, the objective function for the dual problem at time T for a given ω is U1 (cT , T ) + U2 (XT ) − y (cT + XT ) Y (T ). The first order conditions for cT and XT are: U  (cˆ ,T ) U  (Xˆ ) and Yˆ (T ) = 2 T . Yˆ (T ) = 1 T (2) But, Xˆ T = XT , which implies Yˆ (T ) = Y2 (T ), and, cˆT = cT , which implies Yˆ (T ) = Y1 (T ). These use y1 = y2 = yˆ and yˆ = v  (x), which completes the proof.

Appendix

267

12.5 Notes Excellent references for solving the investor’s optimization problem in an incomplete market are Dana and Jeanblanc [42], Duffie [52], Karatzas and Shreve [124], Merton [147], and Pham [156].

Appendix Theorem 61 (Numeraire  t Invariance) Let Zt and Yt be semimartingales under P. Consider Vt = V0 + 0 αu dZu . Then V Y = α • (Y Z) where • denotes stochastic integration. Proof For the notation, we use the convention in Protter [158, p. 60], related to time 0 values. By the integration by parts formula Theorem 3 in Chap. 1 we have V Y = Y− • V + V− • Y + [V , Y ]. For the first term we have Y− • V = Y− • (α • Z) = (Y− α) • Z by the associate law for stochastic integrals, Protter [158, Theorem 19, p. 62]. = (αY− ) • Z = α • (Y− • Z). For the second term, V− • Y = (α • Z− ) • Y = α • (Z− • Y ). For the third term we have [V , Y ] = [α • Z, Y ] = α • [Z, Y ], see Protter [158, Theorem 29, p. 75]. Combined, V Y = α • (Y− • Z) + α • (Z− • Y ) + α • [Z, Y ]. Or, V Y = α • ((Y− • Z) + (Z− • Y ) + [Z, Y ]). But, Y Z = (Y− • Z) + (Z− • Y ) + [Z, Y ] by the integration by parts formula, Theorem 3 in Chap. 1. Hence, V Y = α • (Y Z). This completes the proof.

Part III

Equilibrium

Overview This part of the book extends the previous analysis to study the notion of an economic equilibrium. The question studied is: how do investors trading in markets determine prices? We employ the competitive market paradigm, meaning that we assume that traders act as price takers. Here investors do not trade strategically anticipating the impact of their trades on the price. This is not a new assumption because it has been utilized in all of the previous chapters. Equilibrium prices are determined such that the outstanding supply of each of the risky assets equals the sum of the traders’ optimal demands. As such, equilibrium endogenously determines the risky asset price processes and, hence, characterizes the equivalent local martingale measure in terms of the economy’s fundamentals (beliefs, preferences, and endowments). This is the last step in our understanding of equivalent local martingale measures (and their generalization—supermartingale deflators). In this characterization of an economic equilibrium, the traditional consumption capital asset pricing model (CCAPM) and the intertemporal capital asset pricing model (ICAPM) are deduced as special cases. Interestingly, the individual optimization problem (Part II) is almost sufficient to characterize an equilibrium when a representative trader economy equilibrium exists that characterizes the equilibrium in the original economy. A representative trader is a hypothetical trader whose trades represent the collective actions of all traders in the economy. A representative trader economy equilibrium exists when the representative trader’s optimal demands exactly equal supply, clearing the market. If such a representative trader economy equilibrium exists, then characterizing the equilibrium is straight forward. First, we solve for the representative trader’s optimal demands. Second, we impose one additional constraint—supply equals the representative trader’s demand. This market clearing constraint endogenously determines prices, and hence endogenously determines the equilibrium local martingale

270

III

Equilibrium

measure in terms of the representative trader’s beliefs, preferences, and endowment. In this characterization, aggregate market wealth—the market portfolio—plays a crucial role. This reason is because the representative trader’s wealth is the market wealth and she holds the market portfolio. If a representative trader economy equilibrium does not exist, then to characterize the equilibrium one must first aggregate the different traders’ optimal demands. Second, equating these aggregate demands to aggregate supply endogenously determines the equilibrium price process. Given this equilibrium price process, each trader’s optimal demands determines their local martingale measures. Finally, these individual local martingale measures need to be aggregated (under strong assumptions) to obtain a “market” based local martingale measure and equilibrium risk return relation. Fortunately, however, we will not need to pursue this alternative and more complex approach for characterizing an economic equilibrium. As shown below, under standard hypotheses on the original economy, a representative trader economy equilibrium that characterizes the original economy’s equilibrium always exists. This is a relatively unused result in asset pricing theory due to Cuoco and He [39], see also Jarrow and Larsson [102].

Chapter 13

Equilibrium

This chapter presents the description of an economy, the definition of an economic equilibrium, and some necessary conditions implied by the existence of an economic equilibrium.

13.1 The Set-Up An economy consists of a normalized frictionless (unless otherwise noted) and competitive market (S, F, P) where the value of a money market account Bt = 1 for all t ≥ 0. Recall that a competitive market is one where all traders act as price takers. This is the same structure used in all the previous chapters. To study equilibrium prices, we need an additional assumption. Assumption (Liquidating Cash Flows) There exists an exogenous random payout vector ξ = (ξ1 , . . . , ξn ) : Ω → Rn++ at time T such that ST = ξ > 0 where Rn++ is the n-fold product of (0, ∞). The vector ξ represents the liquidating cash flows (dividends) to the risky assets. By assumption, the risky assets have no intermediate cash flows (dividends) over [0, T ). As noted in Sect. 2.3 of Chap. 2, this is without loss of generality. The existence of liquidating dividends are needed in an equilibrium setting to determine market prices prior to time T . This assumption is only used below when proving the existence of an equilibrium.

© Springer Nature Switzerland AG 2021 R. A. Jarrow, Continuous-Time Asset Pricing Theory, Springer Finance, https://doi.org/10.1007/978-3-030-74410-6_13

271

272

13 Equilibrium

In addition, to characterize an economy we also need structure that identifies the outstanding supply of the traded assets, the preferences of the traders in the market, and the traders’ endowments. These quantities are necessary to characterize traders’ optimal demands and the aggregate supply.

13.1.1 Supply of Shares The supply of shares outstanding for the money market account and the risky assets are N0 (t) > 0 Nj (t) = Nj > 0, j = 1, . . . , n for all t ∈ [0, T ] a.s. P. We assume that the supply of the mma and the risky assets is positive. For the mma, the interpretation is that there is a government that exogenously supplies the “riskless” asset. For the risky assets, the interpretation is that these represent share ownership in physical assets that exist in positive quantities. For simplicity, we assume that the supply of shares outstanding is constant across time. This assumption could be relaxed, but with additional (albeit minor) complications.

13.1.2 Traders in the Economy There are I traders in the economy. We let the trader’s beliefs be represented by the probability measure Pi defined on (Ω, F ), which is equivalent to P for all i, i.e. all traders agree on zero probability events, and these zero probability events are the same as those under the statistical probability measure P. This is a necessary condition to impose on an economy, otherwise traders will disagree on the notions of arbitrage opportunities and dominated assets, in particular NA, NUPBR, NFLVR, and ND, which are all invariant with respect to a change in equivalent probability measures (see Chap. 2) and an economic equilibrium will not exist. The filtration F corresponds to the information set possessed by each trader, hence, we assume that there is symmetric information in the economy. The generalization of this economy to include differential information is given when discussing market efficiency in Chap. 16. Differential information economies are more complex to analyze because the set of admissible s.f.t.s.’s in the definitions of NA, NUPBR, NFLVR, and ND all change with a change in the filtration F. ρ

The traders preferences #i are assumed to have a state dependent EU representation with respect to Pi . The utility functions Ui : (0, ∞) × Ω → R are defined over terminal wealth for i = 1, . . . , I . The last section in this chapter discusses the

13.1 The Set-Up

273

necessary changes needed to include intermediate consumption. We assume that the traders’ utility functions satisfy the properties given in Chap. 9, repeated here for convenience. Assumption (Utility Function) The utility function Ui : (0, ∞) × Ω → R is such that for all ω ∈ Ω a.s. P, (i) (ii) (iii) (iv)

Ui (x, ω) is B(0, ∞) ⊗ FT measurable, Ui (x, ω) is continuous and differentiable in x ∈ (0, ∞), Ui (x, ω) is strictly increasing and strictly concave in x ∈ (0, ∞), and (Inada Conditions) lim Ui (x, ω) = ∞ and lim Ui (x, ω) = 0. x→∞

x↓0

In addition, we assume that preferences satisfy the reasonable asymptotic elasticity condition AE(Ui ) < 1 for all i, i.e. Assumption (Reasonable Asymptotic Elasticity (AE(Ui ) < 1 for all i)) AE(Ui , ω) = limsup x→∞

xUi (x, ω) 0. A trader’s time T wealth is denoted by XTi ≥ 0. Last, we add the following assumption. Assumption (Finite Value Functions) Given the price process S, let  vi (xi , S) =

sup E

XT ∈C (xi )

dPi Ui (XT ) dP



be the ith traders’ value function for a given initial wealth xi > 0. We assume that there exists a xi > 0 such that vi (xi , S) < ∞ for all i = 1, . . . , I . Note the dependence of the trader’s value function on a given price process S. In conjunction, as shown in Chaps. 10–12, these three assumptions are almost sufficient to guarantee the existence and uniqueness of the solution to each trader’s optimization problem. Omitted is the assumption that the price process S satisfies NFLVR. This assumption is omitted at this point in the analysis because the price process is to be endogenously determined in equilibrium (to be defined below).

274

13 Equilibrium

Given a price process S, an accounting identity relates the time 0 wealth to the value of the endowed shares xi =

e0i

+ e · S0 = i

e0i

+

n 

eji Sj (0) > 0.

(13.1)

j =1

In addition, it must be the case that in aggregate, the total endowed shares equal the outstanding supply, i.e. N0 = I e0i i=1 (13.2) I Nj = i=1 eji for j = 1, . . . , n.

13.1.3 Aggregate Market Wealth Given a price process S, the aggregate market wealth at time t ∈ [0, T ] is given by N0 +

n 

(13.3)

Nj Sj (t).

j =1

The aggregate market wealth represents the value of the mma and the market portfolio, where the market portfolio is that portfolio whose percentage holding in each risky asset is equal to the proportion of wealth that asset represents in the economy, i.e. market portfolio weight j (t) =

Nj Sj (t) mt

for j = 1, . . . , n

N S (t) where mt = nj=1 Nj Sj (t) and nj=1 jmjt = 1. For subsequent use, we point out that the aggregate wealth of all the traders in the economy equals the aggregate market wealth. Lemma 29 (Time 0 Aggregate Market Wealth) I 

xi = N0 + m0

i=1

Proof xi = e0i + ei · S0 = e0i + nj=1 eji Sj (0) I I i I n i i=1 xi = i=1 e0 + i=1  j =1 ej Sj (0)  I i n i = I i=1 e0 + j =1 i=1 ej Sj (0) n I i i = N0 + j =1 Nj Sj (0) = N0 + m0 since N0 = I i=1 e0 and Nj = i=1 ej . This completes the proof.

13.1 The Set-Up

275

13.1.4 Trading Strategies Superscripts identify the different trader’s nonnegative wealth s.f.t.s.’s (α0i , α i ) ∈ N (xi ) for complete and incomplete markets (see the Chaps. 10 and 11) where N (xi ) = {(α0 , α) ∈ (O, L (S)) : Xt = α0 (t) + αt · St , t Xt = xi + 0 αu · dSu ≥ 0, ∀t ∈ [0, T ] and   α0i (t), αti = α i (t) = α1i (t), . . . , αni (t) ∈ Rn for i = 1, . . . , I . Hence, given a price process S, the trader’s time t ∈ [0, T ] budget constraints are xi = α0i (0) + α0i · S0 = α0i (0) +

n 

αji (0)Sj (0) > 0

(13.4)

j =1

Xti = α0i (t) + αti · St = α0i (t) +

n 

αji (t)Sj (t) ≥ 0

j =1

where t Xti = xi + 0 αui · dSu or for all t ∈ [0, T ]. dXti = αti · dSt

(13.5)

To simplify the subsequent analysis, we add the following assumption. Assumption (Non-redundant Assets) The risky assets S are non-redundant. This assumption removes redundant assets from the economy, and consequently is without-loss-of-generality (see Chap. 2, Sect. 2.4). In the context of the finite dimensional Brownian motion market in Chap. 2, Sect. 2.8, this just implies that the volatility matrix in expression (2.30) is a n × D matrix where n ≤ D with a column rank equal to n. Here, n corresponds to the number of risky assets and D the number of independent Brownian motions in the market.

13.1.5 An Economy Given this additional structure, we can now define an economy. An economy is a collection consisting of a market, excluding the specification of the price process, in

276

13 Equilibrium

conjunction with a listing of the traders’ beliefs, utility functions, and endowments. We omit the price process S from the definition of an economy because the price process will be determined endogenously. In symbols, the following collection represents an economy. (An Economy)  I    i i (F, P) , (N0 , N) , Pi , Ui , e0 , e i=1

For future use, we note that the utility functions in such an economy are assumed to satisfy the assumptions given above in order to guarantee that each trader’s optimization problem has a solution.

13.2 Equilibrium The subsequent definition for an equilibrium applies to the two types of market structures—complete and incomplete markets.  Definition 41 (An Equilibrium) Fix an economy (F, P), (N0 , N), (Pi , Ui , (e0i ,  ei ))I i=1 . The economy is in equilibrium if there exists a price process S and demands (α0i (t), αti ) : i = 1, . . . , I such that for all t ∈ [0, T ] a.s. P, (i) (α0i (t), αti ) are optimal for i = 1, . . . , I , (ii) N0 = I α0i (t), and i=1 i (iii) Nj = I i=1 αj (t) for j = 1, . . . , n. An economy is in equilibrium if there exists a price process S such that given this price process, each trader’s demands for the traded assets are optimal, and the supply of shares equals aggregate demand for all times t ∈ [0, T ] and all ω ∈ Ω a.s. P. This is a very strong condition. In essence, for all future dates and realizations of uncertainty, the economy is in equilibrium. It provides the conceptual ideal against which actual economies can be contrasted. This is called a Radner [162] equilibrium.

13.3 Theorems This section proves two theorems related to the existence of an economic equilibrium. The first relates to an economy’s aggregate optimal wealth, and the second relates the existence of an equilibrium to the First and Third Fundamental Theorems 13 and 16 of asset pricing in Chap. 2.

13.3 Theorems

277

Theorem 62 (Equilibrium Aggregate Optimal Wealths) Fix an economy 

I  (F, P) , (N0 , N) , Pi , Ui , e0i , ei i=1 . Let S be an equilibrium price process with XTi the optimal time T wealth of trader i = 1, . . . , I . Then, I  i=1

XTi = N0 +

n 

Nj Sj (T ) = N0 + mT .

j =1

I i I n I i i Proof i=1 XT = i=1 α i=1  j =1 αj (T )Sj (T ) 0 (T ) + I i n i = I i=1 α0 (T ) + j =1 i=1 αj (T ) Sj (T ) = N0 + nj=1 Nj Sj (T ) by conditions (ii) and (iii) in the definition of an equilibrium. This completes the proof. Theorem 63 (Necessary Conditions for an Equilibrium) Suppose there exists an equilibrium with price process S, then (i) NFLVR and (ii) ND hold. Proof This proof is based on Jarrow and Larsson [100]. First, we need a definition. An admissible s.f.t.s. (α0 (t), αt ) ∈ A (x) with initial wealth x ≥ 0 is maximal  T if for T every other admissible s.f.t.s. (β0 (t), βt ) ∈ A (x) such that 0 βt ·dSt ≥ 0 αt ·dSt , T T 0 βt · dSt = 0 αt · dSt holds. The proof proceeds via a sequence of lemmas. Lemma 1 (Delbaen and Schachermayer [46], Theorem 5.12) (α0 (t), αt ) ∈ t A (0) is maximal if and only if there exists a Q ∈ Ml such that 0 αs · dSs is a Q martingale. This theorem in Delbaen and Schachermayer [46] applies to our market because local martingales and σ -martingales coincide when the risky asset prices are nonnegative. Lemma 2 (Delbaen and Schachermayer [45], Theorem 2.14) Finite sums of maximal strategies are again maximal. This theorem in Delbaen and Schachermayer [45] applies to our market because the proof in Delbaen and Schachermayer [45] never uses the local boundedness assumption. Lemma 3 Let (α0 (t), αt ) ∈ N (x) be an optimal trading strategy for investor i, then (α0 (t), αt ) is maximal in N (x) ⊂ A (x). The proof is by contradiction. If (α0 (t), αt ) is not maximal in N (x), then there T T is a (β0 , β) ∈ N (x) such that VT = x+ 0 βt · dSt ≥ 0 αt · dSt + x = XT and T T P x + 0 βt · dSt > x + 0 αt · dSt > 0. Hence, E [Ui (VT )] > E [Ui (XT )],

278

13 Equilibrium

contradicting the optimality of (α0 (t), αt ) ∈ N (x). This completes the proof of Lemma 3. Lemma 4 Suppose investor i has an optimal trading strategy (α0 , α) ∈ N (x) with terminal wealth XT . Then S satisfies NFLVR. The proof uses Theorem 12 in Chap. 2 that NFLVR holds if and only if NA and NUPBR hold. (Case 1) Show NA. Assume not NA in the nonnegative wealth s.f.t.s., i.e. there exists a simple arbitrage opportunity (β0 , β) ∈ N (0) with initial wealth 0 and time T value process T VT = 0 βt · dSt ≥ 0, P (VT > 0) > 0. Then, the s.f.t.s. (α0 + β0 , α + β) ∈ N (x) is feasible and E [Ui (XT + VT )] > E [Ui (XT )], contradicting the optimality of the trading strategy. Note that NA in N (x) implies NA in A (x). Indeed, if the lower bound in the admissible s.f.t.s. is c < 0, add c units of the mma to any simple arbitrage opportunity in A (x) to get one in N (x). This completes the proof of Case 1. (Case 2) Show NUPBR. By Karatzas and Kardaras [121, Proposition 4.19, p. 476], S satisfies NUPBR. The same argument as in Case 1 for NA implies that NUPBR in N (x) implies NUPBR in A (x) as well. In conjunction, Case (1) and (2) prove NFLVR in A (x). This completes the proof of Lemma 4. Lemma 5 In equilibrium, the market portfolio represents a maximal trading strategy.

Proof in equilibrium, each trader has an optimal portfolio α0i (t), αti ∈ N (x).

By Lemma 3, α0i (t), αti is maximal for all i. I i I i  Consider the market portfolio: (N0 , N) = α (t), i=1 0 i=1 αt . The market portfolio represents a buy and hold nonnegative wealth s.f.t.s. which is the sum of the individual traders’ maximal s.f.t.s. By Lemma 2 the market portfolio is maximal. This completes the proof of Lemma 5. Lemma 6 In equilibrium, the nonnegative wealth s.f.t.s.  j j γ0 (t), γt = (0, (0, . . . , 1, . . . , 0)) with a 1 in the j th place is maximal for all j. Note that this lemma implies, by the definition of ND, that ND holds in the economy. The proof: first, when j = 0, the definition of NA is equivalent to the statement that this trading strategy is maximal. Next, consider j ∈ {1, . . . , n}.

13.3 Theorems

279

  I i (t), I α i representBy Lemma 5, the trading strategy (N0 , N) = α i=1 0 i=1 t ing the market portfolio is maximal. Consider the nonnegative wealth s.f.t.s. obtained after dividing by Nj = I i i=1 αj (t), a positive constant.   I I  i i I i  I 1 i i=1 α0 (t) i=1 αt for j = 1, . . . , n. , I i I i i i=1 α0 (t), i=1 αt = I i=1 αj (t)

i=1 αj (t)

i=1 αj (t)

This trading strategy is maximal because multiplying a maximal trading strategy by a positive constant maintains maximality. The time t value process  of this maximal trading  strategy  is I i α Vt = N0 I 1 i + Sj (t) + nk=1,k=j Ii=1 i s · Sk (t) i=1 αj (s) i=1 αj (s) I i because in equilibrium N0 = i=1 α0 (t). By Lemma 4, NFLVR holds. By the First Fundamental Theorem 13 of asset pricing in Chap. 2, Ml is nonempty. And, under any such Q ∈ Ml , Sj (t) is a Q local martingale. ˜ ∈ Ml such that the value process Vt is a Q ˜ By Lemma 1, there exists a Q martingale.    I  αi +Sj (t)+ nk=1,k=j Ii=1, i s ·Sk (t) ≥ Sj (t) ≥ 0. But, Vt = N0 I 1 i i=1 αj (s)

˜ By Theorem 5 in Chap.  1, Sj (t)is a Q martingale. j

i=1 αj (s)

j

By Lemma 1 again, γ0 (t), γt is maximal for all j . This completes the proof of Lemma 6. Thus, both NFLVR and ND hold in equilibrium. This completes the proof. This theorem shows that NFLVR and ND are necessary conditions for the existence of an equilibrium. Indeed, if NFLVR is violated in an economy, then every trader would see a FLVR, implying the nonexistence of an optimal portfolio, contradicting the existence of an equilibrium. Similarly, if ND is violated, aggregate supply would exceed aggregate demand for the dominated risky asset, again contradicting the existence of an equilibrium. Hence, the results of Part I of this book are implied by equilibrium pricing models. The results of Part I are more robust in that they do not require the additional structure necessary to define an economy, nor the existence of an economic equilibrium. Applying the First and Third Fundamental Theorems of asset pricing (Theorems 13 and 16 in Chap. 2) we obtain the following corollary. Corollary 12 (Necessary Conditions for an Equilibrium) Suppose there exists an equilibrium with price process S, then there exists a Q ∈ M for the price process S where M = {Q ∼ P : S is a Q martingale}. Remark 81 (Trader’s Supermartingale Deflator) It is important to note that in an incomplete market, the above theorem shows that an equilibrium implies that both NFLVR and ND hold for the risky asset price process S, which implies that there

280

13 Equilibrium

exists an equivalent martingale probability measure Q ∈ M. This does not imply, however, that a trader’s supermartingale deflator Y is the probability density of a martingale measure. It need not be a probability density nor a martingale deflator (see Sect. 11.6 in Chap. 11). The reason for this difference is that in an incomplete market, there are potentially an infinite number of local martingale measures, local martingale deflators, and supermartingale deflators. The investor’s supermartingale deflator may be one of these and not the martingale probability measure Q. The implication of this observation is that in equilibrium, trader’s may view the risky asset price process as exhibiting a price bubble (see Chap. 3), even though a martingale measure exists. The existence of asset price bubbles for an equilibrium risky asset price process in an incomplete market will be discussed more thoroughly when the equilibrium price process is characterized in Chap. 15 below. In a complete market, since there is only one local martingale measure (uniqueness by the Second Fundamental Theorem 14 of asset pricing in Chap. 2), the trader’s supermartingale deflator Y is a probability density with respect to Q and a martingale deflator (see Sect. 10.6 of Chap. 10). Hence, in a complete market, no trader believes that asset price bubbles exist. This completes the remark.

13.4 Intermediate Consumption This section discusses the changes in the model’s structure necessary to include intermediate consumption. For the complete details, see Jarrow [94]. When we add consumption flows, we need to add the supply of consumption goods that the traders consume. This is done by adding an exogenous endowment of consumption goods for each trader, which can be a stochastic process. The utility function also needs to be augmented to include utility from intermediate consumption. Here, in addition to reasonable asymptotic elasticity, an additional assumption on the boundedness of the utility function with respect to consumption is needed, see Chap. 12.

13.4.1 Supply of the Consumption Good Define ti ≥ 0 as a stochastic endowment flow of the consumption good for trader i. Aggregate endowment of the consumption good at time t is I  i=1

ti .

13.4 Intermediate Consumption

281

13.4.2 Demand for the Consumption Good Of course, the time t trader i’s demand for the consumption good is cti . Aggregate demand for the consumption good at time t is I 

cti .

i=1

13.4.3 An Economy Given a price process S, trader i  s time 0 wealth is xi = α0i (0) + α0i · S0 . The value process at time t is Xti = α0i (t) + αti · St . The trader’s nonnegative wealth s.f.t.s. in the mma and the risky assets finances both the value process and the net cumulative consumption, i.e. Xti +

t

t i i 0 cu dt − 0 u du = xi dXti = αti · dSt − cti dt

t + 0 αui · dSu or + ti dt for all t ∈ [0, T ].

An economy is defined to be a market in conjunction with a complete specification of the trader’s utility functions and endowments, i.e. (An Economy)   I    (F, P) , (N0 , N) , Pi , Ui , e0i , ei , ti i=1

Finally, an equilibrium is defined by including the consumption and endowment processes.  Definition 42 (An Equilibrium) Fix an economy (F, P) , (N0 , N), (Pi , Ui ,  (e0i , ei ), ti )I i=1 . The economy i

is in equilibrium if there exists a price process S and demands (α0 (t), αti ), cti such that for all t ∈ [0, T ] a.s. P,

(i) (α0i (t), αti ), cti are optimal for i = 1, . . . , I , i (ii) N0 = I i=1 α0 (t),

282

13 Equilibrium

i (iii) Nj = I i=1 α (t) for j = 1, . . . , n, and I i jI i (iv) i=1 t = i=1 ct . This is the consumption goods market clearing. Expression (iv) is the consumption goods market clearing condition. With these modifications, the previous theorems can all be extended.

13.5 Notes Additional references for the determination of economic equilibrium in dynamic markets include Dana and Jeanblanc [42], Duffie [52], Karatzas and Shreve [124], and Merton [147].

Chapter 14

A Representative Trader Economy

To characterize the equilibrium and to facilitate existence proofs, we introduce the notion of a representative trader. A representative trader is a hypothetical individual whose trades, in a sense to be made precise below, reflect the aggregate trades of all individuals in the economy. A representative trader is defined by her beliefs, utility function, and endowments, which are constructed by aggregating the beliefs, utility functions, and endowments of all the traders in the economy. For simplicity of presentation, this chapter focuses on traders having preferences only over terminal wealth. The last section in this chapter discusses the necessary changes needed to include intermediate consumption.

14.1 The Aggregate Utility Function The representative trader’s utility function over terminal wealth is the aggregate utility function defined next. It reflects the optimal behavior of the “average” trader in the economy. 

I  Given is an economy (F, P) , (N0 , N) , Pi , Ui , e0i , ei i=1 . For use within the next few chapters, recall that an economy assumes that the traders’ utility functions satisfy the regularity, asymptotic elasticity, and finite value function conditions given in Sect. 13.1 of Chap. 13. These are imposed to guarantee that each trader’s expected utility of wealth maximization problem has a solution. Definition 43 (Aggregate Utility Function with Weightings λ) Define the aggregate utility function U : (0, ∞) × L0++ × Ω → R by 7 U (x, λ, ω) =

sup {x1 ,...,xI }∈RI

I  i=1

I

 dPi Ui (xi , ω) : x = λi (ω) xi dP

© Springer Nature Switzerland AG 2021 R. A. Jarrow, Continuous-Time Asset Pricing Theory, Springer Finance, https://doi.org/10.1007/978-3-030-74410-6_14


0 for all i = 1, . . . , I , and λ = (λ1 , . . . , λI ). As defined, the aggregate utility function sums the individual traders utility functions, weighting each differently according to λ and the traders’ beliefs. In addition, the aggregate utility function chooses the largest such sum by allocating the aggregate wealth x to the various traders. In this sense, one can interpret the aggregate utility function as that of a central planner attempting to maximize the “welfare” of all the traders in the economy by allocating the economy’s wealth according to the given “welfare” function. The first lemma studies properties of the aggregate utility function. Lemma 30 (The Aggregate Utility Function with Weightings λ) Given the aggregate utility function U : (0, ∞) × L0++ × Ω → R. For fixed λ ∈ L0++ , U (·, λ, ·) satisfies the properties of a utility function from Chap. 9, i.e. (i) (ii) (iii) (iv)

U (x, λ, ω) is B(0, ∞) ⊗ FT measurable, U is continuous and differentiable in x on (0, ∞) a.s. P, U is strictly increasing and strictly concave in x on (0, ∞) a.s. P, (Inada Conditions) U  (0, λ, ω) = lim U  (x, λ, ω) = ∞, and x↓0

U  (∞, λ, ω) = lim U  (x, λ, ω) = 0 a.s. P. x→∞

Proof Property (i) follows from the fact that Ui are jointly measurable for all i and dPi dP is FT -measurable. I dPi Note that i=1 λi (ω) dP Ui (xi , ω), considered as a function of xi > 0, is continuous, differentiable, strictly increasing, and strictly concave. To prove the remaining properties we consider the Lagrangian L(x, μ) =

I  i=1

I

λi

 dPi Ui (xi ) + μx (x − xi ) dP i=1

where μx ≥ 0 is the Lagrangian multiplier which depends on x. For fixed (λi )i∈I , a saddle point exists (see Ruszczynski [169, p. 127]) and the first order conditions uniquely characterize the solution.  i λi dP dP Ui (xi ) = μx f or i = 1, . . . , I and x= I i=1 xi .

(14.1)

Standard results (see Ruszczynski [169, p. 192]) imply properties (ii) and (iii). In addition, we also have that U  (x, λ, ω) = μx a.s. P (see Ruszczynski [169, p. 151]). Note that given the properties of U proven above, μx > 0. This implies λi

dPi  U (xi ) = U  (x, λ) dP i

(14.2)

14.1 The Aggregate Utility Function

285

a.s. P for all i. Hence, condition (iv) holds as well. This completes the proof. Given the individual trader’s utility functions satisfy AE(Ui ) < 1 for all i = 1, . . . , I , the aggregate utility function will also. Lemma 31 (Asymptotic Elasticity) Given AE(Ui , ω) < 1 for all i = 1, . . . , I , AE(U, λ, ω) = limsup x→∞

xU  (x, λ, ω) 0 and ρ  with ρ < ρ  < 1 such that

xU  (x) U (x)

x→∞

≤ ρ  for all x > X. Note that ρ > 0

by the properties of U . Conversely, if for some X > 0 and ρ < 1, x > X, then

 (x) limsup xU U (x) x→∞

xU  (x)

x > X if and only if U (x) ≤ ρ for all x > X. Proof of Lemma: (⇒) Assume for some X > 0 and ρ ∈ (0, 1) that x > X. Since it is differentiable, U (x) xρ

 =

≤ ρ for all

≤ ρ < 1. This completes the proof of (Step 2).

(Step 3) Lemma: For any fixed X > 0 and ρ ∈ (0, 1),



xU  (x) U (x)

U (x) xρ

U (x) xρ

is decreasing for all

is decreasing for all

U  (x)x ρ − ρx ρ−1 U (x) xU  (x) − ρU (x) = ≤ 0. x 2ρ x ρ+1

This implies xU  (x) − ρU (x) ≤ 0 and thus

xU  (x) U (x)

≤ ρ for all x > X.

286

14 A Representative Trader Economy

(⇐) The argument above is reversible. This completes the proof of the lemma. (Step 3) Given AE(Ui ) < 1 for i = 1, 2, we know there exists a Xi > 0 and xUi (x) Ui (x)

ρi ∈ (0, 1) such that

≤ ρi for all x > Xi .

Choose a ρ < 1 such that ρ ≥ max(ρ1 , ρ2 ) and a X ≥ max(X1 , X2 ). are decreasing for all x > X and i = 1, 2. The above lemma implies that Uxi (x) ρ Ui (x) Ui (x) Indeed, if x p is decreasing then x q is decreasing for all q > p. (Step 4) Show Ux(x) ρ is decreasing for all x > X where U (x) = sup {U1 (x1 ) + U2 (x2 ) : x = x1 + x2 } . {x1 ,x2 }∈RI

Choose any y ≥ x > X. In this optimization problem with y = y1 + y2 , the feasible region is compact ([0, y]), hence the supremum is attained, i.e. there exists y1 , y2 such that y = y1 + y2 and 1 ) + U2 (y2 ). Hence, ρU (y) = U1 (y ρ U (y) U1 (y1 )+U2 (y2 ) y1 y2 U1 (y1 ) U1 (y2 ) = = + ρ ρ . yρ yρ y y y y 1

2

Now yi ≥ yyi x for each i, hence by monotonicity of Uxi (x) ρ :      ρ U1 y1 x  ρ U1 y2 x y y U (y) y1 y2 yi x     y1 x ρ + y2 x ρ . Define xi = y , substitution yields yρ ≤ y y y

U (y) yρ

≤

U (y) yρ

≤

y

U1 (x1 ) ρ x−

+ U1x(xρ 2 ) . But x1 + x2 = x, hence   U1 (x1 ) U1 (x2 ) = Ux(x) sup + ρ . This proves xρ xρ

x1 +x2 =x

U (x) xρ

is decreasing for all

x > X. (Step 4) Using (Step 3), applying the lemma and the characterization of  (x) AE(U ) < 1, proves limsup xU U (x) < 1. x→∞

This completes the proof. Remark 82 (Normalization of Aggregate Utility Function Weightings λ) Without loss of generality, the aggregate utility function can be modified to ∗

U (x, λ, ω) =

7I 

sup {x1 ,...,xI }∈RI

i=1

I

 dPi Ui (xi , ω) : x = ξ(ω)λi (ω) xi dP


0 is a strictly positive FT -measurable random variable without ∗ } ∈ RI to changing the optimal solution {x1∗ , . . . , xI 7 U (x, λ, ω) =

sup {x1 ,...,xI }∈RI

I  i=1

< I  dPi Ui (xi , ω) : x = λi (ω) xi . dP i=1

(14.3)

14.1 The Aggregate Utility Function

287

∗ } ∈ RI be the optimal solution to Indeed, to prove this fact let {x1∗ , . . . , xI expression (14.3). Note that

∗

U (x, λ, ω) =

7I 

sup {x1 ,...,xI }∈RI

& = ξ(ω) 7 = ξ(ω)

i=1

7 sup

{x1 ,...,xI }∈RI

I 

i=1 I  i=1

I

 dPi ξ(ω)λi (ω) xi Ui (xi , ω) : x = dP


0.

(14.10)

The first order condition, which is necessary and sufficient for a solution, is x ρ−1

λ 1

(1 + λ ρ−1 )ρ−1 1

−y =0

or

−1

x ∗ = y ρ−1 k ρ−1 where k =

λ

. Substitution and algebra gives the conjugate aggregate

1

(1+λ ρ−1 )ρ−1

utility function ! ρ

U˜ (y, λ) = −y ρ−1

1

1 + λ ρ−1

"

1

λ ρ−1 1

with U˜  (y, λ) = −y ρ−1



1

1+λ ρ−1 1

λ ρ−1



ρ−1 ρ

 (14.11)

1   2−ρ   ρ−1 1 < 0 and U˜  (y, λ) = − ρ−1 > y ρ−1 1+λ 1

λ ρ−1

0. This completes the example.

14.2 The Portfolio Optimization Problem A representative trader is defined as a collection (P, U (λ), (N0 , N)) where N = (N1 , . . . , Nn ). A representative trader has beliefs P (the statistical probability measure), the aggregate utility function U (x, λ) depending on λ, and an initial endowment of shares equal to the outstanding supply of shares in the economy for the mma and stocks (N0 , N). Given a price process S, the representative trader’s initial wealth equals the aggregate initial wealth in the economy, i.e. x = N0 + N · S0 = I i=1 xi (total aggregate wealth), see Lemma 29 in Chap. 13. Using the aggregate utility function, the representative trader’s optimal portfolio problem is analogous to that of any trader. Problem 13 (Choose the Optimal Trading Strategy (α0 , α)) Given a price process S, v(x, λ, S) =

sup (α0 ,α)∈N (x)

E [U (XT , λ)]

where

14.2 The Portfolio Optimization Problem

291

N (x) = {(α0 , α) ∈ (O, L (S)) : Xt = α0 (t) +  αt · St , t Xt = x + 0 αu · dSu ≥ 0, ∀t ∈ [0, T ] . As before we solve this problem by splitting the solution into two steps. The first step determines the optimal time T wealth, and the second step determines the trading strategy that achieves this wealth. For the first step we solve the following problem. Problem 14 (Choose the Optimal Derivative XT ) Given a price process S, v(x, λ, S) =

sup E [U (XT , λ)]

XT ∈C (x)

where

(14.12)

(complete market)   C (x) = XT ∈ L0+ : E Q [XT ] ≤ x (incomplete market) C (x) = {XT ∈ L0+ : E [XT YT ] ≤ x for all YT ∈ Ds } To prove the existence of the representative trader’s optimal portfolio, we need the following lemma. Lemma 32 (Finite Value Function) v(x, λ, S) < ∞ for some x > 0. Proof Given S, by assumption we have that for all i = 1, . . . , I , vi (xi , S) < ∞ for some xi > 0. We use the normalization I i=1 λi = 1 in the definition of the {x aggregate utility function. Define x = minI i=1 i }. Then, v(x) = sup E [U (XT , λ)] XT ∈C (x) & '   I I dPi = sup E sup i=1 λi (ω) dP Ui (Xi (T ), ω) : XT = i=1 Xi (T ) XT ∈C (x) {X1 ,...,XI }∈RI & '   I I dPi sup ≤E sup i=1 λi (ω) dP Ui (Xi (T ), ω) : XT = i=1 Xi (T ) XT ∈C (x){X1 ,...,XI }∈RI   i ≤ I sup E dP i=1 dP Ui (Xi (T ), ω) X ∈C (x )

T i v(x , S) < ∞. = I i i=1 The last inequality holds because λi < 1 and x < xi implies C (x) ⊂ C (xi ). This completes the proof.

292

14 A Representative Trader Economy

Given Lemma 30, Eq. (14.12), the aggregate utility function satisfies the regularity conditions of a state dependent utility function, (i) (reasonable asymptotic elasticity) AE(U, λ, ω) = limsup x→∞

xU  (x,λ,ω) U (x,λ,ω)

< 1 a.s.

P, and (ii) (non-trivial optimization) v(x, λ, S) < ∞ for some x > 0. We can now prove that a solution to the representative trader’s optimization problem exists under the following additional assumption: (iii) S satisfies NFLVR, i.e. Ml = ∅ where Ml = {Q ∼ P : S is a Q local martingale}. Indeed, a unique solution exists to the representative trader’s optimization problem under these assumptions by applying the theorems from the individual trader’s optimization problems to the aggregate utility function (i.e., Theorem 50 in Chap. 10 or Theorem 53 in Chap. 11). This yields the following characterization theorem for the solution to the representative trader’s optimization problem. Theorem 64 (Representative Trader’s Optimal Wealth) The representative trader’s optimal wealth exists and is unique. The optimal solution is characterized by the following expressions. (complete market) Xˆ Tλ = I (yY ˆ T , λ)

with I = (U  )−1 = −U˜ 

where U˜ (y, λ, ω) = sup [U (x, λ, ω) − xy] f or all

y > 0,

x>0

YT ∈ Ml is the unique local martingale measure, and yˆ is the solution to the budget constraint   ˆ T , λ) = x. E YT I (yY Finally, the optimal portfolio wealth process Xˆ λ ∈ X (x) when multiplied by the local martingale density, Xˆ tλ Yt , is a P martingale. (incomplete market) Xˆ Tλ = I (yˆ YˆTλ , λ)

with I = (U  )−1 = −U˜ 

where YˆTλ ∈ Ds is the solution to v(y, ˜ λ, S) = inf E[U˜ (yYT , λ)] YT ∈Ds

(14.13)

14.2 The Portfolio Optimization Problem

293

where U˜ (y, λ, ω) = sup [U (x, λ, ω) − xy] f or all

y > 0,

x>0

and yˆ is the solution to the budget constraint   E YˆTλ I (yˆ YˆTλ , λ) = x. It is important to note, for subsequent use, that the solution YˆTλ ∈ Ds depends on the utility function U (x, λ). Finally, the optimal portfolio wealth process Xˆ λ ∈ X (x) when multiplied by the supermartingale deflator, Xˆ tλ Yˆtλ , is a P martingale. The solution to the second step, the optimal trading strategy, is given in the next theorem. Theorem 65 (Representative Trader’s Optimal Trading Strategy) (complete and incomplete markets) Given a price process S, there exists a unique (α0 , α) ∈ N (x) such that Xˆ Tλ = x +



T

αt · dSt .

(14.14)

0

This is the optimal nonnegative wealth s.f.t.s. Proof This follows directly from the uniqueness of the optimal terminal wealth and the assumption that the traded assets are non-redundant (see Chap. 13, Sect. 13.1). This completes the proof. For the representative trader, buy and hold trading strategies will play a key role. The next theorem characterizes buy and hold trading strategies for the representative trader. Theorem 66 (Buy and Hold Optimal Trading Strategies) Given a price process S, suppose that the representative trader’s optimal portfolio is a buy and hold strategy, i.e. Xˆ tλ = θ + η · St

f or all t ∈ [0, T ]

where θ ∈ R and η = (η1 , . . . , ηn ) ∈ Rn++ . If (i) θ = 0 or (ii) if θ = 0 and YˆTλ is a probability density with respect to P, then YˆTλ is the probability density for an equivalent martingale probability measure.

294

14 A Representative Trader Economy

Proof Because the optimal portfolio is a buy and hold strategy, by Theorem 64, Xˆ tλ Yˆtλ = (θ + η · St ) Yˆtλ is a martingale under P. Note that θ Yˆtλ , (η · St )Yˆtλ , and Sj (t)Yˆtλ for all j are supermartingales under P because each is the wealth process of a particular buy and hold trading strategy. The first represents buying just θ units of the mma, the second is just buying η units of the risky assets, and the third is buying one share of the j th risky asset. If θ = 0, then this implies θ Yˆtλ and (η ·St )Yˆtλ are martingales under P. θ Yˆtλ being a martingale under P implies that Yˆtλ is a probability density with respect to P. If θ = 0, let YˆTλ be a probability density with respect to P. Note that in both cases, we have that (η · St )Yˆtλ is a martingale under P. Since ηi > 0 for all i, the only way (η · St )Yˆtλ can be a martingale under P is if each Sj (t)Yˆtλ is a martingale under P. Hence, YˆTλ is the density for an equivalent martingale probability measure. This completes the proof. This theorem has the following corollary. Corollary 13 (Buy and Hold Trading Strategies) Let the representative trader’s optimal portfolio be a buy and hold trading strategy with strictly positive holdings in all of the risky assets and where YˆTλ is a probability density with respect to P if the holdings in the mma are zero. Then, (Existence of an Equivalent Martingale Measure) there exists a Q ∈ M for the price process S where M = {Q ∼ P : S is a Q martingale}, and (No Bubbles)   ˆ λ , there is no under the representative trader’s risk adjusted beliefs dQ = Y T dP asset price bubble for S. ˆλ Proof Define dQ dP = YT . By the previous theorem, this defines a probability density with respect to P under which S is a martingale. Hence, there is no bubble under this equivalent probability measure. This completes the proof. Remark 83 (Bubbles Require Retrading) The above corollary is consistent with the result that an asset price bubble exists if and only if after purchasing the asset at time 0, there is a re-trading time before time T that strictly increases the value of the asset above liquidating the position at time T (see Corollary 4 in Chap. 3). This completes the remark.

14.3 Representative Trader Economy Equilibrium This section studies an economy where there is only one trader—the representative trader. We have an analogous definition of an equilibrium for this economy. Here, the representative trader must be endowed with the market’s aggregate wealth at

14.3 Representative Trader Economy Equilibrium

295

time 0, and the representative trader’s time T wealth equals (by construction) the aggregate market wealth as well. Since the representative trader cannot trade with anyone else in the market, the notion of an equilibrium must reflect this fact. This implies that in such an equilibrium, no trades can take place. This is called a notrade equilibrium. With this insight, the following definitions are straightforward. In symbols, the following collection represents a representative trader economy indexed by the aggregate utility function’s weights λ. (A Representative Trader Economy) {(F, P) , (N0 , N) , (P, U (λ), (N0 , N))} Note that in this collection, there is only one trader, represented by the triplet (P, U (λ), (N0 , N)) and indexed by the aggregate utility function’s weights λ. And, in the definition of this economy, the price process S is not assumed to satisfy NFLVR because its properties are endogenously determined in equilibrium. Definition 44 (Representative Trader Economy Equilibrium) Fix a representative trader economy {(F, P) , (N0 , N) , (P, U (λ), (N0 , N))}. The representative trader economy is in equilibrium if there exists a price process S and demands (α0λ (t), αtλ ) such that for all t ∈ [0, T ] a.s. P, (i) (α0λ (t), αtλ ) are optimal for (P, U (λ), (N0 , N)), (ii) N0 = α0λ (t), and (iii) Nj = αjλ (t) for j = 1, . . . , n. In this definition, the representative trader’s endowment is given by (N0 , N), which is equal to all the shares outstanding in the market. Second, this is a no-trade equilibrium. This is because the representative trader’s optimal trading strategy (α0λ (t), αtλ ) = (N0 , N) represents the same holdings for all t ∈ [0, T ] a.s. P. Note that this trading strategy generates a time T wealth equal to the aggregate market wealth, starting from an initial endowment that is also equal to aggregate wealth, i.e. XTλ

= N0 +

n 

 Nj Sj (0) +

T

N · dSt = N0 +

0

j =1

n 

Nj Sj (T )

(14.15)

j =1

where N = (N1 , . . . , Nn ) ∈ Rn . A key observation is that in a representative trader economy equilibrium, the endowment of the representative trader at times 0 and T are equal to the aggregate market wealths at times 0 and T , i.e. N0 + m0 = N0 +

n  j =1

Nj Sj (0)

and

N0 + mT = N0 +

n  j =1

Nj Sj (T ).

296

14 A Representative Trader Economy

We now study the relation between equilibrium in the economy  I    (F, P) , (N0 , N) , Pi , Ui , e0i , ei i=1

and equilibrium in a hypothetical representative trader economy purposefully constructed to characterize the original economy’s equilibrium. To facilitate understanding, we introduce two related definitions. Definition 45 (Representative Trader Economy Reflects the Economy’s Optimal Wealths and Asset Demands) Given an economy {(F, P) , (N0 , N) , i i

I  Pi , Ui , e0 , e i=1 with price process S, we say that the representative trader economy {(F, P) , (N0 , N) , (P, U (λ), (N0 , N))} reflects the economy’s aggregate optimal wealths with price process S if Xˆ Tλ =

I 

Xˆ T i a.s. P

i=1

where Xˆ Tλ , Xˆ T i correspond to the optimal wealth of the representative trader and the ith trader, respectively for i = 1, · · · , I . We say that the representative trader economy {(F, P) , (N0 , N) , (P, U (λ), (N0 , N))} reflects the economy’s aggregate optimal asset demands for the price process S if αˆ 0λ (t)

=

I  i=1

αˆ 0 (t) and i

αˆ tλ

=

I 

αˆ t i

for all t ∈ [0, T ] a.s. P

(14.16)

i=1

where (αˆ 0λ (t), αˆ tλ ), (αˆ 0i (t), αˆ t i ) correspond to the optimal trading strategy of the representative trader and the ith trader, respectively for i = 1, · · · , I . Note that if the representative trader economy reflects’s the economy’s aggregate optimal asset demands, then it will also reflect the economy’s aggregate optimal wealths, which are generated by the admissible s.f.t.s.’s. Definition 46 (Representative Trader Economy’s Equilibrium Reflecting an Economy’s Equilibrium) Given are an economy {(F, P) , (N0 , N) , i i

I  Pi , Ui , e0 , e i=1 , an equilibrium price process S with demands (αˆ 0i (t), αˆ t i )I i=1 for all t ∈ [0, T ], and a representative trader economy {(F, P) , (N0 , N) , (P, U (λ) , (N0 , N))} with equilibrium price process S ∗ .

14.3 Representative Trader Economy Equilibrium

297

We say that the economy’s equilibrium is reflected by this representative trader economy’s equilibrium if 1. S ∗ = S and 2. the representative trader economy reflects the original economy’s aggregate optimal asset demands. These definitions make precise when a representative trader economy reflects the economy’s optimal wealths, asset demands, and equilibrium. We now apply these definitions to prove two theorems, which in conjunction, guarantee the existence of a representative trader economy equilibrium that reflects the original economy’s equilibrium. These theorems are formulated for an incomplete market, with a complete market a special case. Theorem 67 (Existence of a Representative Trader Economy That Reflects the Original Economy’s Optimal Wealths) Given an economy {(F, P) , (N0 , N) , i i

I  Pi , Ui , e0 , e i=1 with price process S. Assume S satisfies NFLVR. Then, the representative trader economy 



 (F, P) , (N0 , N) , P, U (λ∗ ), (N0 , N)

reflects the economy’s aggregate optimal wealths where λ∗i =

v1 (x1 , S)YT1 (S) vi (xi , S)YTi (S)

(14.17)

with YTi (S) the solution to trader i’s dual problem and vi (xi , S) trader i’s value function, given the initial wealth xi for all i = 1, . . . , I . This theorem shows that given any economy, the price process S satisfying NFLVR is sufficient to guarantee the existence of a representative trader economy that reflects the economy’s aggregate optimal wealths. This is a very mild sufficient condition, which shows that robustness of the concept of a representative trader. Note that this theorem relates to the economy’s aggregate optimal wealths, and not the economy’s aggregate asset demands. Second, note the dependence of the individual and representative trader’s value functions on the price process S which is exogenously specified in the statement of this theorem. This makes sense because all traders act as price takers when forming their optimal demands. We emphasize that the price process S given in the hypothesis to this theorem is not assumed to be an equilibrium price process, it is only assumed to satisfy NFLVR. Simply stated, this theorem guarantees that if the market satisfies NFLVR, there always exists a representative trader economy indexed by the aggregate utility function weightings λ∗ as given expression (14.17) such that the representative

298

14 A Representative Trader Economy

trader’s optimal wealth reflects the aggregate optimal wealths of the individuals in the economy. Proof For the given price process S satisfying NFLVR, from Sect. 11.9  in Chap.  11, for each trader i the optimal wealth XTi , which maximizes Ei Ui (XTi ) = i i E dP dP Ui (XT ) , is characterized by the expression dPi   i  U XT = vi (xi , S)YTi (S) dP i

(14.18)

where vi (xi , S) is the value function for the investor’s optimization problem, YTi (S) is the solution to    dP dPi ˜ v˜i (y, S) = inf E Ui yYT dP dPi YT ∈Ds (S) where Ds (S) is the set of supermartingale deflators with respect to P. We use this change of measure in the investor’s objective function because Ds (S) depends on the probability measure P (see the discussion in Chap. 11 following the definition of Ds ). The optimal supermartingale deflator YTi (S) depends on the price process S because Ds (S) depends on S. Consider the collection of representative trader economies {(F, P) , (N0 , N) , (P, U (λ), (N0 , N))} indexed by λ. Let S be the price process in all of these economies. Optimizing the representative trader’s wealth, we have U  (XTλ , λ) = v  (x, λ, S)YTλ (S) where the individual components X˜ Ti satisfy λi

dPi  ˜ i U (X ) = μXλ for i = 1, . . . , I T dP i T XTλ =

I 

X˜ Ti

i=1

and (the shadow price of the constraint) U  (XTλ , λ) = μXλ . T

(14.19)

14.3 Representative Trader Economy Equilibrium

299

See expression (14.2) above. Combined these give U  (XTλ , λ) = λi

dPi  ˜ i U (X ) dP i T

for all i.

(14.20)

We want to show XTi = X˜ Ti for all i. Using expression (14.18), this will be satisfied if and only if dPi  ˜ i U (X ) = vi (xi , S)YTi (S) dP i T

for all i

because the right side uniquely determines XTi via expression (14.18). Using expression (14.20), this will be satisfied if and only if λ1 v1 (x1 , S)YT1 (S) = λi vi (xi , S)YTi (S)

for i = 2, . . . , I .

It is easy to show that a FT -measurable solution λ∗ = (λ∗1 , . . . , λ∗I ) to this system of (I − 1) functional equations in I unknowns is λ∗i =

v1 (x1 , S)YT1 (S)

for i = 1, 2, . . . , I .

vi (xi , S)YTi (S)

This completes the proof. Remark 84 (Complete Market with Respect to Q ∈ Ml (S)) Suppose the market is complete with respect to any Q ∈ Ml (S). Note that we make the dependence of the equivalent local martingale measure on the price process S explicit in the notation for the set of local martingale measures. Then, by the Second Fundamental Theorem 14 of asset pricing in Chap. 2, Q ∈ Ml (S) is unique. Define YT = dQ dP . Recall that in Chap. 11,  v˜i (y, S) =  =

inf

YT ∈Ml (S)

E

inf

YT ∈Ds (S)

E

  dP dPi ˜ Ui yYT dP dPi

     dPi ˜ dP dP dPi ˜ =E . Ui yYT Ui yYT dP dPi dP dPi

The last equality holds in this set of equations because Ml (S) is a singleton set. Combined, this implies that YTi (S) = YT .

300

14 A Representative Trader Economy

Consequently, in a complete market, the aggregate utility function’s weights in Theorem 67 simplify to λ∗i =

v1 (x1 , S) vi (xi , S)

for i = 1, 2, . . . , I .

This completes the remark. We can now derive the key theorem in this section which provides sufficient conditions for the existence of a representative trader economy equilibrium that reflects the original economy’s equilibrium. Theorem 68 (Existence of a Representative Trader Economy Equilibrium Reflecting the Original Economy’s Equilibrium) Given an economy  i i

I  P) , , , N) , P , U , e , e (F, (N0 i i 0 i=1 let S be an equilibrium price process with the optimal trading strategies (αˆ 0 i (t), αˆ t i )I i=1 . Then, the representative trader economy 



 (F, P) , (N0 , N) , P, U (λ∗ ), (N0 , N)

reflects the original economy’s equilibrium, where λ∗i =

v1 (x1 , S)YT1 (S) vi (xi , S)YTi (S)

with YTi the solution to trader i’s dual problem and vi (xi , S) trader i’s value function, given the initial wealth xi and the price process S, for all i = 1, . . . , I . Proof By Theorem 67, given the equilibrium price process S for the original economy, the representative trader economy 



 (F, P) , (N0 , N) , P, U (λ∗ ), (N0 , N)

∗ ˆ i reflects the original economy’s optimal wealths, i.e. Xˆ Tλ = I i=1 XT a.e. P. The application of this theorem requires that S satisfies NFLVR, but this follows from Theorem 63 in Chap. 13. A trading strategy that achieves this wealth is ∗ αˆ 0λ (t)

=

I 

αˆ 0i (t)

and

∗ αˆ tλ

i=1

=

I 

αˆ t i for all t ∈ [0, T ] a.s. P.

i=1

In the original economy’s equilibrium N0 =

I  i=1

αˆ 0i (t) and

N=

I  i=1

αˆ t i for all t ∈ [0, T ] a.s. P.

14.4 Pareto Optimality

301 ∗

∗

This implies that the buy and hold trading strategy αˆ 0λ (t) = N0 and αˆ tλ = N for all t ∈ [0, T ] a.s. P generates the representative trader’s optimal wealth. By uniqueness of the trading strategy generating any terminal wealth (by the nonredundant assets assumption in Chap. 13), this is the representative trader’s optimal trading strategy. Hence, the representative trader economy that reflects the original economy’s optimal demands is in equilibrium with price process S. This completes the proof. The difference in the hypothesis of this Theorem 68, as contrasted with Theorem 67, is that the price process S is assumed to be an equilibrium price process as distinct from just satisfying NFLVR. The price process being an equilibrium price process enables the representative trader economy to reflect not only the aggregate optimal wealth, but the optimal aggregate demands and the price process. This is an important theorem because it enables one to characterize an economic equilibrium in a complex economy via an equivalent equilibrium in a simpler representative trader economy. Simply stated, given an economic equilibrium, there always exists a representative trader economy that reflects the original economy’s equilibrium (with the market weightings λ∗ as given in the theorem).

This characterization is used extensively in Chap. 15 below.

14.4 Pareto Optimality This section relates the notion of a representative trader and economic equilibrium to Pareto optimality. To facilitate this discussion, we need to introduce some definitions. First, fix an economy  I    i i (F, P) , (N0 , N) , Pi , Ui , e0 , e i=1

with price process S. A wealth allocation (XTi : i = 1, . . . , I ) is called feasible if market I i wealth equals the aggregate allocation, i.e. N0 + mT = i=1 XT where mT = n N S (T ). j =1 j j A feasible wealth allocation (XTi : i = 1, . . . , I ) is Pareto optimal (efficient) if there exists no alternative feasible allocation (X˜ Ti : i = 1, . . . , I ) such that Ei [Ui (X˜ Ti )] ≥ Ei [Ui (XTi )] for all i = 1, . . . , I and Ei [Ui (X˜ Ti )] > Ei [Ui (XTi )] for at least one i = 1, . . . , I . It is easy to see that this definition is equivalent to the following restatement.

302

14 A Representative Trader Economy

A given wealth allocation (XTi : i = 1, . . . , I ) is Pareto optimal if for all i = 1, . . . , I , (XT1 , . . . , XTI ) maximizes Ei [Ui (X˜ Ti )]

subject to

Ej [Uj (X˜ T )] ≤ Ej [Uj (XT )] for all j, j = i I ˜j j =1 X = N0 + mT . j

j

T

To solve this problem, for all i = 1, . . . , I , define the Lagrangian   j i E [U (X j )] − E [U (X ˜ γ )] j j j j j =1,j =i j T T I j i +μ (N0 + mT − j =1 X˜ T )

Li (X˜ T1 , . . . , X˜ TI ) = Ei [Ui (X˜ Ti )] +

n

for γji ≥ 0 all j = 1, . . . , I and μi ≥ 0. We note that the Lagrangian multipliers γji and μi are constants. The first order conditions for the optimal solution (XT1 , . . . , XTI ) are necessary and sufficient under our hypotheses (see Ruszczynski [169], p. 127). These conditions are for all i = 1, . . . , I , the constraints are satisfied with equality, ∂Ei [Ui (XTi )] ∂XTi

− μi = 0,

and

j

γji

∂Ej [Uj (XT )] j ∂XT

− μi = 0

for j = 1, . . . , I , j = i.

Or, for all i = 1, . . . , I , ∂Ei [Ui (XTi )] ∂XTi j

∂Ej [Uj (XT )]

= γji > 0

for j = 1, . . . , I , j = i.

(14.21)

j

∂XT

Next, consider the optimization problem max

I 

(XT1 ,...,XTI ) i=1

λ˜ i Ei [Ui (XTi )] I 

subject to

XTi = N0 + mT

i=1

where λ˜ i > 0 for i = 1, . . . , I are constants.

(14.22)

14.4 Pareto Optimality

303

To solve this problem, define the Lagrangian L(XT1 , . . . , XTI ) =

I 

λ˜ i Ei [Ui (XTi )] + δ(N0 + mT −

I 

i=1

XTi )

i=1

for δ ≥ 0. The first order conditions for the optimal solution (XT1 , . . . , XTI ), which are necessary and sufficient under our hypotheses, are that the constraint is satisfied with equality and λ˜ i

∂Ei [Ui (XTi )] ∂XTi

−δ =0

for i = 1, . . . , I .

Or, for all i = 1, . . . , I , ∂Ei [Ui (XTi )] ∂XTi j ∂Ej [Uj (XT )] j ∂XT

=

λ˜ j >0 λ˜ i

for j = 1, . . . , I , j = i.

(14.23)

Note that expressions (14.21) and (14.23) are identical. This proves that an allocation is Pareto optimal if and only if it solves the optimization problem in expression (14.22) where λ˜ i > 0 for i = 1, . . . , I are constants. It is convenient to rewrite the objective function in expression (14.22) as I  i=1

λ˜ i Ei [Ui (XTi )]

&I '    dPi dPi i i λ˜ i Ui (XT ) =E = E λ˜ i Ui (XT ) . dP dP i=1 i=1 (14.24) I 

As written, the Pareto optimality objective function is almost identical to the aggregate utility function as given in Definition 43 for the representative trader. The key difference is that the aggregate utility function’s weights λi (ω) are FT measurable random variables, where in the Pareto optimality objective function the weights λ˜ i are constants. Otherwise, the objective functions and the optimization problems are identical (see expression (14.12)). This implies that we can apply our insights from economic equilibrium to study when an equilibrium allocation is Pareto optimal. First, we assume that the economy is in equilibrium. By Theorem 63 in Chap. 13, the market satisfies NFLVR, hence, there exists an equivalent local martingale measure Q ∈ Ml . Second, under the hypotheses of Theorem 68 we have that there exists a representative trader economy equilibrium with the aggregate utility function weights λ∗i (ω) as given in the theorem, i.e. for all i, λ∗i =

v1 (x1 , S)YT1 (S) vi (xi , S)YTi (S)

.

304

14 A Representative Trader Economy

From the above characterization of a Pareto optimal allocation, we have that the equilibrium allocation is Pareto optimal if and only if the aggregate utility function weights λ∗i (ω) given in the previous expression are constants. We investigate the implications of this observation. (i) Suppose the market is complete with respect to any Q ∈ Ml . Then, by the Second Fundamental Theorem 14 of asset pricing in Chap. 2, Q ∈ Ml is unique. i Define YT = dQ dP . Then, for all investors i = 1, . . . , I , YT = YT . Substitution v  (x ,S)

yields λ∗i = v1 (x1,S) for all i = 1, . . . , I . These are constants, hence, the i i equilibrium allocation is Pareto optimal. (ii) In an incomplete market, an equilibrium allocation is in general not Pareto efficient. Indeed, in an incomplete market (see Chap. 11), λ∗i =

v1 (x1 ,S)YT1 (S) vi (xi ,S)YTi (S)

for all i = 1, . . . , I are random due to YTi (S) being random and different across i. This implies that in an incomplete market, an equilibrium allocation is Pareto optimal if and only if YTi (S) = YT1 (S) for all i = 1, . . . , I . This restrictive condition is difficult to satisfy in an incomplete market economy. This condition is satisfied, for example, trivially if all traders have identical beliefs, preferences, and initial endowments. If traders have identical beliefs and preferences, but different initial endowments, then this condition is satisfied if YTi (S) is independent of the initial wealth (because all other investor characteristics are identical). State independent utility functions that satisfy this condition include the logarithmic and power utility functions (see Pham [156, p. 198]). These utility functions exhibit linear risk tolerance (see Example 13 in Chap. 9) and they are related to Gorman aggregation (see Back [5, p. 55]), which is a well known sufficient condition that generates Pareto optimal equilibrium allocations.

14.5 Existence of an Equilibrium This section uses the notion of a representative trader to prove the existence of an equilibrium in the original economy under two different sets of sufficient conditions. The first theorem is the converse of Theorem 68. Theorem 69 (Existence of an Equilibrium) Given an economy  i i

I  P) , . , N) , P , U , e , e (F, (N0 i i 0 i=1 Let S be an equilibrium price process for the representative trader economy {(F, P) , (N0 , N) , (P, U (λ), (N0 , N))} with aggregate utility function weightings λi =

v1 (x1 , S)YT1 (S) vi (xi , S)YTi (S)

14.5 Existence of an Equilibrium

305

with YTi (S) the solution to trader i’s dual problem and vi (xi , S) trader i’s value function, given the initial wealth xi and the price process S, for all i = 1, . . . , I . Then, there exists an equilibrium in the original economy with price process S. Proof As in the proof of Theorem 67, these weightings imply that the optimal I dPi demands XTi generated by the aggregate utility function U (λ) = i=1 λi dP Ui are optimal for the individual traders as well. We note that the individual traders’ time 0 endowments sum to the representative trader’s initial endowment. In addition the individual traders’ optimal T time i wealths sum to the representative trader’s time T wealth, i.e. XTλ = I i=1 XT . In the representative trader economy equilibrium, x = N0 + m0 and XTλ = N0 + mT . For the economy, the individual trader’s asset holdings (α0i (t), αti ) : i = 1, . . . , i i

I ) are uniquely implied by their initial endowments (e0 , e ) : i = 1, . . . , I and optimal wealths (XTi : i = 1, . . . , I ). This is due to the assumption of I i non-redundant assets in Chap. 13. Since XTλ = i=1 XT , the individual share holdings must sum to those of the representative trader for all t ∈ [0, T ], i.e. I I i (t) and N = i N0 = α j i=1 0 i=1 αj (t) for j = 1, . . . , n for all t ∈ [0, T ]. This implies that the economy is in equilibrium. This completes the proof. This theorem provides a convenient method to prove that an equilibrium exists in an economy. As shown, to prove an equilibrium exists in an economy, it is sufficient to prove that an equilibrium exists in the representative trader economy with the aggregate utility function weightings as specified in Theorem 67 (e.g., see Karatzas and Shreve [124], Cuoco and He [39], Basak and Cuoco [8], and Hugonnier [74]). We now provide a set of sufficient conditions for the existence of an equilibrium in the original economy using the previous theorem in the proof. This set of sufficient conditions applies to both a complete and incomplete market. Theorem 70 (Sufficient Conditions for an Equilibrium) Given an economy  i i

I  (F, P) , (N0 , N) , Pi , Ui , e0 , e i=1 .   (i) Assume that for all λ, E U  (N0 + N · ξ, λ) < ∞, and E [U (N0 + N · ξ, λ)] < ∞ where ξ is the exogenous liquidating dividends for the risky assets and U (x, λ) is the aggregate utility function. Define Stλ = Eλ [ξ |Ft ]

where

U  (N0 + N · ξ, λ) dQλ = >0 dP E [U  (N0 + N · ξ, λ)] (14.25)

for all t ∈ [0, T ] with Eλ [·] denoting expectation with respect to Qλ , (ii) Assume that there exists a λ˜ such that λ˜ i =

˜

˜

v1 (x1 , S λ )YT1 (S λ ) vi (xi , S λ˜ )YTi (S λ˜ )

(14.26)

306

14 A Representative Trader Economy ˜

˜

with Y i (S λ ) the optimal solution to trader i’s dual problem and vi (xi , S λ ) ˜ trader i’s value function, given the initial wealth xi and the price process S λ , for all i = 1, . . . , I . Then, ˜

(1) S λ is an equilibrium price representative trader economy  process for the ˜ (F, P) , (N0 , N) , P, U (λ), (N0 , N) , (2) this representative trader economy reflects the economy’s equilibrium, and ˜ (3) S λ is an equilibrium price process for the economy. Proof (Step 1) Consider the collection of representative trader economies {(F, P) , (N0 , N) , (P, U (λ), (N0 , N))} indexed by λ. By the definition of S λ , M = ∅. Hence, Ml = ∅ for the price process S λ . By the first fundamental theorem of asset pricing, S λ satisfies NFLVR. By our assumptions on the traders’ utility functions there exists a xi > 0 such that vi (xi , S λ ) < ∞ for all i, and by Lemma 32 that v(x, λ, S λ ) < ∞ for some x > 0. This implies that a solution (optimal wealth process) to the individual and representative traders’ optimization problems exist and are unique. We next prove using the price process S λ that the representative trader’s optimal trading strategy is a buy and hold with zero units in the mma and N shares in the stocks, i.e. αˆ 0λ (t, S λ ) = N0 , and αˆ tλ (S λ ) = N for all t ∈ [0, T ] a.s. P where Xˆ Tλ (S λ ) denotes the representative trader’s optimal demands. Note the explicit dependence of all the previous quantities on λ and S λ . Because S λ is a Qλ martingale, any nonnegative wealth self-financing trading strategy X ∈ X (x) is a supermartingale under Qλ . Consider the buy and hold trading strategy (N0 , N). Note that the wealth of this trading strategy N0 + N · S λ ∈ X (x). Since  N0 , N are constants,  U (N0 +N ·ξ,λ) λ = N · S is a Qλ martingale. This implies that x = E E[U  (N +N ·ξ,λ)] N · ξ 0    U (N0 +N ·ST ,λ) E E[U  (N0 +N ·ST ,λ)] N · ST . We now show that this buy and hold trading strategy is optimal. Consider an arbitrary X ∈ X (x). Then,  E [U (XT , λ) − U (N0 + N · ST , λ)] ≤ E U  (N0 + N · ST , λ)(XT − N0 λ −N · ST )] by the concavity of U . But, using the definition of dQ dP , the right side can  be written as:  E U  (N0 + N · ST , λ)(XT −N0 − N · ST )    λ (X − N − N · S ) ≤ 0 since = E U  (N0 + N · ST , λ) E dQ T 0 T   dP dQλ E dP (XT − N0 − N · ST ) ≤ x − x = 0. The last inequality is due to the fact   λ that XT must satisfy the budget constraint and E dQ (N + N · S ) = x. This 0 T dP completes the proof of Step 1.

14.5 Existence of an Equilibrium

307

(Step 2) Fix a λ. Consider an economy  I    (F, P) , (N0 , N) , Pi , Ui , e0i , ei i=1

with the price process S λ . For each λ, let XTi (S λ ) denote the optimal demands for trader i in the economy with the price process S λ . Note the explicit dependence of the optimal demands on the price process S λ . (Step 3) Fix the λ˜ that satisfies hypothesis (ii). Jointly consider the economy  I    (F, P) , (N0 , N) , Pi , Ui , e0i , ei i=1

and the collection of representative trader economies {(F, P) , (N0 , N) , (P, U (λ), (N0 , N))} ˜ indexed by λ using the same price process S˜ = S λ . Note that this is a fixed and identical price process for all the representative trader economies indexed by λ. ˜ ˜ Since S˜ = S λ , Ml = ∅ for the price process S. By Theorem 67, λ˜ is such that the representative trader economy



(F, P) , (N0 , N) , (P, U (λ˜ ), (N0 , N))



reflects the aggregate optimal wealths in the original economy given the price ˜ i.e. process S, ˜ ˜ Xˆ Tλ (S) =

I 

˜ a.s. P. Xˆ T i (S)

(14.27)

i=1

  I I ˜ ˜ We note that the trading strategy ˆ 0 i (t, S), ˆ t i (S) i=1 α i=1 α ˆ λ˜ ˜ ˆ i ˜ generates I i=1 XT (S) and, therefore, XT (S). ˜ (Step 4) We claim that S˜ = S λ is an equilibrium price for the economy and the representative trader economy 

  ˜ (N0 , N) . (F, P) , (N0 , N) , P, U (λ), ˜

(Part a) First, by (Step 1) for the price process S λ the representative trader with ˜ trading strategy is a buy and hold with zero units in the mma aggregate utility U (λ)’s

308

14 A Representative Trader Economy ˜

˜

˜

˜

and N shares in the stocks, i.e. αˆ 0λ (t, S λ ) = N0 , and αˆ tλ (S λ ) = N for all t ∈ [0, T ] ˜

λ a.s. P. This implies price process for the representative    that S is an equilibrium trader economy (F, P) , (N0 , N) , P, U (λ˜ ), (N0 , N) . (Part b) By expression (14.27), we get

˜ ˜ Xˆ Tλ (S λ ) =

I 

˜ Xˆ T i (S λ ) a.s. P.

i=1

And, by the assumption of non-redundant assets in Chap. 13, uniqueness of the ˜ ˜ representative trader’s optimal trading strategy generating Xˆ Tλ (S λ ) implies N0 =

I 

˜

αˆ 0 i (t, S λ ) and

i=1

N=

I 

˜

αˆ t i (S λ ) for all t ∈ [0, T ] a.s. P,

i=1

i.e. aggregate demand equals aggregate supply in the original economy. This proves ˜ that S λ is an equilibrium price process for the original economy  I    i i . (F, P) , (N0 , N) , Pi , Ui , e0 , e i=1

This completes the proof. In this theorem, sufficient condition (i) is a mild restriction on an economy. This hypotheses is needed for the integrability of the representative trader’s utility and marginal utility for aggregate wealth. The remaining sufficient condition (ii) is quite strong, effectively assuming the existence of a set of aggregate utility function weights λ˜ for which an equilibrium exists in the economy. Independently verifying this hypothesis involves solving a difficult fixed point problem in the infinite dimensional space of nonnegative adapted and cadlag processes. Finding a set of sufficient conditions on the primitives of the economy such that solution λ to expression (14.26) exists is an important open research question. Section 14.7 below provides examples of economies and sufficient conditions in those economies such that aggregate utility function weightings λ˜ satisfying expression (14.26) can be shown to exist, proving that the set of solutions to expression (14.26) is not vacuous (see also the existence proof for the CAPM in Chap. 17 for another example). Recent papers providing sufficient conditions for the existence of an equilibrium in complete and incomplete markets, but using different methods than those discussed above, include Karatzas and Shreve [124], Kardaras, Xing, and Zitkovic [127], Kramkov [131], Larsen [138], Choi and Larsen [31], Christensen and Larsen [32], Larsen and Sae-Sue [139], and Zitkovic [188, 189]. The theorem presented above is for a more general economy then the existence proofs contained in Karatzas and Shreve [124] and Kramkov [131], both of which study a Brownian motion based economy.

14.6 Uniqueness of the Equilibrium

309

14.6 Uniqueness of the Equilibrium Important in characterizing an equilibrium is uniqueness of the price process and optimal supermartingale deflator. These issues are discussed in this section.

14.6.1 Uniqueness of the Equilibrium Price Process The equilibrium price process given in expression (14.25) is unique. Indeed, given an economy  I    , (F, P) , (N0 , N) , Pi , Ui , e0i , ei i=1

let S be an equilibrium price process. Then, by Theorem 68, we know there always exists a representative trader economy {(F, P) , (N0 , N) , (P, U (λ), (N0 , N))} with the same equilibrium price process S that reflects the economy’s equilibrium. In this representative trader economy equilibrium, the representative trader’s optimal trading strategy is α0λ (t) = N0 and αtλ = N for all t ∈ [0, T ]. By Corollary 13, the representative trader’s supermartingale deflator YˆTλ is a probability density with respect to P and S is a Qλ martingale. Noting that ST = ξ yields Stλ = Eλ [ξ |Ft ]

where

U  (N0 + N · ξ, λ) dQλ = >0 dP E [U  (N0 + N · ξ, λ)]

for all t ∈ [0, T ], which completes the uniqueness proof.

14.6.2 Uniqueness of the Supermartingale Deflators Given an equilibrium price process S, we now discuss the uniqueness of the individual and representative trader’s supermartingale deflators. There are two cases to consider: complete and incomplete markets. (Complete Market) In a complete market with respect to any Q ∈ Ml (S), from Chap. 10, these supermartingale deflators are all local martingale deflators which are probability densities with respect to P. By Remark 84 above, they are all equal since there is a unique local martingale measure Q ∈ Ml (S) by the Second Fundamental

310

14 A Representative Trader Economy

Theorem 14 of asset pricing in Chap. 2. Using Corollary 12 in Chap. 13, we have that the local martingale measure is also a martingale measure, i.e. Q ∈ M(S). Hence, the expression for investor i’s supermartingale deflator is given by expression (11.29) in Sect. 11.9 of Chapter 11, i.e. dPi  U (Xi (S)) dQ  ∈ M(S) for all i = 1, . . . , I . = YTi (S) = dP i T i  dP i E dP dP Ui (XT (S))

The change of measure in the numerator is needed since the set of martingale deflators M(S) depends on the probability measure P. The representative trader’s optimal supermartingale deflator is given by U  (N0 + mT , λ) dQ = YTλ (S) = ∈ M(S) dP E[U  (N0 + mT , λ)] where the aggregate market wealth is XTλ =

I 

XTi = N0 +

n 

Nj Sj (T ) = N0 + mT .

j =1

i=1

(Incomplete Market) In an incomplete market, the supermartingale deflators are YTλ (S) = YTi (S) =

U  (N0 +mT ,λ) vi (N0 +m0 ,λ,S) dPi dP

Ui (XTi (S)) vi (xi ,S)

∈ Ds (S)

and

∈ Ds (S) for all i = 1, . . . , I

n i where the aggregate market wealth is XTλ = I i=1 XT = N0 + j =1 Nj Sj (T ) = I n N0 + mT and x = i=1 xi = N0 + j =1 Nj Sj (0) = N0 + m0 . These supermartingale deflators can all be different. We note that the set of supermartingale deflators Ds (S) depends on the probability measure P (see the discussion in Chap. 11 after the definition of Ds ). This is the reason investor i’s supermartingale deflator has the change of measure in the numerator, see expression (11.29), Sect. 11.9 in Chap. 11. Second, since the representative trader’s optimal trading strategy in the representative trader economy equilibrium is a buy and hold trading strategy with positive units  of the mma, by Theorem 66, YTλ is a probability density with respect to P, i.e. E YTλ = 1, YTλ =

U  (N0 + mT , λ) ∈ M(S) E [U  (N0 + mT , λ)]

14.7 Examples

311

is a martingale deflator, and  Ytλ = E

U  (N0 + mT , λ) |Ft E [U  (N0 + mT , λ)]



for all t ∈ [0, T ] is a martingale deflator process (see Theorem 3 in Chap. 1).

14.7 Examples This section provides, as examples, two economies where sufficient condition (iii) in Theorem 70 can be shown to have a solution, thereby proving that the set of economies satisfying expression (14.26) is not vacuous. Another example proving the existence of an equilibrium in an incomplete market, using the approach of Theorem 70, is contained in Chap. 17 on the static CAPM, Theorem 90.

14.7.1 Identical Traders The first economy studied in one in which the market is incomplete, but all traders have identical beliefs, utility functions, and endowments, i.e.       Pi , Ui , e0i , ei = P, U1 , e01 , e1

for all i.

Assume hypothesis (i) in Theorem 70 is satisfied. Then, since all trader’s are identical, expression (14.26) has the trivial solution λ˜ i = 1

for all for all i = 1, . . . , I

U (x, 1, ω) =

I  i=1

Ui

or

  x x , ω = I · U1 ,ω I I

where 1 = (1, . . . , 1) is the I -tuple consisting of all ones. Each trader is effectively a “representative trader in this economy.” Trader i’s optimal nonnegative wealth s.f.t.s. is the buy and hold trading strategy (α0 (t) = N0 N i I , αt = I ). This implies that the equilibrium price process Stλ = E Q [ξ |Ft ]

312

14 A Representative Trader Economy

where U  (N0 + N · ξ, 1) dQ = YTλ (S λ ) = dP E [U  (N0 + N · ξ, 1)] is independent of λ as well. Hence, we have just proven that an equilibrium exists for this incomplete market economy with identical traders, under hypothesis (i) alone, with the price process St = E Q [ξ |Ft ] for all t ∈ [0, T ].

14.7.2 Logarithmic Preferences The second economy studied is an incomplete market with  I I    Pi , Ui , e0i , ei = Pi , ln(·), e0i , ei i=1

i=1

where traders have state independent logarithmic utility functions Ui (x) = ln(x) with x > 0 for all i = 1, . . . , I . The traders have different beliefs and endowments. Assume hypothesis (i) in Theorem 70 is satisfied. In this economy, given weights λ, the aggregate utility function satisfies ! U (x, λ, ω) =

sup

x=x1 +···+xI

I  i=1

" dPi ln (xi ) . λi dP

As discussed in Remark 82 above, without loss of generality we normalize the aggregate utility function weights such that I  i=1

λi

dPi = 1. dP

(14.28)

To determine the aggregate utility function, define the Lagrangian L =

I  i=1

λi

dPi ln (xi ) + μ (x − x1 − · · · − xI ) dP

with μ ≥ 0. The first order conditions, which are necessary and sufficient for an optimum, are λi

dPi 1 =μ dP xi

for all i.

14.7 Examples

313

The solution is i λi dP dP

xi =

1 λ1 dP dP

x1

for all i = 1, . . . , I .

(14.29)

Using the constraint x − x1 − · · · − xI = 0, substitution and algebra yields λ1 dP1 x dP1 x x1 = I dP dP = λ1 i dP i=1 λi dP where the last equality uses expression (14.28). Substituting back into expression (14.29) gives the solution xi = λi

dPi x dP

for all i = 1, . . . , I .

Hence, the aggregate utility function is U (x, λ) =

I 

λi

i=1

     I dPi dPi dPi dPi ln λi x = ln λi + ln(x) λi dP dP dP dP

(14.30)

i=1

where we have used expression (14.28) again. Note that U  (x, λ) =

1 x

is independent of λ. The conjugate aggregate utility function is U˜ (y) = sup x>0

&I  i=1

'   dPi dPi ln λi + ln(x) − xy . λi dP dP

The first order condition, which is necessary and sufficient for an optimal solution, is 1 U˜  (y) = − y = 0. x This gives x =

1 y

and U˜ (y) =

I  i=1

λi

  dPi dPi ln λi − ln (y) − 1. dP dP

314

14 A Representative Trader Economy

The representative trader’s supermartingale deflator is 1

dQλ U  (N0 + N · ξ, λ) N +N ·ξ , = 0 = YTλ = 1 dP E [U  (N0 + N · ξ, λ)] E N0 +N ·ξ which is independent of λ. The supermartingale deflator process is    Ytλ = E YTλ |Ft =

E

1 N0 +N ·ξ



E

 |Ft  ,

1 N0 +N ·ξ

which is independent of λ. Consider the price process & Stλ = Eλ [ξ |Ft ] = E ξ

YTλ Ytλ



' |Ft

=

E E





ξ N0 +N ·ξ Ft 1 N0 +N ·ξ

|Ft

 .

Since the price process is independent of λ, we write Stλ = St . Now, from Example 17 and using the modification for differential beliefs in Sect. 9.8 in Chap. 9, we have that the ith trader’s conjugate utility function is       dP dP dPi ˜ dPi = −1 . Ui yi YTi −ln(yi ) − ln(YTi ) + ln dP dPi dP dPi Given yi , the supermartingale deflator YTi (S) is the solution to 

dPi inf E dP Y i ∈Ds (S) T

    dP i −1 . −ln(yi ) − ln(YT ) + ln dPi

This observation proves that the optimal solution YTi (S) is independent of yi . Next, given YTi (S), yi is determined as the solution to xi = where Ii (y) =

1 y

e0i (0) + e0i

· S0λ

    dP i YTλ = E Ii yi YT (S) dPi

for y > 0. Substitution for S0λ and YTλ yields 

e0i (0) + e0i ·

E E



ξ N0 +N ·ξ 1 N0 +N ·ξ



⎛

⎡

 =E⎣

1 dP yi YTi (S) dP i

ξ N0 +N ·ξ

⎝

 E

1 N0 +N ·ξ

⎞⎤  ⎠⎦ .

14.8 Intermediate Consumption

315

Algebra gives the solution ⎜ yi = ⎜ ⎝





⎛ e0i (0)E

E 

ξ 1 dP N0 +N ·ξ YTi (S) dP

1 N0 +N ·ξ



i

+ e0i

·E



ξ N0 +N ·ξ

⎞ ⎟ ⎟ ⎠,

which is independent of λ. Next, we examine expression (14.26) to see if a solution exists. Recall that yi = vi (xi , S λ ). In this case λ needs to satisfy λi =

y1 YT1 (S)

for all i = 1, . . . , I .

yi YTi (S)

Since the right side is independent of λ, this is the explicit solution. Hence, we have just proven that an equilibrium exists for this incomplete market economy under hypothesis (i) alone, with the price process  St =

E E





ξ N0 +N ·ξ Ft 1 N0 +N ·ξ

|Ft

 

for all t ∈ [0, T ]. This completes the set of examples proving the existence of aggregate utility function weightings λ that satisfy expression (14.26).

14.8 Intermediate Consumption To include intermediate consumption, the representative trader’s utility function U as defined earlier needs to be extended to include intermediate consumption U (c, t, λ) as given in Chap. 12. With this extension, analogous results to those obtained in the previous section can be derived by replacing the representative trader’s supermartingale deflator from optimizing the expected utility of terminal wealth with the representative trader’s supermartingale deflator from optimizing the expected utility of intermediate consumption, see Jarrow [94]. The characterization of the representative trader’s supermartingale deflator is analogous to that given in Chap. 12. Indeed, the representative trader’s optimal supermartingale deflator process is Ytλ (S) =

U  (ct , t, λ) ∈ Ds (S) v  (x, λ, S)

(14.31)

316

14 A Representative Trader Economy

i for all t ∈ [0, T ] where ct = I i=1 ct is the representative trader’s optimal consumption. If S is an equilibrium price process, then in equilibrium, the representative trader’s optimal consumption equals aggregate consumption.

14.9 Notes For references discussing a representative trader in the context of Pareto optimality see Back [5], Huang and Litzenberger [73], Dana and Jeanblanc [42], and Duffie [52]. For existence proofs in a complete market using the representative trader approach see Dana and Jeanblanc [42] and Karatzas and Shreve [124].

Chapter 15

Characterizing the Equilibrium

Assuming that an equilibrium exists, this chapter characterizes the economic equilibrium. For simplicity of presentation, this chapter focuses on traders having preferences only over terminal wealth. The last section in this chapter discusses the necessary changes needed to include intermediate consumption. The key result in this chapter is a characterization of the equilibrium supermartingale deflator as a function of the economy’s primitives: beliefs, preferences, and endowments. Indeed, using a representative trader economy equilibrium that reflects the equilibrium in the original economy, an equilibrium supermartingale deflator is characterized as a function of the representative trader’s (aggregate) utility function and aggregate market wealth. Finally, this chapter derives the intertemporal capital asset pricing model (ICAPM) and the consumption capital asset pricing model (CCAPM) as special cases of this characterization.

15.1 The Set-Up 

I  Given is an economy (F, P) , (N0 , N) , Pi , Ui , e0i , ei i=1 . As in Chap. 13, we ρ

assume that there are I traders in the economy with preferences #i assumed to have a state dependent EU representation with utility functions Ui defined over terminal wealth that satisfy the properties of a utility function in Chap. 9 including reasonable asymptotic elasticity AE(Ui ) < 1 for all i. We assume that an equilibrium price process S exists for this economy. Then, by Theorem 67 in Chap. 14, there exists a representative trader economy {(F, P) , (N0 , N) , (P, U (λ), (N0 , N))}

© Springer Nature Switzerland AG 2021 R. A. Jarrow, Continuous-Time Asset Pricing Theory, Springer Finance, https://doi.org/10.1007/978-3-030-74410-6_15

317

318

15 Characterizing the Equilibrium

such that the representative trader economy equilibrium reflects the original economy’s equilibrium. In particular, this implies that both the optimal asset demands and the equilibrium price process S is the same in the representative trader economy equilibrium as in the original economy’s equilibrium. This chapter characterizes the economic equilibrium in the original economy using this representative trader economy’s equilibrium.

15.2 The Supermartingale Deflator This section characterizes the equilibrium supermartingale deflator. Remark 85 (Choice of Characterization) Given an equilibrium price process S, there are many possible characterizations of the equilibrium supermartingale deflator. The different characterizations correspond to the representative trader’s and the individual traders’ supermartingale deflators, determined from their portfolio optimization problems. In this regard, we note that the representative trader’s optimal trading strategy in the representative trader economy equilibrium is necessarily a buy and hold trading strategy with positive units in the mma. Consequently, by Theorem 66 in Chap. 14, letting YTλ denote the representative trader’s optimal supermartingale deflator, λ YTλ is a probability density with respect to P and dQ dP = YT is an equivalent martingale measure, i.e. YTλ ∈ M. We will use this observation repeatedly below. (Complete Market) In a complete market with respect to any Q ∈ Ml where Ml = {Q ∼ P : S is a Q local martingale}, by the Second Fundamental Theorem 14 of asset pricing in Chap. 2, the local martingale measure Q ∈ Ml is unique. By Sect. 14.6.2 in Chap. 14, all of the traders’ and representative trader’s supermartingale deflators are martingale deflators that are probability densities with respect to P (i.e. they are in the set M). Hence, all of these martingale densities are equal, i.e. dPi  i dQ U  (N0 + mT , λ) dP Ui (XT )   = YT = = ∈M i  dP E[U  (N0 + mT , λ)] i E dP dP Ui (XT )

for n all i = 1, . . . , I where the aggregate market wealth N0 + mT = N0 + j =1 Nj Sj (T ). This expression for investor i’s supermartingale deflator is from expression (11.29), Sect. 11.9 in Chap. 10. The change of measure in the numerator is needed since the set of martingale deflators depend on the probability measure P. (Incomplete Market)

15.2 The Supermartingale Deflator

319

In an incomplete market, the supermartingale deflators need not be unique, hence each may yield a different characterization of an equilibrium supermartingale deflator, YTλ = YTi =

U  (N0 +mT ,λ) v  (N0 +m0 ,λ,S) dPi dP

Ui (XTi (S)) vi (xi ,S)

∈ Ds

and

∈ Ds for all i = 1, . . . , I

where the aggregate market wealth N0 + mt = N0 + By the observation above YTλ =

n

j =1 Nj Sj (t)

for t ∈ {0, T }.

U  (N0 + mT , λ) ∈ M. E [U  (N0 + mT , λ)]

For purposes of empirical implementation, the characterization of the supermartingale deflator in terms of the representative trader’s aggregate utility function is preferred. This is for two reasons. First, for the supermartingale deflators of the individual traders, their wealth at times 0 and T , (xi , XTi ), are unobservable. In contrast, for the supermartingale deflator of the representative trader, the market wealth process (N0 + m0 , N0 + mT ) is observable. In terms of preferences, both utility function representations U and Ui are equally unobservable. Second, the representative trader’s supermartingale deflator is a martingale deflator, in contrast to those of the individual traders. This observation has important implications for the existence of asset price bubbles in equilibrium under the representative trader’s martingale probability density. This is discussed shortly. This completes the remark. Given these remarks and using Theorem 64 in Chap. 14, the next theorem summarizes the properties an equilibrium supermartingale deflator. Theorem 71 (An Equilibrium Supermartingale Deflator (and State Price Density)) Given an equilibrium price process S, an equilibrium supermartingale deflator is (Complete Market) YT =

U  (N0 + mT , λ) ∈M E[U  (N0 + mT , λ)]

YTλ =

U  (N0 + mT , λ) ∈M E [U  (N0 + mT , λ)]

(Incomplete Market)

where the aggregate market wealth N0 + mt = N0 + nj=1 Nj Sj (t) for t ∈ {0, T }, and U (x, λ) is the representative trader’s utility function.

320

15 Characterizing the Equilibrium

As shown in this theorem, an equilibrium supermartingale deflator is characterized as a function of the representative trader’s   marginal utility and aggregate market wealth at times 0 and T . Because E YTλ = 1, the next corollary follows from Theorem 3 in Chap. 1. Corollary 14 (An Equilibrium State Price Density Process) state price density process is given by  E[U  (N0 + mT , λ) |Ft ]  Ytλ = E YTλ |Ft = ∈M E [U  (N0 + mT , λ)]

An equilibrium

(15.1)

for all t ∈ [0, T ]. These expressions manifest the importance of the market portfolio in the pricing of all risky assets (and derivatives) in an economic equilibrium. Remark 86 (Pricing Derivatives) When pricing derivatives in an incomplete market using risk-neutral valuation (see Sect. 2.7 in Chap. 2), the set of equivalent martingale measures typically contains an infinite number of elements. Derivatives that are non-attainable via an admissible s.f.t.s. whose value process is a Q martingale do not have a unique price. Since the equilibrium supermartingale deflator is not unique in an incomplete market, choosing the representative trader’s state price density (as in the above theorem), gives a rule for obtaining an element in this set. However, the derivative price determined in this fashion is different from that obtained by choosing an individual trader’s supermartingale deflator. The individual traders would want to use their own supermartingale deflators to price derivatives, and not the representative trader’s supermartingale deflator. We emphasize this insight with the following observation: the representative trader’s supermartingale deflator does not provide the relevant probability measure for a trader to use to price derivatives.

The representative trader’s supermartingale deflator is useful, but only for characterizing the equilibrium asset price process S. This completes the remark.

15.3 Asset Price Bubbles This section studies whether asset price bubbles can exist in a rational equilibrium. The answer is: (1) no in a complete market (for the representative trader and all individual traders), and (2) no for the representative trader in an incomplete market, but yes for individual traders in an incomplete market. To prove this claim, we start with an observation. By Theorem 63 in Chap. 13, for either a complete or incomplete market, we know that both NFLVR and ND hold. Next, by the Third Fundamental Theorem 16 of asset pricing in Chap. 2, we get that there exists an equivalent martingale measure Q ∈ M.

15.3 Asset Price Bubbles

321

15.3.1 Complete Markets For a complete market with respect to any Q ∈ Ml , since the equivalent local martingale measure is unique by the Second Fundamental Theorem 14 of asset pricing in Chap. 2, this implies that the unique local martingale measure is a martingale measure since YT =

U  (N0 + mT , λ) ∈ M ⊂ Ml . E[U  (N0 + mT , λ)]

Hence,   E [YT ] =  1 and its supermartingale deflator process Yt = U (N0 +mT ,λ) is a martingale deflator (see Theorem 3 in Chap. 1). E E[U  (N +m ,λ)] |Ft T 0 Thus, there can be no asset price bubbles for either individual or the representative trader. This provides an alternative proof, in the context of an equilibrium economy of Theorem 31 in Chap. 3.

15.3.2 Incomplete Markets For an incomplete market, the local martingale measure is not necessarily unique. By the observation at the start of this section, since YTλ is a probability density for P, the representative trader’s supermartingale deflator is a martingale deflator that generates an equivalent martingale measure (see Theorem 66 in Chap. 14). This implies that the representative trader sees no price bubble. In contrast, an individual trader’s optimal supermartingale deflator need not be a martingale measure. Hence, individual traders may believe that the asset price exhibits a price bubble. In addition, it is possible that some investors see the price process as exhibiting an asset price bubble while others do not (see Jarrow [93] for an elaboration). Remark 87 (Implications for Empirical Testing) The implications of these observations about asset price bubbles for the empirical testing of equilibrium asset price models are two. First, when estimating the equilibrium market price processes for the traded risky assets using time series data, one can use the representative trader’s supermartingale deflator to evaluate risk. Because YTλ is a probability density for P, the representative trader’s supermartingale deflator is a martingale deflator that generates an equivalent martingale measure. Under this equivalent martingale measure, the equilibrium asset price process exhibits no price bubble. Hence, empirical estimation of the equilibrium systematic risk return relation (discussed in the next section) excludes the consideration of asset price bubbles when this martingale measure is used. Second, when evaluating a trader’s optimal trading strategy, the trader’s supermartingale deflator is the relevant supermartingale deflator, even in an economic equilibrium. This implies that each trader must explicitly consider asset price

322

15 Characterizing the Equilibrium

bubbles when forming their optimal portfolio. Hence, the trader must also explicitly consider asset price bubbles when valuing derivatives in equilibrium in an incomplete market. It is also interesting to note that in an incomplete market the equilibrium systematic risk return relation is not unique. Any trader’s supermartingale deflator YTi ∈ Dsi or the representative trader’s supermartingale deflator YTλ can be used to characterize systematic risk, using Theorem 36 in Chap. 4 (as discussed in the next section). This is because all of these supermartingale deflators are consistent with the equilibrium price process S. This completes the remark.

15.4 Systematic Risk In this section we use the non-normalized market ((B, S), F, P). Given is an equilibrium price process S. By Theorem 67 in Chap. 14, there exists a representative trader economy {(F, P) , (N0 , N) , (P, U (λ), (N0 , N))} such that the representative trader economy equilibrium reflects the original economy’s equilibrium. This section revisits the characterization of systematic risk using the equilibrium state price density. Given the representative trader’s supermartingale deflator is a martingale deflator, we can apply Theorem 36 in Chap. 4. This theorem applies for complete and incomplete markets. Next, we partition [0, T ] into a collection of sub-intervals of length Δ > 0. Fix a time interval [t, t + Δ] ⊂ [0, T ] where t ≥ 0 aligns with one of these partitions. To simplify the presentation we assume that the risky assets contain default-free zerocoupon bonds paying $1 at times t = Δ, . . . , T . From Theorem 38 in Chap. 4, we have that the time t conditional expected return of any traded asset satisfies. Theorem 72 (The Equilibrium Risk Return Relation)

 

E[U  (N0 + mT , λ) |Ft+Δ ] Bt

Ft E [Ri (t) |Ft ] = r0 (t)−cov Ri (t), (1 + r (t)) 0

 E[U (N0 + mT , λ) |Ft ] Bt+Δ

(15.2)

for all i = 1, . . . , n and t ∈ [0, T ] where r0 (t) =

1 p(t,t+Δ)

− 1 is the return on a

default-free zero-coupon bond that matures at time t + Δ and Ri (t) = is the return on the ith risky asset.

Si (t+Δ)−Si (t) Si (t)

15.5 Consumption CAPM

323

We note that when the time step is small, i.e. Δ ≈ dt, then and this relation simplifies to

Bt Bt+Δ (1

+ r0 (t)) ≈ 1,

 E[U  (N0 + mT , λ) |Ft+Δ ]

Ft . E [Ri (t) |Ft ] ≈ r0 (t) − cov Ri (t), E[U  (N0 + mT , λ) |Ft ] 

(15.3)

This expression is the equilibrium risk return relation for traded assets. Systematic risk is measured of any asset’s return with the equilibrium state  by the covariance  E[U (N0 +mT ,λ)|Ft+Δ ] price density E[U  (N0 +mT ,λ)|Ft ] , which depends on aggregate market wealth and the representative trader’s marginal utility.    (N +m ,λ)|F t+Δ ] T 0 • When E[U is large, systematic risk is low (remember the E[U  (N0 +mT ,λ)|Ft ] negative sign). This occurs when the marginal utility of wealth is large, hence aggregate market wealth is scarce. An asset whose return is large when market wealthis scarce is valuable. Such an asset is “anti-risky.”  E[U  (N0 +mT ,λ)|Ft+Δ ] • When E[U  (N0 +mT ,λ)|Ft ] is small, systematic risk is high. This occurs when the marginal utility of wealth is small, hence aggregate market wealth is plentiful. An asset whose return is large when market wealth is plentiful is less valuable. Such an asset is “risky.”

15.5 Consumption CAPM This section provides a restatement of Theorem 72 to obtain the consumption capital asset pricing model (CCAPM). As in the previous section, we partition [0, T ] into a collection of sub-intervals of length Δ > 0. Fix a time interval [t, t + Δ] ⊂ [0, T ] where t ≥ 0 aligns with one of these partitions. Theorem 73 (CCAPM) Define μm (t) = E[N0 + mT |Ft ]. Assume U is state independent and three times differentiable. Then, E [Ri (t) |Ft ] ≈ r0 (t) −

U  (μm (t), λ) cov [Ri (t), (μm (t + Δ) − μm (t)) |Ft ] U  (μm (t), λ) (15.4)

for all i = 1, . . . , n and t ∈ [0, T ] where r0 (t) =

1 p(t,t+Δ)

− 1 is the return on a

default-free zero-coupon bond that matures at time t + Δ and Ri (t) = is the return on the ith risky asset.

Si (t+Δ)−Si (t) Si (t)

Proof Use a Taylor series expansion of U  around μm (t). U  (N0 + mT , λ) = U  (μm (t), λ) + U  (μm (t), λ)(N0 + mT − μm (t)) + 12 U  (θ, λ)(N0 + mT − μm (t))2 for θ ∈ [μm (t), N0 + mT ]. Use the approximation U  (N0 + mT , λ) ≈ U  (μm (t), λ) + U  (μm (t), λ)(N0 + mT − μm (t)). Take conditional expectations at time t + Δ.

324

15 Characterizing the Equilibrium

  E U  (N0 + mT , λ) |Ft+Δ ≈ U  (μm (t), λ) + U  (μm (t), λ)(N0 + E[mT | Ft+Δ ] − μm (t)) = U  (μm (t), λ) + U  (μm (t), λ)(μm (t + Δ) − μm (t)). Take at time t.  conditional expectations  E U  (N0 + mT , λ) |Ft ≈ U  (μm (t), λ) + U  (μm (t), λ)(N0 + E[mT |Ft ] − μm (t)) = U  (μm (t), λ) + U  (μm (t), λ)(μm (t) − μm (t)) = U  (μm (t), λ). Then,    (N +m ,λ)|F t+Δ ] T 0 |F cov Ri (t), E[U  t E[U (N0 +mT ,λ)|Ft ]     m (t),λ)(μm (t+Δ)−μm (t)) |F ≈ cov Ri (t), U (μm (t),λ)+U U(μ (μ t (t),λ) m =

U  (μm (t),λ) U  (μm (t),λ) cov [Ri (t), (μm (t

+ Δ) − μm (t)) |Ft ]. This completes the proof.

To understand this theorem we need to study the various components on the right  (μ (t),λ) m side of expression (15.4). First, the ratio Am (t) = − UU  (μ corresponds to the m (t),λ) absolute risk aversion coefficient for the representative trader, i.e. it is a measure of the market’s risk aversion (see Lemma 23 in Chap. 9). As the market becomes more risk averse, everything else constant, Am (t) increases. This implies that as the market’s risk aversion increases, risk premium increase. Second, the difference Δμm (t) = (μm (t + Δ) − μm (t)) measures the change in expected aggregate wealth, which is consumed at time T . Hence, this measures the change in anticipated “aggregate consumption” over [t, t + Δ]. Any asset that is positively correlated with changes in aggregate consumption is risky because the additional return is less valuable in states where aggregate consumption is large relative to states where it is small. Using this new notation, the above expression becomes E [Ri (t) |Ft ] ≈ r0 (t) + Am (t)cov [Ri (t), Δμm (t) |Ft ] .

(15.5)

In this form, this expression is easily recognized as the CCAPM (see Back [5, p. 265]).

15.6 Intertemporal CAPM This section derives the intertemporal capital asset pricing model (ICAPM). Merton’s [144] ICAPM has implications with respect to the optimal nonnegative wealth s.f.t.s., mutual fund theorems, and the systematic risk return relation among the risky assets. This section focuses only on the systematic risk return relation among the risky assets. In this regard, the ICAPM characterizes the equilibrium relation between the expected return on any asset and the expected returns on the market portfolio and additional (hedging) risk factor returns. This is the relation derived below.

15.6 Intertemporal CAPM

325

As before, we partition [0, T ] into a collection of sub-intervals of length Δ > 0. Fix a time interval [t, t +Δ] ⊂ [0, T ] where t ≥ 0 aligns with one of these partitions. As in Chap. 4, we assume that the risky assets contain default-free zero-coupon bonds paying $1 at times t = Δ, . . . , T . −mt Define rm (t) = mt+Δ to be return on aggregate market wealth, which is the mt return on the market portfolio. For the ICAPM, the only implication of an economic equilibrium used in the subsequent derivation is the fact that the existence of an equilibrium implies the existence of an equivalent martingale measure Q ∈ M (see Theorem 63 in Chap. 13). Then, using Theorem 36 in Chap. 4, we get Theorem 74 (Multiple-Factor Beta Model) Ri (t) − r0 (t) = βim (t) (rm (t) − r0 (t))

+ j ∈Φi βij (t) rj (t) − r0 (t)

(15.6)

for all i = 1, . . . , n and t ∈ [0, T ] where βij (t) = 0 for all i, j ∈ !i , r0 (t) = 1 p(t,t+Δ) − 1 is the return on a default-free zero-coupon bond that matures at time

i (t) t + Δ, Ri (t) = Si (t+Δ)−S is the return on the ith risky asset, and rj (t) is the Si (t) return on the j th basis asset.

Proof This is Theorem 36, adding the market portfolio as a basis asset. This can be done without loss of generality, see Simmons [178, p. 197]. However, for the market portfolio, it may be the case that βim (t) = 0. This completes the proof. Taking time t conditional expectations gives the ICAPM. Corollary 15 (ICAPM) E[Ri (t) |Ft ] − r0 (t) = βim (t) (E[rm (t) |Ft ] − r0 (t))

+ j ∈Φi βij (t) E[rj (t) |Ft ] − r0 (t)

(15.7)

for all i = 1, . . . , n and t ∈ [0, T ]. The risk factors are those basis assets whose expected excess return are nonzero. This corollary is a key implication of Merton’s [144] model (see Back [5, p. 267]). Merton’s derivation was under a more restrictive Markov diffusion process assumption for the risky asset price processes, which implies continuous sample paths and no jumps. The previous derivation does not require the Markov diffusion process assumption.

326

15 Characterizing the Equilibrium

15.7 Intermediate Consumption This section discusses the changes in the previous results from the inclusion of intermediate consumption into a trader’s utility function as given in Chap. 12. Because all of the previous implications are based on a characterization of the representative trader’s equilibrium supermartingale deflator, we replace it with the representative trader’s supermartingale deflator with intermediate consumption to obtain the analogous results (see Jarrow [94] for a complete presentation of the subsequent results). Recall that when including intermediate consumption into an investor’s optimization problem, the optimal supermartingale deflator, when a martingale deflator, is given by  Yti =

E

   (cˆ , T ) |F i Ui (ct , t) U E dP T t dP i   =   dPi   (cˆ , T ) i E dP Ui (ct , t) E dP U T dP i dPi dP Ft



for all t ∈ [0, T ] (see Chap. 12, Sect. 12.4.1.3). The change of measure in the numerator is needed since the set of supermartingale deflators depends on the probability measure P (see Sect. 11.9 in Chap. 11).

15.7.1 Systematic Risk In this section we use the non-normalized market ((B, S), F, P). Given is an equilibrium price process S. The representative trader’s supermartingale deflator process is a martingale with respect to P, i.e. Ytλ =

U  (ct , t, λ) ∈M E [U  (ct , t, λ)]

for all t ∈ [0, T ] where ct is the representative trader’s optimal consumption, which in equilibrium equals aggregate consumption. The next theorem follows using Theorem 38 in Chap. 4. As before, partition [0, T ] into a collection of sub-intervals of length Δ > 0. Fix a time interval [t, t + Δ] ⊂ [0, T ] where t ≥ 0 aligns with one of these partitions. Theorem 75 (The Equilibrium Risk Return Relation)



U  (ct+Δ , t + Δ, λ) Bt E [Ri (t) |Ft ] = r0 (t) − cov Ri (t), (1 + r0 (t))

Ft  U (ct , t, λ) Bt+Δ (15.8) 

15.7 Intermediate Consumption

327

for all i = 1, . . . , n and t ∈ [0, T ] where r0 (t) =

1 p(t,t+Δ)

− 1 is the return on a

default-free zero-coupon bond that matures at time t + Δ and Ri (t) = is the return on the ith risky asset. We note that when the time step is small, i.e. Δ ≈ dt, then and this relation simplifies to

Bt Bt+Δ (1

Si (t+Δ)−Si (t) Si (t)

+ r0 (t)) ≈ 1,

  U  (ct+Δ , t + Δ, λ)

E [Ri (t) |Ft ] ≈ r0 (t) − cov Ri (t),

Ft . U  (ct , t, λ)

15.7.2 Consumption CAPM Using the previous theorem, we can derive the CCAPM. Partition [0, T ] into a collection of sub-intervals of length Δ > 0. Fix a time interval [t, t + Δ] ⊂ [0, T ] where t ≥ 0 aligns with one of these partitions. Theorem 76 (CCAPM) Assume U (c, t, λ) is state independent, three times differentiable in c, and U  (c, t, λ) differentiable in t. Then, E [Ri (t) |Ft ] ≈ r0 (t) −

U  (ct , t, λ) cov [Ri (t), (ct+Δ − ct ) |Ft ] U  (ct , t, λ)

for all i = 1, . . . , n and t ∈ [0, T ] where r0 (t) =

1 p(t,t+Δ)

(15.9)

− 1 is the return on a

default-free zero-coupon bond that matures at time t + Δ, Ri (t) = the return on the ith risky asset, and ct is aggregate consumption. Proof Use a Taylor series expansion of U  around (ct , t). U  (ct+Δ , t + Δ, λ) ≈ U  (ct , t, λ) + U  (ct , t, λ)(ct+Δ − ct ) +

Si (t+Δ)−Si (t) Si (t)

is

∂U  (ct ,t,λ) Δ ∂t



+ 12 U  (ct , t, λ)(ct+Δ − ct )2 . Then, because ∂U (c∂tt ,t,λ) Δ is non-random,      ,t+Δ,λ) U  (ct ,t,λ)+U  (ct ,t,λ)(ct+Δ −ct ) |F |F ≈ cov R (t), cov Ri (t), U (cUt+Δ  (c ,t,λ)  t i t U (ct ,t,λ) t =

U  (ct ,t,λ) U  (ct ,t,λ) cov [Ri (t), (ct+Δ

− ct ) |Ft ] . This completes the proof.

15.7.3 Intertemporal CAPM As noted previously, the ICAPM as contained in Theorem 74 only depends on the existence of an equivalent martingale measure Q ∈ M. And, the existence of an equivalent martingale measure follows directly from the existence of an equilibrium. Hence, the previous derivation of the ICAPM applies unchanged when intermediate consumption is included into the model’s structure, as long as an equilibrium

328

15 Characterizing the Equilibrium

including intermediate consumption exists. This completes the discussion of the ICAPM.

15.8 Notes Characterizing asset market equilibrium expected returns is the realm of traditional asset pricing theory. Hence, there are numerous excellent references on this topic, see Back [5], Bjork [14], Dana and Jeanblanc [42], Duffie [52], Follmer and Schied [63], Huang and Litzenberger [73], Ingersoll [75], Karatzas and Shreve [124], Merton [147], Pliska [157], and Skiadas [179].

Chapter 16

Market Informational Efficiency

Market informational efficiency is a key concept used in financial economics, introduced by Fama [57] in the early 1970s. To formalize this concept, we need the solution to a trader’s portfolio optimization problem (as in Part II) and the meaning of an economic equilibrium (as in Chaps. 13 and 14). Given these insights, a rigorous definition of an efficient market can be formulated. This rigorous definition is contrasted with the intuitive definition originally provided in Fama [57]. It will be shown that this rigorous definition of an efficient market requires only the existence, and not the characterization of an economic equilibrium. Such a rigorous definition allows new insights into the testing of an informationally efficient market, which will be discussed below. This chapter is based on Jarrow and Larsson [100, 103].

16.1 The Set-Up Informational efficiency deals with differential information across the traders in the market. The previous chapters in this book assumed that traders have the same information sets. We need to relax this assumption in this chapter. Consequently, we will briefly set-up the structure for a new economy where traders have differential information. To avoid unnecessary replication, the presentation is brief, pausing only to identify the changes needed to include differential information into the previous formulation. Part I (Arbitrage Pricing Theory) Given is a probability space (Ω, F , P). The traders are represented by the collection i

I F , Pi , Ui , e0i , ei i=1 where the filtration Fi = (Fti )t∈[0,T ] with FTi = F is trader i’s information. To simplify the presentation, as before, this chapter only considers traders with utility functions over terminal wealth. All information in the economy is represented by the filtration F = F1 ∨ · · · ∨ FI , which is the smallest information set that contains all of the trader’s information sets. © Springer Nature Switzerland AG 2021 R. A. Jarrow, Continuous-Time Asset Pricing Theory, Springer Finance, https://doi.org/10.1007/978-3-030-74410-6_16

329

330

16 Market Informational Efficiency

We consider a normalized market where traded is a mma with Bt = 1 for all t ∈ [0, T ] and risky assets with prices St for all t ∈ [0, T ], both in positive supply. Let σ (S) denote the filtration generated by S. We assume that σ (S) ⊂ F1 ∧· · ·∧FI , the smallest information set that is included in all of the trader’s information sets. This means that the traders observe the price process St for all t ∈ [0, T ]. The content in Chaps. 2–8 pertain to an individual trader, and all of the results follow with the information set F and probability measure P replaced by Fi and Pi , respectively. To avoid difficulties in interpreting value process of more informed traders, we require that the admissible s.f.t.s. (α0i , α i ) ∈ A (xi ) of all traders i = 1, . . . , I , which are Fi adapted, be S-integrable with respect to F. Finally, we note that the concepts of NA, NUPBR, NFLVR, and ND all depend on the filtration Fi , but as noted therein, they are invariant to a change in equivalent probability measures, i.e. Pi ∼ P. To identify this dependence, given a filtration G ⊂ F, we will denote these as N A(G), NUPBR(G), NFLVR(G), and ND(G). Part II (Portfolio Optimization Theory) The contents in Chaps. 10–12 pertain to an individual trader, and all of the results follow with the information set F and probability measure P replaced by Fi and Pi , respectively. The optimal nonnegative wealth s.f.t.s.’s (α0i , α i ) ∈ A (xi ) of trader i are adapted to the filtration Fi and S-integrable with respect to F. Part III (Equilibrium) An economy is defined as a collection  I    i i i . (F, P) , (N0 , N) , F , Pi , Ui , e0 , e i=1

The definition of an equilibrium is the same. The economy is in equilibrium if there exists a price process S and demands (α0i (t), αti ) : i = 1, . . . , I such that for all t ∈ [0, T ] a.s. P, (i) (α0i (t), αti ) are optimal for i = 1, . . . , I , (ii) N0 = I α0i (t), and i=1 I (iii) Nj = i=1 αji (t) for j = 1, . . . , n. We now discuss the theorems concerning equilibrium needed for understanding market efficiency in this extended set-up. The necessary conditions for an equilibrium as given in Theorem 63 of Chap. 13 are modified in this economy. The proof of Theorem 63 in Chap. 13 only yields the following result. Theorem 77 (Necessary Conditions for an Equilibrium) Suppose there exists an equilibrium with price process S, then NF LV R(Fi ) holds for all i = 1, . . . , I . This theorem implies by the First Fundamental Theorem 13 of asset pricing in Chap. 2 that there exists a Qi ∈ Ml (Fi ) where S is a Qi local martingale with respect to the filtration Fi for all i = 1, . . . , I . Note the dependence of the set of equivalent local martingale measures on the filtration Fi .

16.1 The Set-Up

331

In a differential information economy, the previous proof cannot be used to show ND(Fi ). The reason is that ND(Fi ) requires that supply equals aggregate demand, and aggregate demand depends on the sum of the traders’ optimal holdings which, in turn, depend on the traders’ information sets Fi . This invalidates the previous proof’s argument because aggregate demand is adapted F, not Fi . And, as just noted, N F LV R(Fi ) only implies the existence of a local martingale measure Qi ∈ Ml (Fi ), not an martingale measure Q ∈ M(F). To obtain this result, we need a different method of proof (see Corollary 16 below). The aggregate utility function U : (0, ∞) × L0++ × Ω → R is defined by 7 U (x, λ, ω) =

sup {x1 ,...,xI }∈RI

I  i=1

I

 dPi Ui (xi , ω) : x = λi (ω) xi dP


0 for all i = 1, . . . , I , and λ = (λ1 , . . . , λI ). A representative trader is defined as a collection (F, P, U (λ), (N0 , N)). As specified, the representative trader has the information set F and beliefs P. The necessity of giving the representative trader the largest information set F is because the representative agent has to optimize considering all the individual traders’ information sets Fi . We note that the optimal nonnegative wealth s.f.t.s. for the representative trader (α0λ , α λ ) ∈ A (x) is therefore adapted to the filtration F. Theorem 66 in Chap. 14 follows immediately with no changes in the proof. We repeat it here for convenience. Theorem 78 (Buy and Hold Optimal Trading Strategies) Given a price process S, suppose that the representative trader’s optimal portfolio is a buy and hold strategy, i.e. Xˆ tλ = θ + η · St

f or all t ∈ [0, T ]

where θ ∈ R and η = (η1 , . . . , ηn ) ∈ Rn++ . If θ = 0 then YˆTλ is the probability density for an equivalent martingale probability measure with respect to F. As before, a representative trader economy is a collection {(F, P) , (N0 , N) , (F, P, U (λ), (N0 , N))} . And, a representative trader economy equilibrium is a price process S and demands (α0λ (t), αtλ ) such that for all t ∈ [0, T ] a.s. P, (i) (α0λ (t), αtλ ) are optimal for (F, P, U (λ), (N0 , N)), (ii) N0 = α0λ (t), and (iii) Nj = αjλ (t) for j = 1, . . . , n.

332

16 Market Informational Efficiency

Last, we recall Theorem 67 from Chap. 14 whose proof also follows without modification. Theorem 79 (Existence of a Representative Trader Economy Equilibrium Reflecting the Original Economy’s Equilibrium) Given an economy  i i i

I  (F, P) , (N0 , N) , F , Pi , Ui , e0 , e i=1 , let S be an equilibrium price process with the optimal trading strategies (αˆ 0 i (t), αˆ t i )I i=1 . Then, the representative trader economy 



 (F, P) , (N0 , N) , F, P, U (λ∗ ), (N0 , N)

reflects the original economy’s equilibrium, where λ∗i =

v1 (x1 , S)YT1 (S) vi (xi , S)YTi (S)

with YTi the solution to trader i’s dual problem and vi (xi , S) trader i’s value function, given the initial wealth xi and the price process S, for all i = 1, . . . , I .

16.2 The Definition This section presents the definition of an informationally efficient market. The original intuitive definition of market efficiency is given by Fama [57, p. 383] in his seminal paper: A market in which prices always ‘fully reflect’ available information is called ‘efficient’.

Three information sets have been considered when discussing efficient markets: (1) historical prices (weak form efficiency), (2) publicly available information (semi-strong efficiency), and (3) private information (strong form efficiency). A market may or may not be efficient with respect to each of these information sets. In quantifying this definition, for its use in testing market efficiency, it is commonly believed that one must first specify an equilibrium model (see Fama [59]). This is called the joint-hypothesis or the bad-model problem. Indeed, Fama states [58, p. 1575], The joint-hypothesis problem is more serious. Thus, market efficiency per se is not testable. It must be tested jointly with some model of equilibrium, an asset pricing model. This point, the theme of the 1970 review (Fama 1970), says that we can only test whether information is properly reflected in prices in the context of a pricing model that defines the meaning of ’properly’.

16.2 The Definition

333

In contrast, we quantify the original definition in such a manner that one can test market efficiency without specifying a particular equilibrium model. As such, our formulation overcomes the bad-model problem. To state our definition, we first recall from Chap. 2 that a market is a collection (S, G, P) where S is G adapted with σ (S) ⊂ G ⊂ F = ∨i Fi We note that because σ (S) ⊂ Fi for all i = 1, . . . , I by assumption, S is also Fi adapted. Combining these two observations, S is G ∨ Fi adapted too, which is possibly a smaller information set. Of course, S is adapted to any larger information set, e.g. F. We can now define an efficient market. Definition 47 (Market Efficiency) The market (S, G, P) is efficient with respect to the information set G ⊂ F = ∨i Fi if there exists a economy 

I (F, P) , (N0 , N) , Fi , Pi , Ui , e0i , ei i=1 for which S is an equilibrium price process. It is important to note that this definition does not require the identification of the equilibrium economy generating the G adapted price process S, only that one such economy exists. That is, we do not have to specify beliefs, preferences, or

I endowments across all traders, i.e. Fi , Pi , Ui , e0i , ei i=1 . This definition is easily seen to be consistent with Fama’s [57] original definition, and the manner in which market efficiency has been tested over the subsequent years using the joint model hypothesis (see Fama [58, 59]). Indeed, as mentioned before, the standard approach to testing efficiency is to assume a particular equilibrium model, and then show the model is consistent or inconsistent with historical observations of the price process and different information sets. If the model is consistent with the data, then informational efficiency is accepted. If inconsistent, then informationally efficiency is not necessarily rejected because of the joint hypothesis. To reject efficiency, one must show that there is no equilibrium model consistent with the data; hence, the definition. Remark 88 (Weak Form, Semi-strong Form, Strong Form Efficiency) When defining an efficient market, the choice of the information set G can vary. The market is said to be weak form efficient if G = σ (S), it is semi-strong form efficient if G = ∧i Fi , and it is strong form efficient if G = F where F = ∨i Fi . Weak form reflects the information σ (S) contained in prices. Semi-strong form reflects publicly known information, which is that known by all the traders, ∧i Fi . And, strong form corresponds to all private information F that is available in the economy. This completes the remark. Remark 89 (Rational Expectations Equilibrium (REE)) Market efficiency is related to the notion of a rational expectations equilibrium. Recall that in our definition of an equilibrium, each trader conjectures a price process S when choosing their optimal trading strategies. An equilibrium price process is one where this conjectured price process is the same for all traders, and it is consistent with trader optimality and market clearing. The equilibrium price process generates an information set,

334

16 Market Informational Efficiency

denoted σ (S). We can now define two types of rational expectations equilibrium (REE). A fully revealing REE is defined to be an equilibrium where the equilibrium price process S is such that σ (S) = F, which is all private information. Since by construction of the economy σ (S) ⊂ Fi for all i, this implies that all of this private information is (fully) revealed by the equilibrium price process. A partially revealing REE is defined to be an equilibrium where the equilibrium price process S is not fully revealing. This completes the remark.

16.3 The Theorem This section provides a characterization of market efficiency. Theorem 80 (Market Efficiency) Let (S, G, P) be a market with G ⊂ F. Then, the market is efficient with respect to G if and only if there exists a Q ∈ M(G) where M(G) = {Q ∼ P : S is a Q martingale w.r.t. G}. Proof (⇒) Suppose the market is efficient with respect to G. By the definition of a market, S is G adapted. Then, because the market is efficient, there exists an equilibrium supporting S. By Theorem 79, there exists a representative trader equilibrium that reflects the original economy’s equilibrium. In this equilibrium, the representative trader’s optimal nonnegative wealth s.f.t.s. is a buy and hold with N0 > 0 units of the mma and N shares of the risky assets. By Theorem 78, the representative trader’s optimal supermartingale deflator YˆTλ is a probability density for an equivalent martingale probability measure with respect to F. Hence, there exists a Q ∈ M(F). Since G ⊂ F, S is also a Q martingale with respect to G. (⇐) Let (S, G, P) be a market with G ⊂ F. Note that S that is G adapted. Suppose there exists a Q ∈ M(G). To prove the existence of an equilibrium supporting S, we need to construct an economy for which the given S is an equilibrium price process. The following construction is based on that in Jarrow and Larsson [101]. Consider the economy  I    (G, P) , (N0 , N) , G, Pi , Ui , e0i , ei i=1

where the trader’s information F = Fi = G, beliefs Pi = P∗ for all i, and 1−ρ ZT   1−ρ Q E ZT

with ZT =

N0 +N ·ST N0 +N ·S0

> 0. Note that Zt =

N0 +N ·St N0 +N ·S0

dP∗ dQ

=

> 0 is a strictly

positive Q martingale with respect to G where E Q [ZT ] = 1 and N0 + N · St is ρ the aggregate market wealth at time t. Let the traders’ preferences be Ui (x) = xρ for all i where 0 < ρ < 1 and x > 0 (see Chap. 9). And, let the traders’ initial N endowments be (e0i (0) = NI0 , e0i = I ) for all i, i.e. each trader is endowed with the same number of shares in the mma and each risky asset.

16.3 The Theorem

335

We claim that each trader’s optimal nonnegative wealth s.f.t.s. is to not trade. If this is the case, then St is an equilibrium price process because at any time t ∈ N [0, T ], letting (αˆ 0i (t) = NI0 , αˆ ti = I ) denote trader i’s optimal s.f.t.s., summing across i gives the market clearing conditions I  i=1

αˆ 0i (t) = N0 ,

I 

αˆ ti = N.

i=1

To prove the claim, first observe that the time T expected utility of this trading strategy is ∗

EP

   N0 N ∗ Ui = E P [Ui (zZT )] + · ST I I

where z = NI0 + NI·S0 . Next, let XT be the terminal wealth of an arbitrary s.f.t.s. (α0 , α) ∈ N (z) with initial wealth z where XT = α0 (T ) + αT · ST .  ∗ ∗ ∗  Then, E P [Ui (XT )] − E P [Ui (zZT )] ≤ E P Ui (zZT ) (XT − zZT ) by the concavity of Ui . But, Ui (zZT ) = (zZT )ρ−1 . Rewriting,   ∗  ∗  E P Ui (zZT ) (XT − zZT ) = zρ−1 E P ZT ρ−1 (XT − zZT )   ∗  1−ρ dQ E P dP = zρ−1 E Q ZT ∗ (XT − zZT )   1−ρ E Q [XT − zZT ]. = zρ−1 E Q ZT But, E Q [XT ] ≤ z since XT is a Q supermartingale with respect to G (a nonnegative Q local martingale with respect to G) with initial value X0 = z. And, E Q [zZT ] = z since E Q [ZT ] = 1. Substitution gives ∗ ∗ E P [Ui (XT )] − E P [Ui (zZT )] ≤ 0, which proves the claim and completes the proof. This theorem generates a number of important implications. The first relates to asset price bubbles. Remark 90 (Bubbles and Market Efficiency) In a complete market with respect to G, if the market is efficient with respect to G, then there can be no asset price bubbles where the price process S is G adapted. The reason is because efficiency implies there exists a Q ∈ M(G), and completeness implies this martingale measure is unique (see Theorem 14 in Chap. 2). Hence, S is a martingale with respect to G, and there is no price bubble. In contrast, if the market is incomplete with respect to G, then bubbles are consistent with an efficient market. Indeed, although efficiency guarantees the existence of a Q ∈ M(G), in an incomplete market there can exist an infinite number of local martingale measures Q ∈ Ml (G). And, if the probability measure

336

16 Market Informational Efficiency

Q ∈ Ml (G) chosen by the market makes S a strict local martingale with respect to G, then there is an asset price bubble. This completes the remark. Useful for empirical testing is the following corollary of Theorem 80. Corollary 16 (Market Efficiency) Let (S, G, P) be a market. Then, (S, G, P) is efficient with respect to G if and only if NFLVR (G) and ND (G) hold. This corollary is a direct application of the Third Fundamental Theorem 16 of asset pricing in Chap. 2. It shows that a market is efficient if and only if there are no “mispriced” assets trading in the economy, where mispriced is interpreted as the existence of a FLVR (G) or dominated assets (G). Remark 91 (Existence of an Equilibrium Implies Market Efficiency with Respect to F) A careful reading of the proof of this theorem shows that the existence of an equilibrium implies the stronger condition that the market is efficient with respect to F, the largest information set. This follows because if any trader sees a NFLVR (Fi ) with their information sets, then they would have no optimal nonnegative wealth s.f.t.s. (α0i , α i ) ∈ A (xi ), and there could be no equilibrium. Because this is true for all Fi , it must be true for F. The converse, of course, is not true. The existence of a Q ∈ M(G) does not imply an equilibrium in an economy with the larger information set F. This completes the remark.

16.4 Information Sets and Efficiency A market (S, G, P) is defined to be efficient with respect to an information set G. This section explores whether the market is also efficient with respect to smaller information sets H ⊂ G and larger information sets G ⊂ J. It is easy to prove that if the market is efficient with respect to G, then it is also efficient with respect to any smaller information set H ⊂ G. Corollary 17 (Smaller Information) Let (S, G, P) be a market that is efficient with respect to G. Consider H ⊂ G. Then, (S, G, P) is efficient with respect to H. Proof Since H ⊂ G, all nonnegative wealth s.f.t.s. using only H are adapted to G, hence NFLVR (H) and ND (H) hold as well. Corollary 16 implies the result. This completes the proof. The converse does not hold. To see this, let (S, G, P) be a market that is efficient with respect to G. Suppose the larger information set J where G ⊂ J includes certain knowledge of a risky asset’s time T price at the earlier time 0 < T which is not included in G. For example, suppose that under G, the risky asset price at time T can increase or decrease from S0 , both with strictly positive probability. But, under J, at time 0 the risky asset’s price is known to increase at time T with probability one. Then, given the price process S, the admissible s.f.t.s. of buying and holding

16.5 Testing for Market Efficiency

337

this risky asset and shorting the money market account to finance this purchase is G adapted and the market satisfies NFLVR (G) (because the market is efficient with respect to G). But, this buy and hold strategy is also J adapted and with respect to J it is a FLVR (J). Consequently, NFLVR (J) does not hold and the market cannot be in an equilibrium where some trader knows J. Thus, the market is not efficient with respect to J. With this observation, the next corollary is immediate. Corollary 18 (Larger Information) Let (S, G, P) be a market that is efficient with respect to G. Consider J where G ⊂ J. Then, (S, G, P) is efficient with respect to J if and only if all admissible s.f.t.s. based on J satisfy both NFLVR (J) and ND (J).

16.5 Testing for Market Efficiency This section discusses three methods that can be used to test for an efficient market. The first two approaches are standard in the literature. The third approach is new and based on Theorem 80 given above.

16.5.1 Profitable Trading Strategies As given in Corollary 16, a market is efficient with respect to G if and only if both NFLVR (G) and ND (G) hold. Hence, one can reject efficiency by finding selffinancing and admissible trading strategies that violate NFLVR (G) or ND (G). This approach has been used in the empirical asset pricing literature to reject market efficiency (see Jensen [118] and references therein). Although this approach can be used to reject efficiency, it is less useful for proving that the market is efficient. To do so, one must prove that all self-financing and admissible trading strategies based on G have been exhausted. To do this, the Third Fundamental Theorem 16 of asset pricing in Chap. 2 can be invoked, see the third method in Sect. 16.5.3 below. We note that this proves that an efficient market with respect to G can be rejected, thereby side-stepping the “curse of a joint hypothesis” noted earlier.

16.5.2 Positive Alphas An application of Theorem 80 provides another method for rejecting market efficiency with respect to G. To do this, we want to consider the returns on the

338

16 Market Informational Efficiency

risky assets. Partition [0, T ] into a collection of sub-intervals of length Δ > 0. Fix a time interval [t, t + Δ] ⊂ [0, T ] where t ≥ 0 aligns with one of these partitions. To simplify the methodology we assume that the risky assets contain default-free zero-coupon bonds paying 1 dollar at times t = Δ, . . . , T . We can use Theorem 37 in Chap. 4, which requires the existence of an equivalent martingale measure Q ∈ M(G), which by Theorem 80 is equivalent to the market being efficient with respect to G. This implies the following corollary. Corollary 19 (Test for Market Efficiency with Respect to G) The market is inefficient with respect to G = (Gt )t∈[0,T ] if αi (t) is non-zero for some risky asset i where Ri (t) − r0 (t) = αi (t) +



βij (t) = 0 for all (i, j ), r0 (t) =

j ∈Φi βij (t) 1 p(t,t+Δ)



rj (t) − r0 (t) + εi (t),

(16.1)

− 1 is the return on a default-free zero-

i (t) coupon bond that matures at time t + Δ, Ri (t) = Si (t+Δ)−S is the return on the Si (t) ith risky asset, rj (t) is the return on the j th basis asset, and E[εi (t) |Gt ] = 0.

This is the standard approach used in the empirical asset pricing literature for testing market efficiency with respect to an information set G via estimating a multiplefactor model and rejecting market efficiency if alpha is nonzero (see Fama [58, 59]). Unlike the models used in the existing empirical literature which assume a particular equilibrium economy, this corollary is derived only assuming the existence of a martingale measure, equivalently, an efficient market with respect to G. The derivation does not require the specification of a particular economy. Hence, this approach to testing efficiency also side-steps the “curse of a joint hypothesis” noted earlier.

16.5.3 Asset Price Evolutions Given a market (S, G, P), to accept market efficiency with respect to G, by Theorem 80, one must show that there exists an equivalent martingale measure Q ∈ M(G). To do this, a two step procedure can be used. Step one in this procedure is to empirically validate a risky asset price process S using historical time series data. Such procedures are commonly invoked when pricing derivatives in Part I of this book. Step two in this procedure is to determine if there exists an equivalent martingale measure for the validated price process S with respect to the filtration G. We note that this is the identical issue studied in Chap. 3 when testing for asset price bubbles. For example, if one shows that the validated price process S can be represented by a geometric Brownian motion as in the BSM model, then as shown in Chap. 5, there exists a Q ∈ M(G). Hence, the market is efficient with respect to G = σ (S). Other examples explored in Chap. 3 include the simple diffusion process in Example 7, the

16.6 Random Walks and Efficiency

339

CEV process in Example 8, and Levy processes as in Theorem 29. As of yet, this method for testing and accepting market efficiency is unexplored in the empirical asset pricing literature.

16.6 Random Walks and Efficiency There is some confusion in the literature regarding the relation between market efficiency and risky asset prices following a random walk. The fact is that there is no relation between these two concepts, i.e. (1) an efficient market does not imply that stock price returns follow a random walk, and (2) stock price returns following a random walk does not imply the market is efficient.

16.6.1 The Set-Up Consider the returns on the risky assets. Partition [0, T ] into a collection of subintervals of length Δ > 0. Fix a time interval [t, t + Δ] ⊂ [0, T ] where t ≥ 0 aligns with one of these partitions. To simplify the methodology we assume that the risky assets contain default-free zero-coupon bonds paying $1 at times t = Δ, . . . , T . 1 Let r0 (t) = p(t,t+Δ) − 1 be the return on a default-free zero-coupon bond that matures at time t + Δ. i (t) Let Ri (t) = Si (t+Δ)−S be the return on the ith risky asset. Si (t)

16.6.2 Random Walk For this discussion, let us first consider the filtered probability space (Ω, G, F , P) with G = (Gt )t∈[0,T ] where GT = F . The risky asset prices, or more formally the risky asset price returns are said to follow a random walk if Ri (t) are measurable with respect to Gt+Δ , (Ri (t), Ri (t + Δ)) are independent with respect to Gt and identically distributed for all t ∈ [0, Δ, 2Δ, . . . , T − Δ] and all i = 1, . . . , n. For empirical testing this implies that an asset’s returns have zero autocorrelation, i.e. cov [Ri (t + Δ), Ri (t) |Gt ] = 0 for all t ∈ [0, Δ, 2Δ, . . . , T − Δ] and i = 1, . . . , n.

(16.2)

340

16 Market Informational Efficiency

Proof First, by the independence of Gt , cov [Ri (t + Δ), Ri (t) |Gt ] = cov [Ri (t + Δ), Ri (t)] = E [Ri (t + Δ)Ri (t)] − E [Ri (t + Δ)] E [Ri (t)]. Since Ri (t) is Gt+Δ -measurable, Ri (t + Δ) is independent of Ri (t), i.e. P ( Ri (t + Δ)| Ri (t)) = P (Ri (t + Δ)). This implies E [Ri (t + Δ)Ri (t)] = E [Ri (t + Δ)] E [Ri (t)]. Substitution gives cov [Ri (t + Δ), Ri (t) |Gt ] = 0, which completes the proof. Remark 92 (Probability Theory and Random Walks) In probability theory, a random walk is a sequence of random variables Y1 , Y2 , . . . , Yk , . . . with Yk = kt=0 Xt such that X0 is a constant and X1 , X2 , . . . are independent and identically distributed random variables with E |Xt | < ∞ for all t, see Ross [166, p. 165]. To see the connection  to risky asset prices, make the identification Si (t+Δ) Xt = log Si (t) = log (1 + Ri (t)). Then, given Ri (t) is independent of Gt , Ri (t) being Gt+Δ -measurable implies that Ri (t + Δ) is independent of Ri (t), i.e. X1 , X2 , . . . are independent. And, Ri (t) being identically distributed k implies X1 , X2 , . .. are identically distributed. Hence, Yk = t=0 Xt =   k Si (t+Δ) Si (k+Δ) = log , the exponent of S log (k + Δ) = Si (0)eYk , i t=0 Si (t) Si (0) follows a random walk. This completes the remark.

16.6.3 Market Efficiency  Random Walk To see what an efficient market with respect to G ⊂ F implies, we need the following math identities. Lemma 33 (Excess Expected Return Math Identities) Let σ (R) ⊂ G = (Gt )t∈[0,T ] . Define the excess returns for risky asset i given G by εi (t) = Ri (t) − E[Ri (t) |Gt ]. Then, (1) E [εi (t) |Gt ] = 0, cov [εi (t + Δ), εi (t) |Gt ] = 0 f or all t, and (2) cov [Ri (t + Δ), Ri (t) |Gt ] = 0 if and only if cov [E [Ri (t + Δ) |Gt+Δ ] , εi (t) |Gt ] = 0. Proof (Step 1) E [εi (t) |Gt ] = E [Ri (t) − E[Ri (t) |Gt ] |Gt ] = E [Ri (t) |Gt ] − E[Ri (t) |Gt ] = 0. And, cov [εi (t + Δ), εi (t) |Gt ] = E [εi (t + Δ)ε(t) |Gt ] = E [E [εi (t + Δ)εi (t) |Gt+Δ ] |Gt ] = E [εi (t)E [εi (t + Δ) |Gt+Δ ] |Gt ] = 0 since E [εi (t + Δ) |Gt+Δ ] = 0. (Step 2) cov [Ri (t + Δ), Ri (t) |Gt ]

16.6 Random Walks and Efficiency

341

= cov [E [Ri (t + Δ) |Gt+Δ ] + εi (t + Δ), E [Ri (t) |Gt ] + εi (t) |Gt ] = cov [E [Ri (t + Δ) |Gt+Δ ] + εi (t + Δ), εi (t) |Gt ] = cov [E [Ri (t + Δ) |Gt+Δ ] , εi (t) |Gt ] + cov [εi (t + Δ), εi (t) |Gt ] = cov [E [Ri (t + Δ) |Gt+Δ ] , εi (t) |Gt ]. This completes the proof. This lemma shows that the excess returns to any risky asset always have zero mean and are uncorrelated across time. These facts have nothing to do with market efficiency. Furthermore, the returns themselves have zero autocorrelation if and only if the correlation between E [Ri (t + Δ) |Gt+Δ ] and εi (t) is zero for all t ∈ [0, Δ, 2Δ, . . . , T − Δ]. Given this lemma, we can now investigate the implications of an efficient market. First, by Theorem 80, the market being efficient with respect to G is equivalent to the existence of an equivalent martingale measure Q ∈ M(G). The only restriction on risky asset returns imposed by this condition is that the risky asset returns (after adjusting for the mma) have zero expected values and zero autocorrelation under Q ∈ M(G). Indeed, to see this note that for all t ∈ [0, Δ, 2Δ, . . . , T − Δ] and i = 1, . . . , n, the martingale condition implies that E

Q



Si (t + Δ) |Gt B(t + Δ)

 =

Si (t) . B(t)

For returns, this implies E Q [Ri (t) |Gt ] = 0 where Ri (t) =

Si (t+Δ) Si (t) B(t+Δ) − B(t) Si (t) B(t)

=

[1 + Ri (t)] −1 [1 + r0 (t)]

is the return on the risky asset, after adjusting for the mma’s return. Then, defining ηi (t) = Ri (t) − E Q [Ri (t) |Gt ] = Ri (t), lemma (condition (1)) implies cov Q [ηi (t + Δ), ηi (t) |Gt ] = cov Q [Ri (t + Δ), Ri (t) |Gt ] = 0. This is the totality of what an efficient market with respect to G implies. It does not imply that the Ri (t) are independent nor identically distributed random variables under Q. It only implies that the returns (after adjusting for the mma) are uncorrelated under Q. But, perhaps more importantly, under the statistical probability P, E [Ri (t) |Gt ] will in general be non-zero, and Ri (t) need not be independent nor identically

342

16 Market Informational Efficiency

distributed. Furthermore, condition (2) of the lemma shows that the correlation P of the returns across time need not be zero either. Zero autocorrelation follows if and only if the correlation between E [Ri (t + Δ) |Gt+Δ ] and εi (t) is zero for all t. Most likely, quite the contrary is true. It would be quite natural for a large excess return at time t to imply that time t + Δ expected returns, E [Ri (t + Δ) |Gt+Δ ], increase. Hence, market efficiency with respect to G does not imply risky asset prices follow a random walk.

16.6.4 Random Walk  Market Efficiency Conversely, if asset returns follow a random walk under the probability measure P, this does not imply that the market is efficient with respect to a given G ⊂ F. The reason is that to be efficient, by Theorem 80, there must exist a martingale measure Q ∈ M(G) for the assets and this may not be the case, as the next example shows. Example 18 (Random Walk in an Inefficient Market) This example gives a market where the risky asset returns follow a random walk, but the market is inefficient. Consider a market (S, F, P) consisting of two risky assets whose price processes follow geometric Brownian motion, i.e. dSi (t) = Si (t) (μi dt + σ dWt ) or μi t− 21 σ 2 t+σ Wt Si (t) = e for i = 1, 2

(16.3)

for all t ∈ [0, T ] where Si (0) = Si (0) = 1, μ1 , μ2 , σ are strictly positive constants, μ1 > μ2 , and Wt is a standard Brownian motion under P with W0 = 0. Let the information set F = σ (S) = σ (W ) be the smallest information set possible for this evolution. First, we show that these risky assets follow a random walk. Indeed, Ri (t) =

Si (t+Δ)−Si (t) Si (t)

=e

=

e

μi (t+Δ)− 21 σ 2 (t+Δ)+σ Wt+Δ

μ t− 1 σ 2 t+σ Wt e i 2 μi Δ− 12 σ 2 Δ+σ (Wt+Δ −Wt )

−1

(16.4)

− 1.

Note that Ri (t), Ri (t + Δ) are independent and identically distributed since the Brownian motion increments (Wt+Δ − Wt ) and (Wt+2Δ − Wt+Δ ) are independent and identically distributed. Second, this market is inefficient with respect to F because there exists an NFLVR(F) (apply Theorem 80 above). Indeed, to see this consider the zero investment, buy and hold (self-financing) trading strategy consisting of 1 unit long S1 (t) and 1 unit short S2 (t) initiated at time 0 and held until time T , i.e. (α0 (t) = 0, α1 (t) = 1, α2 (t) = −1) for all t ∈ [0, T ].

16.7 Notes

343

The value process for this s.f.t.s. is Xt = α1 (t)S1 (t) + α2 (t)S2 (t) = S1 (t) − S2 (t). Note that

1 2 X0 = 0 and XT = eμ1 t − eμ2 t e− 2 σ t+σ Wt > 0 a.s. P, which is a simple arbitrage opportunity (see Chap. 2, Definition 25). This completes the example.

16.7 Notes The definition of market efficiency as contained in this chapter is new to the literature. The use of this formal definition to explore alternative tests of market efficiency is a fruitful area for future research. The extension of Theorem 80 to an economy with transaction costs is contained in Jarrow and Larsson [101, 103].

Chapter 17

Epilogue (The Fundamental Theorems and the CAPM)

This chapter studies the static fundamental theorems of asset pricing and the CAPM for two reasons. First, because they are of historical interest. Second, because they highlight the advances and insights obtained from the dynamic models studied in this book. This chapter provides a new derivation of the static CAPM using the martingale approach.

17.1 The Fundamental Theorems This is a discrete time model with two times t ∈ {0, 1}. Trading takes place at time 0 and all trades are liquidated at time 1. We are given a complete probability space (Ω, F , P). Given in a normalized market (S, F , P) with Bt = 1 for t ∈ {0, 1}. To simplify the notation, we let the time 0 price of risky asset prices be denoted Sj (0) = sj > 0 and the time 1 prices be Sj (1) = Sj ≥ 0 for j = 1, . . . , n. Returns S over [0, 1] are defined by Rj = sjj −1 for j = 1, . . . , n. As before, one can interpret the time 1 asset price as a liquidating dividend. We will also consider the non-normalized market below in which case the

notation is sj , Sj for the risky assets j = 1, . . . , n and (1, B) = (1, 1 + r) for the money market account where r is the default-free spot rate of interest. Note that since the money market account’s value is unity at time 0, the normalized and nonnormalized risky asset prices are identical at time 0. In the non-normalized market, S the returns on the risky assets are defined by R˜j = sjj − 1 for j = 1, . . . , n. Algebra yields the following result, which will prove useful below. Rj =

R˜ j − r 1+r

for j = 1, . . . , n.

© Springer Nature Switzerland AG 2021 R. A. Jarrow, Continuous-Time Asset Pricing Theory, Springer Finance, https://doi.org/10.1007/978-3-030-74410-6_17

(17.1)

345

346

17 Epilogue (The Fundamental Theorems and the CAPM) S

S (1+r)

j j −(1+r) s −1−r sj R˜ j −r Proof 1+r = j 1+r = = Rj . 1+r since Sj = Sj (1 + r). This completes the proof.

The second equality follows

We add the following assumption on the risky assets.

  Assumption (Non-redundant Assets) E[Sj2 ] < ∞, var Sj > 0 for all j = 1, . . . , n, and ⎛

cov[S1 , S1 ] ⎜ ⎜ cov[S , S ] 2 1 ⎜ ⎜ .. ⎜ ⎜ . ⎝

cov[S1 , S2 ] cov[S2 , S2 ] .. .

⎞

· · · cov[S1 , Sn ] ⎟ · · · cov[S2 , Sn ] ⎟ ⎟ ⎟ .. ⎟ .. ⎟ . . ⎠ cov[Sn , S1 ] cov[Sn , S2 ] · · · cov[Sn , Sn ] n×n

is nonsingular. In this assumption, E[Sj2 ] < ∞ implies that E[Sj ] < ∞ for all j (see Ash [3, p. 226]) and by the Cauchy–Schwartz inequality (see Ash [3, p. 82]) that E[Si Sj ] < ∞ for all i, j . Hence, cov[Si , Sj ] = E[Si Sj ] − E[Si ]E[Sj ] 0 for all j ) means that the assets are indeed “risky” and none are equivalent to the mma. The non-singularity of the covariance matrix implies that the risky assets are nonredundant as defined in Sect. 2.4 of Chapter 2. The next lemma follows from this assumption. Lemma 34 Under the non-redundant assets assumption, (1) dim [span(1, S1 , . . . , Sn )] = n + 1, i.e. the asset prices (considered as random variables Si (ω)) are linearly independent, and (2) |Ω| ≥ n + 1, i.e. the cardinality of the set of states is greater than or equal to the number of traded securities. Proof (Step 1) Show (1, S1 , . . . , Sn ) are linearly independent. Let (γ0 , γ1 , . . . , γn ) ∈ Rn+1 be such that γ0 +

n  j =1

γj Sj = 0.

(17.2)

17.1 The Fundamental Theorems

347

We will show that (γ0 , γ1 , . . . , γn ) = 0, which gives the desired result. First, taking the covariance of expression (17.2) with respect to S1 , then S2 ,. . . , then Sn gives the system of equations n

j =1 γj cov[S1 , Sj ]

n

=0

j =1 γj cov[S2 , Sj ]

=0 .. . n γ cov[S j n , Sj ] = 0 j =1 In matrix form ⎛

cov[S1 , S1 ] cov[S1 , S2 ] ⎜ cov[S2 , S1 ] cov[S2 , S2 ] ⎜ ⎜ .. .. ⎝ . . cov[Sn , S1 ] cov[Sn , S2 ]

⎞⎛ ⎞ ⎛ ⎞ γ1 0 · · · cov[S1 , Sn ] ⎜ γ2 ⎟ ⎜ 0 ⎟ · · · cov[S2 , Sn ] ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎟⎜ . ⎟ = ⎜ . ⎟. .. .. ⎠ ⎝ .. ⎠ ⎝ .. ⎠ . . · · · cov[Sn , Sn ] 0 γn

The nonsingularity of the covariance matrix implies that (γ1 , . . . , γn ) = 0. Second, substitution of (γ1 , . . . , γn ) = 0 into expression (17.2) gives γ0 = 0, which completes the proof of (Step 1). (Step 2) If |Ω| = ∞, then the result follows. If |Ω| = m < ∞, then write Ω = {1, 2, . . . , m}. Consider the matrix ⎛

⎞

1 ··· 1 ⎟ S1 (2) · · · S1 (m) ⎟ ⎟ .. . . .. ⎟ ⎟ . . ⎟ . ⎠ Sn (1) Sn (2) · · · Sn (m) (n + 1) × m 1

⎜ ⎜ S (1) ⎜ 1 ⎜ ⎜ .. ⎜ . ⎝

By (Step 1) the row rank of this matrix is (n + 1). Since the row rank and column rank of a matrix are equal (see Theil [183, p. 11]), m ≥ n + 1. This completes the proof of (Step 2) and the Lemma. A self-financing trading strategy (s.f.t.s.) in a static model is just a buy and hold trading strategy (α0 , α) = (α0 , α1 , . . . , αn ) ∈ Rn+1 with time 0 and 1 values x = α0 +

n 

αj sj

and

j =1

X = α0 +

n  j =1

αj Sj .

348

17 Epilogue (The Fundamental Theorems and the CAPM)

The trading strategy is self-financing because there is no cash inflow or outflow until the assets are liquidated at time 1. Remark 93 (Admissibility) In this setting, we do not need an admissibility condition because no doubling trading strategies are possible. This is because one cannot trade more than once in this static market. This completes the remark.

17.1.1 The First Fundamental Theorem In this static model, the First Fundamental Theorem 13 of asset pricing in Chap. 2 simplifies. Before stating the simplified theorem, we need to repeat some definitions. Definition 48 (No Arbitrage (NA)) A trading strategy (α0 , α) ∈ Rn+1 with initial wealth x and time 1 value X is a (simple) arbitrage opportunity if (i) x = 0, (zero investment) (ii) X ≥ 0 with P probability one, and (iii) P (X > 0) > 0.



In the context of a static model, we say that a risky asset price sj , Sj is a Q   martingale if sj = E Q Sj . Given these definitions, we can now state the theorem. Theorem 81 (The First Fundamental Theorem) NA if and only if M = ∅ where M = {Q ∼ P : S is a Q martingale}, i.e. there exists an equivalent martingale probability Q. Proof This proof is based on Follmer and Schied [63, Theorem 1.6, p. 7]. An arbitrage is (α0 , α) ∈ Rn+1 such that α0 + α · s = 0, α0 + α · S ≥ 0, P (α0 + α · S > 0) > 0. Define Zi = Si − si ≥ 0 for all i where Z = (Z1 , . . . , Zn ) ∈ Rn . In this case, the mma’s difference in value becomes identically zero. Equivalently, an arbitrage is α ∈ Rn such that α · Z ≥ 0, P (α · Z > 0) > 0. Then, NA is equivalent to: for all α ∈ Rn , α · Z ≥ 0 implies α · Z = 0. Since Zi ≥ −si , under any probability measure Q, E Q [Zi ] is well defined for all i, although it may be +∞. Then, Q ∈ M if and only if E Q [Zi ] = 0 for all i, i.e. E Q [Z] = 0. (Step 1) To show there exists a Q ∼ P such that E Q [Zi ] = 0 for all i implies NA. Suppose NA is violated, i.e. there exists a α ∈ Rn such that α · Z ≥ 0 and P (α · Z > 0) > 0. Because Q ∼ P, Q (α · Z > 0) > 0. Hence E Q [α · Z] > 0. But, E Q [α · Z] = α · E Q [Z] = 0 by assumption, which is a contradiction. (Step 2) To show NA implies that there exists a Q ∼ P such that E Q [Zi ] = 0 for all i.

17.1 The Fundamental Theorems

349

Let P denote the set of all probability measures Q equivalent to P such that E Q [Zi ] < ∞ for all i. Note that P is  a convex set. Theset is nonempty since P ∈ P. Define Z = E Q [Z] : Q ∈ P ∈ Rn since Z ∈ Rn . Note that Z is a convex set in Rn . To see this, choose E Q1 [Z] , E Q1 [Z] ∈ Z and a ∈ [0, 1]. Then, aE Q1 [Z] + (1 − a)E Q1 [Z] = E aQ1 +(1−a)Q2 [Z] ∈ Z , which proves convexity. We claim 0 ∈ Z , which proves that there exists a Q ∼ P such that E Q [Z] = 0. To prove this, suppose not, i.e. 0 ∈ / Z. Then, by a separating hyperplane theorem, we obtain a vector α ∈ Rn such that α · x ≥ 0 for all x ∈ Z and α · x0 > 0 for some x0 ∈ Z (see Follmer and Schied [63, Proposition A1, p. 399.]). Written out, there exists a Q0 ∈ P such that α · E Q0 [Z] = E Q0 [α · Z] > 0, which implies Q0 (α · Z > 0) > 0. We claim that α · Z ≥ 0 and α · Z = 0 , which implies that α is an arbitrage opportunity, yielding the contradiction. Given Q0 (α · Z > 0) > 0, we only need to prove that α · Z ≥ 0. Let A = {ω ∈ Ω: α · Z < 0}. Define fn (ω) = 1 − n1 1A (ω) + n1 1Ac (ω) > 0, which is F measurable and dQn dP (ω)

=

fn (ω) E[fn ]

> 0 for n = 2, 3, . . ., which are densities for probability

measures Qn ∼ P .   fn (ω) ≤ Given that Zi fn (ω) ≤ Zi because 0 < fn ≤ 1, 0 ≤ E Qn [Zi ] = E Zi E[f n] E[Zi ] E[fn ]

< ∞, which implies Qn ∈ P.

By the dominated convergence theorem,   lim E [α · Z fn (ω)] = E lim α · Z fn (ω) = E [α · Z 1A ] ≥ 0.

n→∞

n→∞

By the definition of A, α · Z 1A ≤ 0, which implies P (α · Z < 0) = 0, i.e. α · Z ≥ 0. This proves the claim, which completes the proof. Remark 94 (NFLVR and Bubbles) In this static model, the simpler NA is used and NFLVR is not needed. There are no local martingale measures in this single period setting, hence there is no analogue for the Third Fundamental Theorem 16 of asset pricing in Chap. 2 and no asset price bubbles can exist (see Chap. 3). Given there is substantial empirical evidence that price bubbles exist, this is a limitation of the static model, not shared by the dynamic model. This completes the remark.

17.1.2 The Second Fundamental Theorem A derivative is any nonnegative random payoff at time 1, X ∈ L0+ . As before, it is easy to value a derivative in a complete market. The definition of market completeness simplifies to the following in a static model.

350

17 Epilogue (The Fundamental Theorems and the CAPM)

Definition 49 (Complete Market with Respect to Q ∈ Ml ) Given NA, i.e. M = ∅. Choose a Q ∈ M. The market is complete with respect to Q if given any X ∈ L1+ (Q), there exists a x ≥ 0 and (α0 , α) ∈ Rn+1 such that x = α0 + nj=1 αj sj and α0 +

n  j =1

αj Sj = x +

n 

αj [Sj − sj ] = X.

j =1

Note that in the static model, the fact that Sj are Q martingales for all j implies that X is a Q martingale. Hence, we do not have to include this restriction in the definition itself, as was necessary in the continuous-time setting of Chap. 2. Given this definition, the second fundamental theorem of asset pricing can now be stated. Theorem 82 (The Second Fundamental Theorem) Assume NA, i.e. M = ∅. The market is complete with respect to Q ∈ M if and only if M is a singleton, i.e. the equivalent martingale measure is unique. This is the same as Theorem 3 in Chap. 2. The following proof is based on Follmer and Schied [63, Theorem 1.40, p. 24]. Proof (Completeness ⇒ M a singleton) Consider two measures Q1 , Q2 ∈ M. Choose an arbitrary A ∈ F . Consider Z = 1A ≥ 0. Since the market is complete with respect to both Q1 , Q2 ∈ M, there exists i x > 0, α i ∈ Rn such that x i = α i · s and α i · S = Z for i = 1, 2. We claim that x 1 = x 2 . We prove this by contradiction. Suppose, without loss of generality, that x 1 > x 2 at time 0. Then, NA is violated, which contradicts the hypothesis of the theorem. To see that NA is violated, consider the following trading strategy: short x 1 > 0 and α 1 ∈ Rn , go long short x 2 > 0 and α 2 ∈ Rn , and invest x 1 − x 2 > 0 in a buy and hold position in the m.m.a. The initial value of this trading strategy is 0 and its time 1 value is x 1 − x 2 > 0 with probability one. This is a arbitrage opportunity, which proves the claim. Hence, we have 1 1 x 1 = E Q [Z] = E Q [1A ] = Q1 (A), 2 2 x 2 = E Q [Z] = E Q [1A ] = Q2 (A), and x 1 = x 2 , which yields Q1 (A) = Q2 (A). Since this is true for all A ∈ F , we have Q1 = Q2 . Hence, M is a singleton. (M a singleton ⇒ Completeness ) We need two lemmas. Lemma 1 Given a probability space (Ω, F , P), X, Y ∈ L0+ . The following are equivalent. (i) X(ω) ≤ Y (ω) a.s. for all Q ∼ P where E Q (Y ) = y. (ii) E Q (X) ≤ E Q (Y ) for all Q ∼ P where E Q (Y ) = y.

17.1 The Fundamental Theorems

351

(iii) E Q (X) ≤ y for all Q ∼ P where E Q (Y ) = y. (iv) sup E Q (X) ≤ y for all Q ∼ P where E Q (Y ) = y. Q∈M

Proof of Lemma 1 The equivalence of (ii)–(iv) is obvious. (i) implies (ii) follows by taking expectations. Show (ii) implies (i). Assume not (i). Define A1 , A2 , B1 , B2 ∈ F by A1 = {ω ∈ Ω : Y > y}, A2 = {ω ∈ Ω : y ≤ Y }, B1 = {ω ∈ Ω : X > Y }, B2 = {ω ∈ Ω : X ≤ Y }. Note that A1 ∪ A2 = Ω and B1 ∪ B2 = Ω. We have Q(A1 ∩ B1 ) + Q(A1 ∩ B2 ) + Q(A2 ∩ B1 ) + Q(A2 ∩ B2 ) = 1. Consider a probability measure Q ∼ P such that the distribution of Y is symmetric about y. Then, Q is such that E Q (Y ) = y and Q(A1 ) = Q(A2 ) = 1/2. Thus, Q(A1 ∩ B1 ) + Q(A1 ∩ B2 ) = 1/2 and Q(A2 ∩ B1 ) + Q(A2 ∩ B2 ) = 1/2. The choice of these probabilities Q(A1 ∩B1 ), Q(A1 ∩B2 ), Q(A2 ∩B1 ), Q(A2 ∩B2 ) are arbitrary, except for the two constraints. Choose one of these symmetric Q ∼ P where Q(A1 ∩B2 ) = Q(A2 ∩ B2 ) = 0. This implies Q(B2 ) = 0, i.e. Y < X a.s. Hence, y = E Q [Y ] < E Q [X], which is the contradiction. This completes the proof of Lemma 1. Note that if X > Y a.s. for all Q ∼ P where E Q (Y ) = y, then applying the above lemma to −X < −Y and using −sup{−r} = inf {r} for r ∈ R gives E Q (X) ≥ y for all Q ∼ P where E Q (Y ) = y. Lemma 2 Assume NA. Define C1 (x) =   Z ∈ L0+ : ∃x7> 0, α0 , α ∈ Rn+1 , x = α0 + < α · s, α0 + α · S ≥ Z , and C2 (x) = Z ∈ L0+ : sup E Q [Z] ≤ x . Then C1 (x) = C2 (x). Q∈M

Proof of Lemma 2 Show C1 (x) ⊂ C2 (x). Given Z ∈ C1 (x), ∃x > 0, α0 , α ∈ Rn+1 , x = α0 + α · s, α0 + α · S ≥ Z. Take expectations with Q ∈ M, then α0 + E Q [α · S] = α0 + α · s = x ≥ E Q [Z]. This is true for all Q ∈ M, hence, Z ∈ C2 (x). Show C2 (x) ⊂ C1 (x). Given Z ∈ C2 (x), then sup E Q [Z] ≤ x where x = α0 + α · s for α0 , α ∈ Rn+1 . Q∈M

Note that Q ∈ M is equivalent to the statement: Q ∼ P where E Q (α0 + α · S) = α0 + α · s. By Lemma 1, α0 + α · S ≥ Z where x = α0 + α · s for α0 , α ∈ Rn+1 . That is, Z ∈ C1 (x). This completes the proof of Lemma 2. 0 Define the  super-replication price. Given Z ∈ L+ ,  c¯ = inf x > 0 : ∃α0 , α ∈ Rn+1 , x = α0 + α · s, α0 + α · S ≥ Z = inf {x ∈ R : Z ∈ C1 (x)} = inf {x ∈ R : Z ∈ C2 (x)}

352

17 Epilogue (The Fundamental Theorems and the CAPM)

= inf {x ∈ R : sup E Q [Z] ≤ x} = sup E Q [Z]. Q∈M

Q∈M

The third equality follows by Lemma 2. Define the Given Z ∈ L0+ ,  sub-replication price.  n+1 c = sup x > 0 : ∃α0 , α ∈ R , x = α0 + α · s, α0 + α · S ≤ Z   = −inf −x > 0 : ∃α0 , α ∈ Rn+1 , −x = α0 + α · s, −α0 − α · S ≥ −Z   = −inf y > 0 : ∃α˜ 0 , α˜ ∈ Rn+1 , y = α˜ 0 + α˜ · s, α˜ 0 + α˜ · S ≥ −Z = − sup E Q [−Z] Q∈M

= inf E Q [Z]. Q∈M

M a singleton implies c¯ = c. Given Z ∈ C1 (c), ¯ Z ∈ C2 (c) ¯ by Lemma 2. Thus, ∃α0 , α ∈ Rn+1 such that Q Q E [Z] ≤ c¯ where c¯ = E [α0 + α · S] for all Q ∈ M. By Lemma 1, this implies α0 + α · S ≥ Z. Using c¯ = c, we have by a symmetric argument applied to Z using the infimum that ∃α0 , α ∈ Rn+1 such that E Q [Z] ≥ c = c¯ where c¯ = E Q [α0 + α · S] for all Q ∈ M. By Lemma 1 again, this implies α0 + α · S ≤ Z. Combined, this yields α0 + α · S = Z, which proves the market is complete.

17.1.3 Risk Neutral Valuation If the market is complete, the arbitrage free value of the derivative is given using risk neutral valuation. This is the analogue of Theorem 17 in Chap. 2. Theorem 83 (Risk Neutral Valuation) Assume NA, i.e. M = ∅. Let the market be complete with respect to Q ∈ M. 1 n+1 such that Given any n X ∈ L+ (Q), there exists a x ≥ 0 and (α0 , α) ∈ R x = α0 + j =1 αj sj and α0 +

n  j =1

αj Sj = x +

n 

αj [Sj − sj ] = X.

j =1

The arbitrage-free value of the derivative is x = E Q [X]. If the market is incomplete, then the arbitrage-free prices lie between the super- and sub-replication prices, defined as follows. Given X ∈ L1+ (Q) for all Q ∈ M, the super-replication price is   c¯ = inf x > 0 : ∃α0 , α ∈ Rn+1 , x = α0 + α · s, α0 + α · S ≥ X ,

17.1 The Fundamental Theorems

353

and the sub-replication price is   c = sup x > 0 : ∃α0 , α ∈ Rn+1 , x = α0 + α · s, α0 + α · S ≤ X . We have the following theorem. This is the analogue of Theorems 43 and 44 in Chap. 8. The proof is contained in the proof of Theorem 82 above. Theorem 84 (Super- and Sub-replication) Assume NA, i.e. M = ∅. Given X ∈ L1+ (Q) for all Q ∈ M, the arbitrage free price range for the derivative X is '

& Q

Q

c = inf E [X], c¯ = sup E [X] . Q∈M

Q∈M

17.1.4 Finite State Space Market We now characterize conditions on a finite state space market for the first and second fundamental theorems of asset pricing to hold. Let |Ω| = m < ∞ and write Ω = {1, 2, . . . , m}. Consider the payoff matrix ⎛

⎞

1 1 ··· 1 ⎜ ⎟ ⎜ S (1) S (2) · · · S (m) ⎟ 1 1 ⎜ 1 ⎟ ⎜ .. . . .. ⎟ ⎜ .. ⎟ ⎜ . . . ⎟ . ⎝ ⎠ Sn (1) Sn (2) · · · Sn (m) (n + 1) × m where the first row corresponds to the payoffs to the mma and the last n rows correspond to the payoffs to the risky assets. By Lemma 34, we have that m ≥ n+1. Using the First Fundamental Theorem 81 of asset pricing, we have that this market satisfies NA if and only if there exists a (q1 , q2 , . . . , qm ) ∈ Rm ++ such that ⎛

⎞⎛

⎞

⎛

⎞

··· 1 1 q1 ⎟ ⎜ ⎟⎜ ⎟ ⎜s ⎟ ⎜q ⎟ · · · S1 (m) ⎟ ⎜ 1⎟ ⎟⎜ 2 ⎟ ⎟ ⎜ ⎜ ⎟ . ⎟ ⎟⎜ .. ⎟ =⎜ .. ⎟ .. ⎜ . ⎟ ⎜ ⎟ . .. ⎟ ⎠ ⎝ ⎠⎝ . ⎠ sn Sn (1) Sn (2) · · · Sn (m) qm (n + 1) × m m × 1 (n + 1) × 1 1

⎜ ⎜ S (1) ⎜ 1 ⎜ ⎜ .. ⎜ . ⎝

1 S1 (2) .. .

m where Rm ++ denotes the strictly positive subspace of R . The first equation in this matrix equation makes (q1 , q2 , . . . , qm ) probabilities. The remaining n equations are the martingale conditions, i.e. E q (Si ) = si for all i = 1, . . . , n where E q (·) is expectation with respect to (q1 , q2 , . . . , qm ) ∈ Rm ++ . Whether or not a solution exists to this matrix equation depends on the elements within the payoff matrix. For

354

17 Epilogue (The Fundamental Theorems and the CAPM)

example, if n = 1 and m = 2, called a binomial model, a solution exists to this matrix equation if and only if S1 (1) > 1 > S1 (2), see Jarrow and Chatterjea [97]. Using the Second Fundamental Theorem 82 of asset pricing, assuming such a solution exists (i.e. NA holds), the market is complete if and only if n + 1 = m. This follows because the probabilities (q1 , q2 , . . . , qm ) will be unique if and only if the matrix is invertible. We note that this last statement also proves the following stronger result. If there are a finite of securities trading in a static model, then if m = |Ω| > n + 1, the market is incomplete. This applies when the state space is uncountably or countably infinite |Ω| = ∞. Hence, in a static model, unless the state space is finite and equal to the number of assets trading, the market is incomplete. Recall that we have assumed the risky assets are non-redundant so that m ≥ n + 1. We next provide an example of an incomplete market where the state space is uncountably infinite. Example 19 (Lognormally Distributed Return Market) This is an example where the state space |Ω| = ∞ (uncountably infinite) and the market is incomplete. Consider a market consisting of the mma and only one traded risky asset with time 0 and 1 prices denoted s and S, respectively. We assume that the risky asset’s payoffs are S = sez where z is normally distributed (E [z] = μ, V ar [z] = σ 2 > 0) under P. Note that, using the moment generating function of a normal random variable (see Mood et al. [150, p. 541]), we have 1

2

E [S] = seμ+ 2 σ . Consider the random variable Y = ey > 1, where y is normally distributed (E [y] = − 12 ν 2 , V ar [y] = ν 2 > 0) under P with √VCov(z,y) = ρ. Then, ar[y]V ar[z] 1 2 1 2 +2ν

E [Y ] = E [ey ] = e− 2 ν

= 1. Define the equivalent change of measures dQ(ν,ρ) = Y (ν,ρ) > 0 dP

where the superscript indicates the parameters of y’s joint normal distribution with z. We note that z + y is normally distributed (E [z + y] = μ − 12 ν 2 , V ar [z + y] = σ 2 + 2ρσ ν + ν 2 ). Using these equivalent measures we have that EQ

(ν,ρ)

    1 2 [S] = E SY (ν,ρ) = E sez+y = seμ+ 2 σ +ρσ ν .

Using the First Fundamental Theorem 81 of asset pricing, this economy satisfies NA because Y (ν,ρ) is a martingale deflator when ρν = − σ1 (μ+ 12 σ 2 ), i.e. E SY (ν,ρ) = s.

17.2 Basis Assets, Multi-Factor Beta Models, and Systematic Risk

355

By the Second Fundamental Theorem 82 of asset pricing, the economy is incomplete since there are many (ρ, ν) pairs such that this is true, hence the equivalent martingale measure is not unique. This completes the example. Remark 95 (Third Fundamental Theorem of Asset Pricing) In a static model, there is no analogue of the third fundamental theorem of asset pricing (see Theorem 16 in Chap. 2) because NA and ND are equivalent. This follows because there is no admissibility condition. If a risky asset’s time 0 and 1 prices are dominated by some traded portfolio, then shorting the more expensive and buying the cheaper creates an arbitrage opportunity. This completes the remark.

17.2 Basis Assets, Multi-Factor Beta Models, and Systematic Risk This section studies the systematic risk return relation derived only using the assumption of NA. In the finite dimensional static model, the mma and the n risky assets form a collection of basis assets, as defined in Chap. 4. This follows from Lemma 34 above. Indeed, (1, S1 , . . . , Sn ) are a set of basis assets because for the traded portfolios X = span(1, S1 , . . . , Sn ). This follows due to the non-redundant risky asset assumption. Without the non-redundant asset assumption, the set of traded assets would exceed the number of basis assets. The return on any s.f.t.s. RX = Xx − 1 can therefore be written as a linear combination of the returns on these basis assets as shown in the following theorem. This is the analogue of Theorem 36 in Chap. 4. Theorem 85 (Multiple-Factor Model) RX = β0 · 0 +

n 

βj Rj

(17.3)

j =1

α s βj = jx j for j = 1, . . . , n, and β0 + nj=1 βj = 1. α α s S Proof Xx = αx0 + nj=1 xj Sj = αx0 + nj=1 jx j sjj . α0 + nj=1 αj sj α s S α s S . Then Xx − 1 = αx0 + nj=1 jx j sjj − 1 = αx0 + nj=1 jx j sjj − x Algebra yields n αj sj Sj n αj sj α0 α0 X j =1 x sj − j =1 x . x −1= x − x + Algebra completes the proof.

where β0 =

α0 x ,

356

17 Epilogue (The Fundamental Theorems and the CAPM)

Remark 96 (Non-normalized Market) In the non-normalized market, using expression (17.1) and noting that the same expression applies to a portfolio, substitution yields   n R˜ j −r R˜ X −r = β j =1 j 1+r 1+r . Or,   R˜ X − r = nj=1 βj R˜ j − r , or   R˜ X = r 1 − nj=1 βj + nj=1 βj R˜ j . Noting that β0 = 1 − nj=1 βj yields the final result R˜ X = β0 r +

n 

βj R˜ j .

j =1

This completes the remark. To understand which basis assets in Theorem 85 are risk factors, i.e. have non-zero excess expected returns, we need to add the following assumption. Assumption (NA)

M = ∅

dQ dP

Let Y = ∈ M be a fixed martingale deflator, which is also called the state price density (see Chap. 2). We now prove the analogue of Theorem 38 in Chap. 4. Theorem 86 (The Risk Return Relation) Assume NA. A traded portfolio’s X ∈ X expected return satisfies E[RX ] = −cov [RX , Y ] .

(17.4)

Proof E [Y  X] = x. E Y Xx = 1.

  E Y Xx − 1 = 0 since E [Y ] = 1. E [Y ] E (RX ) + cov [RX , Y ] = 0. Algebra completes the proof. Remark 97 (Non-normalized Market) For future reference, we give the risk return relation in the non-normalized market. Using expression (17.1), substituting into expression (17.4) gives    ˜ X −r ˜ X −r E R1+r = −cov R1+r , Y . Algebra gives the final result.   E[R˜ X ] = r − cov R˜ X , Y . This completes the remark.

(17.5)

17.2 Basis Assets, Multi-Factor Beta Models, and Systematic Risk

357

We can now characterize which basis assets in expression (17.3) have nonzero risk premium. Since the basis assets are themselves traded portfolios, applying Theorem 86 we have that the expected return on any basis asset satisfies   E[Rj ] = −cov Rj , Y .

(17.6)

  If cov Rj , Y = 0, then the j th basis asset has a nonzero risk premium and is a risk factor. This completes the characterization. If the martingale deflator Y ∈ M trades, i.e. Y ∈ span(1, S1 , . . . , Sn ) we can prove a beta model. Before this, however, we need the following lemma.   Lemma 35 (Y ∈ M Trades) If Y ∈ span(1, S1 , . . . , Sn ), then E Y 2 < ∞, Y = γ0 +

n 

γj Sj

where

(17.7)

j =1

⎛ ⎞−1 ⎛ ⎞ ⎞ cov[S1 , S1 ] cov[S1 , S2 ] · · · cov[S1 , Sn ] E[S1 − s1 ] γ1 ⎜ cov[S2 , S1 ] cov[S2 , S2 ] · · · cov[S2 , Sn ] ⎟ ⎜ E[S2 − s2 ] ⎟ ⎜ γ2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟, ⎜ . ⎟ = −⎜ .. .. .. .. .. ⎝ ⎠ ⎝ ⎠ ⎝ .. ⎠ . . . . . γn cov[Sn , S1 ] cov[Sn , S2 ] · · · cov[Sn , Sn ] E[Sn − sn ] ⎛

γ0 = 1 −

n 

γj E[Sj ],

and

j =1 n    E Y 2 = γ0 + γj sj . j =1

  Proof (Step 1) Show E Y 2 < ∞ . If Y ∈ span(1, S1 , . . . , Sn ), then there exists (γ0 , γ1 , . . . , γn ) ∈ Rn+1 such that Y = γ0 +

n 

γj Sj .

j =1

Thus, 



⎡⎛

⎢ E Y 2 = E ⎣⎝γ0 +

n  j =1

⎞2 ⎤ ⎥ γj Sj ⎠ ⎦ < ∞

(17.8)

358

17 Epilogue (The Fundamental Theorems and the CAPM)

since  E[S  i Sj ] < ∞ all i, j . Because Y trades, its time 0 price equals E [Y · Y ] = E Y 2 = γ0 + nj=1 γj E[Sj Y ] = γ0 + nj=1 γj sj . This completes the proof of (Step 1). (Step 2) Taking the covariance of expression (17.8) with respect to S1 , then S2 ,. . . , then Sn gives the system of equations n γ cov[S1 , Sj ] = cov[S1 , Y ] jn=1 j j =1 γj cov[S2 , Sj ] = cov[S2 , Y ] .. . n γ cov[S , n Sj ] = cov[Sn , Y ] j =1 j Using expression (17.6), we have E[Rj ] = − since cov[Rj , Y ] = cov



Sj sj

 − 1, Y =

1 cov[Sj , Y ] sj

1 sj

  cov Sj , Y . Algebra gives

cov[Sj , Y ] = −E[Rj ]sj = −E[Sj − sj ]. Substitution into the system of equations, in matrix form, we have ⎛

cov[S1 , S1 ] cov[S1 , S2 ] ⎜ cov[S2 , S1 ] cov[S2 , S2 ] ⎜ ⎜ .. .. ⎝ . . cov[Sn , S1 ] cov[Sn , S2 ]

⎞⎛ ⎞ ⎛ ⎞ γ1 −E[S1 − s1 ] · · · cov[S1 , Sn ] ⎜ ⎟ ⎜ ⎟ · · · cov[S2 , Sn ] ⎟ ⎟ ⎜ γ2 ⎟ ⎜ −E[S2 − s2 ] ⎟ ⎟⎜ . ⎟ = ⎜ ⎟. .. . .. .. ⎠ ⎝ .. ⎠ ⎝ ⎠ . . · · · cov[Sn , Sn ] γn −E[Sn − sn ]

Multiplying both sides of this equation by the inverse of the covariance matrix yields the first part of expression (17.7). Second, taking the expectation of expression (17.8) gives γ0 +

n 

γj E[Sj ] = E[Y ] = 1.

j =1

This completes the proof of (Step 2). This completes the proof.   Note that E Y 2 = γ0 + nj=1 γj sj is the time 0 price of the traded asset Y ∈ M. Then, the return on Y can be written as RY =

Y − 1. E[Y 2 ]

We can now derive the beta model, which is the analogue of Theorem 39 in Chap. 4.

17.3 Utility Functions

359

Theorem 87 (Beta Model) If Y ∈ span(1, S1 , . . . , Sn ), then E[Rj ] =

cov[Rj , RY ] E [RY ] . var[RY ]

(17.9)

Proof Fromexpression (17.4), applied to Y , we get E(RY ) = −cov[RY , Y ] = Y 2 2 −E[Y 2 ]cov RY , E[Y 2 ] = −E[Y ]cov [RY , RY ] = −E[Y ]var[RY ]. E[RY ] . Algebra gives −E[Y 2 ] = var[R Y]   (17.6), we have E[Rj ] = −cov Rj , Y = −E[Y 2 ]cov   Using expression   Y 2 2 Rj , E[Y 2 ] = −E[Y ]cov Rj , RY . Substitution for E[Y ] completes the proof.

This theorem can be used to determine which basis assets are risk factors, i.e. have non-zero excess returns.

17.3 Utility Functions This section introduces the traders’ utility functions. Let all traders have the same beliefs, i.e. Pi = P for i = 1, . . . , I . We assume that all investors have meanvariance utility functions defined over terminal wealth, i.e. Ui : (0, ∞) → R with Ui (x) = x −

bi 2 x 2

(17.10)

for i = 1, . . . , I where bi > 0 is a risk aversion parameter. This utility function is unique up to an affine transformation (see Lemma 19 in Chap. 9). We note that ⎧ 1 ⎪ ⎨ > 0 if 0 < x < bi 1  Ui (x) = 1 − bi x = = 0 if x = bi ⎪ ⎩ < 0 if x > 1 . bi

(17.11)

This utility function exhibits decreasing marginal utility for x ≥ b1i . This is a well-known deficiency of mean-variance preferences. For the subsequent analysis to apply, we need to restrict ourselves to 0 < x < b1i . The inverse of the derivative Ui over 0 < x < Ii (y) =

1−y bi

1 bi

is

for 0 < y < 1.

(17.12)

360

17 Epilogue (The Fundamental Theorems and the CAPM)

The conjugate function is U˜ i (y) = sup [Ui (x) − xy] 1 bi

= sup [x − 1 bi

bi 2 2x

>x>0

− xy] =

>x>0

1 (1−y)2 2 bi ,

1>y>0

which is strictly convex with first derivative equal to (1 − y) < 0. U˜ i (y) = − bi Proof Solve the optimization problem using standard calculus. Ui (x) − y = 1 − bi x − y = 0, i.e. x = 1−y bi . The second order condition is −bi < 0, so an optimum is obtained. Substitution yields 1 (1−y)2 1 (1−y)2 U˜ i (y) = [ (1−y) − (1−y) bi − 2 bi bi y] = 2 bi . This completes the proof.

17.4 Portfolio Optimization This section solves the investor’s optimization problem. The analysis applies to either a complete or incomplete market. To obtain a solution, we need to assume that the market is arbitrage free. M = ∅

Assumption (NA)

The investor’s optimization problem is the following. Problem 15 (Choose the Optimal Trading Strategy)   bi E X − X2 2 (α0 ,α)∈Rn+1 sup



1 X ∈ 0, bi

 ,

X = α0 +

n 

αj Sj ,

where

and

j =1

xi = α0 +

n 

αj sj .

j =1

To solve this problem, we first simplify it by using the second constraint to remove α0 as a decision variable. Problem 16 (Choose the Optimal Trading Strategy)   bi sup E X − X2 2 α∈Rn

17.4 Portfolio Optimization

361

where   1 X ∈ 0, bi

and

X = xi +

n 

  αj Sj − sj .

j =1

As in the dynamic problem, we divide the solution into two steps. Step 1 solves for the optimal wealth X. Step 2 solves for the optimal trading strategy (α0 , α) that achieves this wealth. Hence, Step 1 is to solve for the following problem. Problem 17 (Choose the Optimal Wealth X)   bi 2 sup E X − X 2 X∈C˜(x ) i

where C˜(xi ) = {X ∈ LS : ∃α ∈ Rn , X = xi +

n 

αj [Sj − sj ]} and

j =1

LS = {X ∈ span(1, S1 , . . . , Sn ),

1 > X > 0}. bi

Note the “tilde” over the constraint set in this optimization problem. Without loss of generality, we consider the above optimization problem with free disposal. Problem 18 (Choose the Optimal Wealth X)   bi sup E X − X2 2 X∈C (xi ) where C (xi ) = {X ∈ LS : ∃α ∈ Rn , xi +

n 

αj [Sj − sj ] ≥ X}.

j =1

Note the “tilde” is removed from the constraint set with free disposal. As the next lemma proves, the solutions to the two problems are equivalent. Lemma 36 (Free Disposal) XT is an optimal solution to Problem 17 if and only if XT is an optimal solution to Problem 18. 1  ∗ bi , U (X) > 0 for all X ∈ C (xi ). Let X be the solution ∗∗ C˜(x i ) ⊂ C (xi ), to Problem 18 and  X ∗∗the  solution to Problem  17. Since  ∗ ∗ E Ui (X ) ≥ E Ui (X ) . Suppose E Ui (X ) > E Ui (X∗∗ ) . This implies that ∃α ∈ Rn , Z = xi + nj=1 αj [Sj −sj ] ≥ X∗ . Then Z ∈ C (xi ) and E [Ui (Z)] >

Proof Given X
X > 0 and E [XY ] ≤ xi . Thus, ∃ (α0 , α) ∈ Rn+1 , X = α0 + nj=1 αj Sj . Multiplying by Y and taking expectations

17.4 Portfolio Optimization

363

yields E [XY ] = α0 + nj=1 αj E[Sj Y ]. Subtracting gives X − E [XY ] =     n n j Sj − sj . Since E [XY ] ≤ xi , we get X = E [XY ] + j =1 α j =1 αj Sj − sj   ≤ xi + nj=1 αj Sj − sj . Hence, X ∈ C (xi ). This completes the proof of Step 1. n (Step n 2) Show C (xi ) ⊂ 1C1 (xi ). Take a X ∈ C (xi ). Then, ∃α ∈ R , X ≤ xi + j =1 αj [Sj − sj ] with bi > X > 0. Multiplying by Y and taking expectations   yields E [XY ] ≤ xi since E Sj = sj . Hence, X ∈ C1 (xi ). This completes the proof of Step 2. This completes the proof. Remark 99 (Correspondence to Chap. 11) In Chap. 11, which studies an incomplete market in continuous-time, the corresponding budget constraint is C (x) = {XT ∈ L0+ : sup E [XT YT ] ≤ x}. YT ∈Ds

The equivalent expression here is C (xi ) = {X ∈ LS : sup E [XY ] ≤ xi }. Y ∈M

But, as noted in Remark 98 above, sup E [XY ] = E[XY ∗ ] for any Y ∗ ∈ M. Y ∈M

Hence, the formula as given in the previous Lemma is an exact correspondence for an incomplete market in continuous-time (as well as for a complete market). This completes the remark. Using this lemma, we rewrite the investor’s optimization problem in its final form. Problem 19 (Choose the Optimal Wealth X)   bi sup E X − X2 2 X∈C (xi ) where C (xi ) = {X ∈ LS : E [XY ] ≤ xi } . To solve this problem, we add the following assumptions. There exists a Y ∈ M

Assumption (The Martingale Deflator Trades) such that Y ∈ span(1, S1 , . . . , Sn ).

This assumption states that there exists a martingale deflator Y ∈ M that trades. Assumption (“Small” Initial Wealth) 0 < xi
0 = sup E [Ui (X)] = vi (xi ). X∈C (xi )

This completes the proof. The dual problem is "

! dual :

inf

sup L (X, y) .

y>0

X∈LS

For optimization problems we know that "

!

"

!

inf

sup L (X, y) ≥ sup

inf L (X, y) .

y>0

X∈LS

y>0

X∈LS

(17.14)

Proof We have L (X, y) = E [Ui (X)] + y (xi − E [XY ]). Note that sup L (X, y) ≥ L (X, y) ≥ inf L (X, y) for all (X, y). X∈LS

y>0

Taking the infimum over y on the left side and the supremum over X on the right side gives " " ! ! inf

sup L (X, y) ≥ sup

inf L (X, y) . This completes the proof.

y>0

X∈LS

y>0

X∈LS

17.4 Portfolio Optimization

365

To prove a solution exists to the primal problem, we show that a solution exists to the dual problem, and that the dual solution is a feasible solution to the primal problem. Hence, by expression (17.14), the dual solution solves the primal problem and there is no duality gap.

17.4.1 The Dual Problem This section provides the solution to the dual problem. "

! inf

sup L (X, y)

y>0

X∈LS

"   bi 2 sup E X − X − yXY + xi y . 2 X∈LS

! = inf y>0

To solve, we first exchange the sup and expectation operator. "  ' bi 2 E sup X − X − yXY + xi y . 2 X∈LS

! & inf y>0

The justification for this step is proven in the appendix to this chapter. We solve this problem working from the inside out. Fixing y, we solve for the optimal X.   bi X − X2 − yXY . 2 X∈LS sup

(17.15)

The first order condition for an optimum gives 1 − bi Xi − yY = 0 Xi =

or

1 − yY = Ii (yY ) bi

where Ii (·) = Ui (·)−1 . Note that since y > 0 and Y > 0, Xi < b1i . In addition, a feasible wealth is xi (corresponding to holding only the mma). We have, by the definition of a maximum that Ui (Xi ) ≥ Ui (xi ). Since Ui (·) is increasing, this implies Xi ≥ xi > 0 a.s. P. Finally, since Y ∈ span(1, S1 , . . . , Sn ), X ∈ span(1, S1 , . . . , Sn ). Thus, Xi ∈ LS and Xi is the optimum solution to expression (17.15). Note that this implies yY < 1 a.s. P.

366

17 Epilogue (The Fundamental Theorems and the CAPM)

Given this optimal Xi ∈ LS , we next solve for y. The conjugate function is     bi bi U˜ i (yY ) = sup X − X2 − yXY = Xi − Xi2 − yXi Y 2 2 X∈LS where 0 < yY < 1. This transforms the problem to "  '   bi 2 E sup X − X − yXY + xi y = inf E[U˜i (yY )] + xi y 2 X∈LS y>0

! & inf y>0

for 0 < yY < 1. Taking the derivative of the right side with respect to y yields E[U˜ i (yY )Y ] + xi = 0. This step requires taking the derivative underneath the expectation operator. The justification for this step is proven in the appendix to this chapter. ) Noting from Lemma 25 in Chap. 9 that U˜ i (yY ) = −Ii (yY ) = − (1−yY bi , we get that the optimal yi satisfies  (1 − yi Y ) Y = xi . E bi 

(17.16)

  Since Y ∈ span(1, S1 , . . . , Sn ), by Lemma 35 we have that E Y 2 < ∞. Algebra and the fact that E[Y ] = 1 gives yi =

1 − bi xi > 0. E[Y 2 ]

Note that yi is strictly positive because by assumption xi < b1i . Expression (17.16) shows that Xi satisfies the budget constraint with an equality, i.e. E [Xi Y ] = xi . Finally, substitution yields the optimal wealth 0 < Xi =

1 yi Y 1 (1 − bi xi ) Y. − = − bi bi bi bi E[Y 2 ]

This completes the solution to the dual problem.

(17.17)

17.4 Portfolio Optimization

367

17.4.2 The Primal Problem This section provides the solution to the primal problem. To complete the proof, we need to show that the solution Xi obtained above is a feasible solution to the " i.e. Xi ∈ C (xi ). Then by expression (17.14), since ! primal problem, inf

sup L (X, y)

y>0

X∈LS

≥ vi (xi ), Xi is the optimal solution to the primal problem

as well. In this regard, we note that: (i) since Y ∈ span(1, S1 , . . . , Sn ), X ∈ span(1, S1 , . . . , Sn ), (ii) by expression (17.17), b1i > Xi > 0, and (iii) E [XY ] = xi . This completes the proof that Xi ∈ C (xi ).

17.4.3 The Optimal Trading Strategy This section determines the optimal trading strategy, which is Step 2 in the solution procedure. This is the trading strategy (α0 , α) such that α0 +

n 

αj Sj = Xi =

j =1

1 (1 − bi xi ) Y. − bi bi E[Y 2 ]

Such a solution exists and is unique since (i) Xi ∈ span(1, S1 , . . . , Sn ) and (ii) dim {span(1, S1 , . . . , Sn )} = n + 1. To obtain this trading strategy, by Lemma 35 expression (17.7), we have α0 +

n  j =1

⎛ ⎞ n  1 (1 − bi xi ) ⎝ γ0 + αj Sj = − γj Sj ⎠ . bi bi E[Y 2 ] j =1

Algebra gives α0 +

n 

 αj Sj =

j =1

  n   1 1 (1 − bi xi ) (1 − bi xi ) γ0 + γj Sj . − − bi bi bi E[Y 2 ] bi E[Y 2 ] j =1

(17.18) Uniqueness of the coordinates to the basis (1, S1 , . . . , Sn ) gives that the optimal trading strategy is

αj =



1 bi

  i xi ) γ0 , α0 = b1i − (1−b 2 bi E[Y ]  (1−bi xi ) − b E[Y 2 ] γj for j = 1, . . . , n i

where

(17.19)

368

17 Epilogue (The Fundamental Theorems and the CAPM)

⎛ ⎞ ⎞−1 ⎛ ⎞ cov[S1 , S1 ] cov[S1 , S2 ] · · · cov[S1 , Sn ] γ1 E[S1 − s1 ] ⎜ cov[S2 , S1 ] cov[S2 , S2 ] · · · cov[S2 , Sn ] ⎟ ⎜ E[S2 − s2 ] ⎟ ⎜ γ2 ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ . ⎟ = −⎜ ⎟ ⎜ ⎟, .. .. .. .. .. ⎝ ⎝ .. ⎠ ⎝ ⎠ ⎠ . . . . . γn cov[Sn , S1 ] cov[Sn , S2 ] · · · cov[Sn , Sn ] E[Sn − sn ] ⎛

γ0 = 1 −

n 

γj E[Sj ].

j =1

Remark 100 (Partial Derivatives with Respect to Sj ) From expression (17.18) we have

=



1 bi

Xi= α0 + nj=1 αj Sj  n  1 (1−bi xi ) i xi ) γj Sj . − b E[Y 2 ] γ0 + j =1 bi − (1−b 2 b E[Y ] i

(17.20)

i

We can alternatively obtain the trading strategy with respect to the risky assets by taking partial derivatives. Indeed ∂Xi = αj = ∂Sj



 1 (1 − bi xi ) γj . − bi bi E[Y 2 ]

The position in the money market account follows by substituting this last equality into expression (17.18) and solving for α0 . We note the correspondence between this alternative derivation of the trading strategy and that given in Chap. 10, Sect. 10.7. This completes the remark. To obtain the next theorem, we first decompose the initial wealth into the position in the mma α0 and that portion in the risky assets, zi = xi − α0 =

n 

αj sj .

(17.21)

j =1

Then, Zi = Xi − α0 =

n 

αj Sj

(17.22)

j =1

is the wealth in the risky asset portfolio at time 1. Assume that zi > 0 and define RZi = Zzii − 1 to be the return on the risky asset portfolio of investor i. The risky asset portfolio’s return can be written in an alternative form.

17.4 Portfolio Optimization

369

Lemma 38 (Risky Asset Portfolio Representation) Let Xi = α0 + with xi = α0 + N j =1 αj sj . Then,

N

j =1 αj Sj

Xi = xi + zi RZi where RZi =

n

=

j =1 θj Rj , θj

αj sj zi

n 

(17.23)

for j = 1, . . . , n, and zi = xi − α0 . Note that

θj =

j =1

n  αj sj j =1

zi

= 1.

Proof Xi = xi + nj=1 αj [Sj − sj ] α s [S −s ] = xi + zi nj=1 jzi j jsj j .

n αj Sj α s S − 1 = nj=1 jzi j sjj − 1. Finally, RZi = Zzii − 1 = j =1zi α s α s S α s Given nj=1 jzi j = 1, we get RZi = nj=1 jzi j sjj − nj=1 jzi j α s [S −s ] = nj=1 jzi j jsj j . Substitution yields Xi = xi + zi RZi . This completes the proof.

Theorem 88 (All Traders Hold the Same Optimal Risky Asset Portfolio) RZi = ηRY =

n 

f or all i = 1, . . . , I

θj Rj

(17.24)

j =1

where η =

γ0 + nj=1 γj sj n j =1 γj sj

, θj =

γ s n j j j =1 γj sj

for all j = 1, . . . , n, and

n

j =1 θj

 Proof (Step 1) By expression (17.20), using the fact that α0 = b1i −     (1−bi xi ) n 1 i xi ) γ gives Zi = nj=1 b1i − (1−b S = − j j 2 j =1 γj Sj bi bi E[Y 2 ] i E[Y ]  b  n i xi ) and zi = b1i − (1−b j =1 γj sj . b E[Y 2 ] i

Hence, RZi =

RY =

−1=

n



1 bi

−

(1−bi xi ) bi E[Y 2 ]

1 bi

−

(1−bi xi ) bi E[Y 2 ]





n

j =1 γj Sj

n

j =1 γj sj

−1=

γj Sj and Y0 = γ0 + nj=1 γj sj . j =1 n n n Y −Y0 j =1 γj Sj − j =1 γj sj j =1 γj sj n = · . Or, Y0 γj sj γ0 + nj=1 γj sj j =1 n γ s j =1 j j RZi · γ + . Algebra completes the proof n 0 j =1 γj sj

But, Y = γ0 + RY =

Zi zi



n γj Sj jn=1 j =1 γj sj

− 1.

Hence,

of (Step 1).

= 1.

(1−bi xi ) bi E[Y 2 ]

 γ0

370

17 Epilogue (The Fundamental Theorems and the CAPM)

α s (Step 2) By Lemma 38, RZi = nj=1 θj Rj where θj = jzi j . From Step 1 we have   i xi ) γj sj . Using this and expression (17.19) gives zi = nj=1 b1i − (1−b b E[Y 2 ] i

 θj = 

1 bi

1 bi

−

−

(1−bi xi ) bi E[Y 2 ]

(1−bi xi ) bi E[Y 2 ]





γj s j

γj sj = n . n j =1 γj sj j =1 γj sj

This completes the proof. This is an important theorem. It shows that all traders hold the same optimal risky asset portfolio (θ1 , . . . , θn ). Their optimal portfolio differs only due to amount of wealth held in the risky asset portfolio zi and in the mma, xi − zi . The difference in these quantities is due to the different risk aversion coefficients bi across the investors.

17.5 Beta Model (Revisited) This section revisits the beta model using the characterization of the martingale deflator obtained from the solution to the investor’s optimization problem. Since the martingale deflator Y ∈ M trades (by assumption), we get Theorem 89 (Beta Model Revisited) E[Rj ] = where RZ =

cov[Rj , RZ ] E [RZ ] var[RZ ]

n

j =1 θj Rj

with θj =

f or j = 1, . . . , n.

γ s n j j j =1 γj sj

for all j and

n

j =1 θj

(17.25) = 1.

cov[R ,R ]

j Y Proof Using Theorem 87, we get E[Rj ] = var[R E [RY ]. Y] 1 Expression (17.24) gives RY = η RZ . Substitution yields   cov[Rj , η1 RZ ] 1 , algebra completes the proof. E R E[Rj ] = Z 1 η

var[ η RZ ]

Remark 101 (Non-normalized Economy) Using expression (17.1) in expression (17.25) gives    cov R˜ j −r , R˜ Z −r    1+r 1+r ˜ R˜ j −r   E RZ −r . Algebra gives E 1+r = 1+r R˜ −r   E R˜ j − r =

Z var 1+r    cov R˜ j −r,R˜ Z −r ˜Z   E R var R˜ Z −r

 − r . Properties of variances and covariances

17.6 The Efficient Frontier

371

yields the final result E[R˜ j ] = r +

 cov[R˜ j , R˜ Z ]  E[R˜ Z ] − r . var[R˜ Z ]

(17.26)

This completes the remark.

17.6 The Efficient Frontier This section introduces the notion of an efficient frontier. To do this, we rewrite the optimization Problem 16 in an alternative form and solve it again.

17.6.1 The Solution (Revisited) Using Lemma 38, we rewrite the optimization problem as follows. Problem 20 (Utility Optimization) 

  bi 2 sup sup E xi + zRθ − [xi + zRθ ] 2 z θ

where

n 

θj = 1.

j =1

We solve the inner problem first, yielding the solution z=

E[Rθ ](1 − bi xi ) . E[Rθ2 ]bi

  Proof E xi + zRθ − b2i [xi + zRθ ]2    = E xi + zRθ − b2i xi2 + 2xi zRθ + z2 Rθ2 = xi − b2i xi2 + zE[Rθ ](1 − bi xi ) − b2i z2 E[Rθ2 ]. Taking the first derivative and setting it equal to zero yields E[Rθ ](1 − bi xi ) − zbi E[Rθ2 ] = 0. Algebra completes the proof. Given the optimal wealth in the risky asset portfolio z, the remaining optimization problem is to choose the proportions in the risky assets themselves, i.e. 7 sup θ

1 (1 − bi xi )2 E[Rθ ]2 2 bi E[Rθ2 ]

< where

n  j =1

θj = 1.

372

17 Epilogue (The Fundamental Theorems and the CAPM)

bi 2 2 2 z E[Rθ ], or bi 2 E[Rθ ](1−bi xi ) bi E[Rθ ]2 (1−bi xi )2 2 xi − 2 xi + E[Rθ ](1−bi xi )− 2 E[Rθ ]. Simplification E[R 2 ]bi E[R 2 ]2 b2

Proof The objective function is xi − yields xi −

bi 2 2 xi

+ zE[Rθ ](1 − bi xi ) −

θ

bi 2 2 xi

+

E[Rθ ]2 (1−bi xi )2 E[Rθ2 ]bi

θ

−

1 E[Rθ ]2 (1−bi xi )2 . 2 E[Rθ2 ]bi

i

Algebra completes the proof.

Finally, this is equivalent to solving ⎧ ⎫ ⎨ E[R ] ⎬ θ sup 3 θ ⎩ E[R 2 ] ⎭ θ

where

n 

θj = 1.

(17.27)

j =1

The solution to this optimization problem does not depend on the ith trader. This yields another proof of Theorem 88 that all traders hold the same optimal risky asset portfolio. In symbols, RZi = Rθ = nj=1 θj Rj is independent of investor i. The solution θ = (θ1 , . . . , θn ) to expression (17.27) is the optimal risky asset portfolio.

17.6.2 Summary The investor’s optimization problem is (i) to E[Ui (xi + zRθ )] where Ui (x) = x − nchoose z to maximize n θ R , and θ = 1. Then, j j j j =1 j =1 (ii) to choose θ to maximize ⎧ ⎫ n ⎨ E[R ] ⎬  θ sup 3 θj = 1. where θ ⎩ E[R 2 ] ⎭ j =1 θ Both (i) and (ii) determine αj = (iii) Finally, α0 is determined such that

zθj sj

xi = α0 +

for j = 1, . . . , n. n 

αj sj .

j =1

The solution θ is independent of the trader’s initial wealth xi .

bi 2 2x ,

Rθ =

17.7 Equilibrium

373

17.6.3 The Risky Asset Frontier and Efficient Frontier This section defines and characterizes both the risky asset frontier and the efficient frontier. Consider the following optimization problem 3 inf θ

E[Rθ2 ]

where E[Rθ ] = μ.

3 Denote the solution by θμ with the minimum being E[Rθ2μ ]. Solving this problem for different μ0maps out what is defined to be the risky asset frontier (“x” − axis, “y” − axis) = ( E[R 2 ], E[R]) space. The tangent line from 0 2 ( E[r ] = 0, r = E[r] = 0), the mma’s coordinates, to the risky asset frontier is determined by that μ∗ such that the slope of this tangent line is maximized, i.e. ⎧ ⎫ ⎨ ⎬ E[Rθμ∗ ] μ∗ μ 3 =3 = sup 3 . μ ⎩ E[R 2 ] ⎭ E[Rθ2μ∗ ] E[Rθ2μ∗ ] θμ This tangent line is defined to be the efficient frontier (including both the mma and the risky assets). To relate the optimal portfolio to the efficient frontier, let θ ∗ denote the optimal risky asset portfolio, i.e. the solution to expression (17.27). Then, θ ∗ yields a (square root of the expected return squared, mean return) that is on the efficient frontier. 7 < 7 < Proof sup θ

3E[Rθ ] E[Rθ2 ]

≥ sup μ

E[Rθμ ] 3 E[Rθ2μ ]

since Rθμ for different μ are a subset of

Rθ . But, letting μ˜ = E[Rθ ∗ ], we have that 7 3 μ˜ E[Rθ2∗ ]

≤

Thus,

E[R ∗ ] 3 θ E[Rθ2∗ ]

7

3μ˜ inf E[Rθ2 ]

=