Stochastic Finance: An Introduction in Discrete Time [3rd rev. and extend. ed.] 9783110218053, 9783110218046

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De Gruyter Graduate

Hans Föllmer Alexander Schied

Stochastic Finance An Introduction in Discrete Time

Third revised and extended edition

De Gruyter

Mathematics Subject Classification 2010: Primary: 60-01, 91-01, 91-02; Secondary: 46N10, 60E15, 60G40, 60G42, 91B08, 91B16, 91B30, 91B50, 91B52, 91B70, 91G10, 91G20, 91G80, 91G99.

The first and second edition of “Stochastic Finance” were published in the series “De Gruyter Studies in Mathematics”.

ISBN 978-3-11-021804-6 e-ISBN 978-3-11-021805-3 Library of Congress Cataloging-in-Publication Data Föllmer, Hans. Stochastic finance : an introduction in discrete time / by Hans Föllmer, Alexander Schied. ⫺ 3rd, rev. and extended ed. p. cm. Includes bibliographical references and index. ISBN 978-3-11-021804-6 (alk. paper) 1. Finance ⫺ Statistical methods. 2. Stochastic analysis. 3. Probabilities. I. Schied, Alexander. II. Title. HG176.5.F65 2011 332.011519232⫺dc22 2010045896

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. ” 2011 Walter de Gruyter GmbH & Co. KG, Berlin/New York Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ⬁ Printed on acid-free paper Printed in Germany www.degruyter.com

Preface to the third edition

This third edition of our book appears in the de Gruyter graduate textbook series. We have therefore included more than one hundred exercises. Typically, we have used the book as an introductory text for two major areas, either combined into one course or in two separate courses. The first area comprises static and dynamic arbitrage theory in discrete time. The corresponding core material is provided in Chapters 1, 5, and 6. The second area deals with mathematical aspects of financial risk as developed in Chapters 2, 4, and 11. Most of the exercises we have included in this edition are therefore contained in these core chapters. The other chapters of this book can be used both as complementary material for the introductory courses and as basis for special-topics courses. In recent years, there has been an increasing awareness, both among practitioners and in academia, of the problem of model uncertainty in finance and economics, often called Knightian uncertainty; see, e.g., [259]. In this third edition we have put more emphasis on this issue. The theory of risk measures can be seen as a case study how to deal with model uncertainty in mathematical terms. We have therefore updated Chapter 4 on static risk measures and added the new Chapter 11 on dynamic risk measures. Moreover, in Section 2.5 we have extended the characterization of robust preferences in terms of risk measures from the coherent to the convex case. We have also included the new Sections 3.5 and 8.3 on robust variants of the classical problems of optimal portfolio choice and efficient hedging. It is a pleasure to express our thanks to all students and colleagues whose comments have helped us to prepare this third edition, in particular to Aurélien Alfonsi, Günter Baigger, Francesca Biagini, Julia Brettschneider, Patrick Cheridito, Samuel Drapeau, Maren Eckhoff, Karl-Theodor Eisele, Damir Filipovic, Zicheng Hong, Kostas Kardaras, Thomas Knispel, Gesine Koch, Heinz König, Volker Krätschmer, Christoph Kühn, Michael Kupper, Mourad Lazgham, Sven Lickfeld, Mareike Massow, Irina Penner, Ernst Presman, Michael Scheutzow, Melvin Sim, Alla Slynko, Stephan Sturm, Gregor Svindland, Long Teng, Florian Werner, Wiebke Wittmüß, and Lei Wu. Special thanks are due to Yuliya Mishura and Georgiy Shevchenko, our translators for the Russian edition. Berlin and Mannheim, November 2010

Hans Föllmer Alexander Schied

Preface to the second edition

Since the publication of the first edition we have used it as the basis for several courses. These include courses for a whole semester on Mathematical Finance in Berlin and also short courses on special topics such as risk measures given at the Institut Henri Poincaré in Paris, at the Department of Operations Research at Cornell University, at the Academia Sinica in Taipei, and at the 8th Symposium on Probability and Stochastic Processes in Puebla. In the process we have made a large number of minor corrections, we have discovered many opportunities for simplification and clarification, and we have also learned more about several topics. As a result, major parts of this book have been improved or even entirely rewritten. Among them are those on robust representations of risk measures, arbitrage-free pricing of contingent claims, exotic derivatives in the CRR model, convergence to the Black–Scholes model, and stability under pasting with its connections to dynamically consistent coherent risk measures. In addition, this second edition contains several new sections, including a systematic discussion of law-invariant risk measures, of concave distortions, and of the relations between risk measures and Choquet integration. It is a pleasure to express our thanks to all students and colleagues whose comments have helped us to prepare this second edition, in particular to Dirk Becherer, Hans Bühler, Rose-Anne Dana, Ulrich Horst, Mesrop Janunts, Christoph Kühn, Maren Liese, Harald Luschgy, Holger Pint, Philip Protter, Lothar Rogge, Stephan Sturm, Stefan Weber, Wiebke Wittmüß, and Ching-Tang Wu. Special thanks are due to Peter Bank and to Yuliya Mishura and Georgiy Shevchenko, our translators for the Russian edition. Finally, we thank Irene Zimmermann and Manfred Karbe of de Gruyter Verlag for urging us to write a second edition and for their efficient support. Berlin, September 2004

Hans Föllmer Alexander Schied

Preface to the first edition

This book is an introduction to probabilistic methods in Finance. It is intended for graduate students in mathematics, and it may also be useful for mathematicians in academia and in the financial industry. Our focus is on stochastic models in discrete time. This limitation has two immediate benefits. First, the probabilistic machinery is simpler, and we can discuss right away some of the key problems in the theory of pricing and hedging of financial derivatives. Second, the paradigm of a complete financial market, where all derivatives admit a perfect hedge, becomes the exception rather than the rule. Thus, the discrete-time setting provides a shortcut to some of the more recent literature on incomplete financial market models. As a textbook for mathematicians, it is an introduction at an intermediate level, with special emphasis on martingale methods. Since it does not use the continuous-time methods of Itô calculus, it needs less preparation than more advanced texts such as [99], [98], [107], [171], [252]. On the other hand, it is technically more demanding than textbooks such as [215]: We work on general probability spaces, and so the text captures the interplay between probability theory and functional analysis which has been crucial for some of the recent advances in mathematical finance. The book is based on our notes for first courses in Mathematical Finance which both of us are teaching in Berlin at Humboldt University and at Technical University. These courses are designed for students in mathematics with some background in probability. Sometimes, they are given in parallel to a systematic course on stochastic processes. At other times, martingale methods in discrete time are developed in the course, as they are in this book. Usually the course is followed by a second course on Mathematical Finance in continuous time. There it turns out to be useful that students are already familiar with some of the key ideas of Mathematical Finance. The core of this book is the dynamic arbitrage theory in the first chapters of Part II. When teaching a course, we found it useful to explain some of the main arguments in the more transparent one-period model before using them in the dynamical setting. So one approach would be to start immediately in the multi-period framework of Chapter 5, and to go back to selected sections of Part I as the need arises. As an alternative, one could first focus on the one-period model, and then move on to Part II. We include in Chapter 2 a brief introduction to the mathematical theory of expected utility, even though this is a classical topic, and there is no shortage of excellent expositions; see, for instance, [187] which happens to be our favorite. We have three reasons for including this chapter. Our focus in this book is on incompleteness, and incompleteness involves, in one form or another, preferences in the face of risk and uncertainty. We feel that mathematicians working in this area should be aware, at

viii

Preface to the first edition

least to some extent, of the long line of thought which leads from Daniel Bernoulli via von Neumann–Morgenstern and Savage to some more recent developments which are motivated by shortcomings of the classical paradigm. This is our first reason. Second, the analysis of risk measures has emerged as a major topic in mathematical finance, and this is closely related to a robust version of the Savage theory. Third, but not least, our experience is that this part of the course was found particularly enjoyable, both by the students and by ourselves. We acknowledge our debt and express our thanks to all colleagues who have contributed, directly or indirectly, through their publications and through informal discussions, to our understanding of the topics discussed in this book. Ideas and methods developed by Freddy Delbaen, Darrell Duffie, Nicole El Karoui, David Heath, Yuri Kabanov, Ioannis Karatzas, Dimitri Kramkov, David Kreps, Stanley Pliska, Chris Rogers, Steve Ross, Walter Schachermayer, Martin Schweizer, Dieter Sondermann and Christophe Stricker play a key role in our exposition. We are obliged to many others; for instance the textbooks [73], [99], [98], [155], and [192] were a great help when we started to teach courses on the subject. We are grateful to all those who read parts of the manuscript and made useful suggestions, in particular to Dirk Becherer, Ulrich Horst, Steffen Krüger, Irina Penner, and to Alexander Giese who designed some of the figures. Special thanks are due to Peter Bank for a large number of constructive comments. We also express our thanks to Erhan Çinlar, Adam Monahan, and Philip Protter for improving some of the language, and to the Department of Operations Research and Financial Engineering at Princeton University for its hospitality during the weeks when we finished the manuscript. Berlin, June 2002

Hans Föllmer Alexander Schied

Contents

Preface to the third edition

v

Preface to the second edition

vi

Preface to the first edition

vii

I Mathematical finance in one period 1

2

3

4

Arbitrage theory 1.1 Assets, portfolios, and arbitrage opportunities . . . 1.2 Absence of arbitrage and martingale measures . . . 1.3 Derivative securities . . . . . . . . . . . . . . . . . 1.4 Complete market models . . . . . . . . . . . . . . 1.5 Geometric characterization of arbitrage-free models 1.6 Contingent initial data . . . . . . . . . . . . . . . .

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Preferences 2.1 Preference relations and their numerical representation 2.2 Von Neumann–Morgenstern representation . . . . . . 2.3 Expected utility . . . . . . . . . . . . . . . . . . . . . 2.4 Uniform preferences . . . . . . . . . . . . . . . . . . 2.5 Robust preferences on asset profiles . . . . . . . . . . 2.6 Probability measures with given marginals . . . . . . .

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50 . 51 . 57 . 67 . 83 . 94 . 113

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121 121 130 139 148 151 159

Optimality and equilibrium 3.1 Portfolio optimization and the absence of arbitrage 3.2 Exponential utility and relative entropy . . . . . . . 3.3 Optimal contingent claims . . . . . . . . . . . . . 3.4 Optimal payoff profiles for uniform preferences . . 3.5 Robust utility maximization . . . . . . . . . . . . 3.6 Microeconomic equilibrium . . . . . . . . . . . .

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Monetary measures of risk 175 4.1 Risk measures and their acceptance sets . . . . . . . . . . . . . . . . 176 4.2 Robust representation of convex risk measures . . . . . . . . . . . . . 186 4.3 Convex risk measures on L1 . . . . . . . . . . . . . . . . . . . . . . 199

x

Contents

4.4 4.5 4.6 4.7 4.8 4.9

Value at Risk . . . . . . . . . . . . . . . . . . . . . . . Law-invariant risk measures . . . . . . . . . . . . . . . Concave distortions . . . . . . . . . . . . . . . . . . . . Comonotonic risk measures . . . . . . . . . . . . . . . . Measures of risk in a financial market . . . . . . . . . . Utility-based shortfall risk and divergence risk measures

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207 213 219 228 236 246

II Dynamic hedging

259

5

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261 261 266 274 287 290 296 302

6

7

Dynamic arbitrage theory 5.1 The multi-period market model . . . . . . . . . . 5.2 Arbitrage opportunities and martingale measures 5.3 European contingent claims . . . . . . . . . . . . 5.4 Complete markets . . . . . . . . . . . . . . . . . 5.5 The binomial model . . . . . . . . . . . . . . . . 5.6 Exotic derivatives . . . . . . . . . . . . . . . . . 5.7 Convergence to the Black–Scholes price . . . . . American contingent claims 6.1 Hedging strategies for the seller 6.2 Stopping strategies for the buyer 6.3 Arbitrage-free prices . . . . . . 6.4 Stability under pasting . . . . . 6.5 Lower and upper Snell envelopes

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321 321 327 337 342 347

Superhedging 7.1 P -supermartingales . . . . . . . . . . . . . . . . 7.2 Uniform Doob decomposition . . . . . . . . . . 7.3 Superhedging of American and European claims 7.4 Superhedging with liquid options . . . . . . . . .

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354 354 356 359 368

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8

Efficient hedging 380 8.1 Quantile hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 8.2 Hedging with minimal shortfall risk . . . . . . . . . . . . . . . . . . 387 8.3 Efficient hedging with convex risk measures . . . . . . . . . . . . . . 396

9

Hedging under constraints 9.1 Absence of arbitrage opportunities 9.2 Uniform Doob decomposition . . 9.3 Upper Snell envelopes . . . . . . 9.4 Superhedging and risk measures .

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404 404 412 417 424

xi

Contents

10 Minimizing the hedging error 10.1 Local quadratic risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Minimal martingale measures . . . . . . . . . . . . . . . . . . . . . . 10.3 Variance-optimal hedging . . . . . . . . . . . . . . . . . . . . . . . .

428 428 438 449

11 Dynamic risk measures 456 11.1 Conditional risk measures and their robust representation . . . . . . . 456 11.2 Time consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 Appendix A.1 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Absolutely continuous probability measures . . . . . . . A.3 Quantile functions . . . . . . . . . . . . . . . . . . . . . A.4 The Neyman–Pearson lemma . . . . . . . . . . . . . . . A.5 The essential supremum of a family of random variables A.6 Spaces of measures . . . . . . . . . . . . . . . . . . . . A.7 Some functional analysis . . . . . . . . . . . . . . . . .

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476 476 480 484 493 496 497 507

Notes

512

Bibliography

517

List of symbols

533

Index

535

Part I

Mathematical finance in one period

Chapter 1

Arbitrage theory

In this chapter, we study the mathematical structure of a simple one-period model of a financial market. We consider a finite number of assets. Their initial prices at time t D 0 are known, their future prices at time t D 1 are described as random variables on some probability space. Trading takes place at time t D 0. Already in this simple model, some basic principles of mathematical finance appear very clearly. In Section 1.2, we single out those models which satisfy a condition of market efficiency: There are no trading opportunities which yield a profit without any downside risk. The absence of such arbitrage opportunities is characterized by the existence of an equivalent martingale measure. Under such a measure, discounted prices have the martingale property, that is, trading in the assets is the same as playing a fair game. As explained in Section 1.3, any equivalent martingale measure can be identified with a pricing rule: It extends the given prices of the primary assets to a larger space of contingent claims, or financial derivatives, without creating new arbitrage opportunities. In general, there will be several such extensions. A given contingent claim has a unique price if and only if it admits a perfect hedge. In our one-period model, this will be the exception rather than the rule. Thus, we are facing market incompleteness, unless our model satisfies the very restrictive conditions discussed in Section 1.4. The geometric structure of an arbitrage-free model is described in Section 1.5. The one-period market model will be used throughout the first part of this book. On the one hand, its structure is rich enough to illustrate some of the key ideas of the field. On the other hand, it will provide an introduction to some of the mathematical methods which will be used in the dynamic hedging theory of the second part. In fact, the multi-period situation considered in Chapter 5 can be regarded as a sequence of one-period models whose initial conditions are contingent on the outcomes of previous periods. The techniques for dealing with such contingent initial data are introduced in Section 1.6.

1.1

Assets, portfolios, and arbitrage opportunities

Consider a financial market with d C 1 assets. The assets can consist, for instance, of equities, bonds, commodities, or currencies. In a simple one-period model, these assets are priced at the initial time t D 0 and at the final time t D 1. We assume that the i th asset is available at time 0 for a price  i  0. The collection  D . 0 ;  1 ; : : : ;  d / 2 RdCC1

4

Chapter 1 Arbitrage theory

is called a price system. Prices at time 1 are usually not known beforehand at time 0. In order to model this uncertainty, we fix a measurable space .; F / and describe the asset prices at time 1 as non-negative measurable functions S 0; S 1; : : : ; S d on .; F / with values in Œ0; 1/. Every ! 2  corresponds to a particular scenario of market evolution, and S i .!/ is the price of the i th asset at time 1 if the scenario ! occurs. However, not all asset prices in a market are necessarily uncertain. Usually there is a riskless bond which will pay a sure amount at time 1. In our simple model for one period, such a riskless investment opportunity will be included by assuming that  0 D 1 and

S0  1 C r

for a constant r, the return of a unit investment into the riskless bond. In most situations it would be natural to assume r  0, but for our purposes it is enough to require that S 0 > 0, or equivalently that r > 1: In order to distinguish S 0 from the risky assets S 1 ; : : : ; S d , it will be convenient to use the notation S D .S 0 ; S 1 ; : : : ; S d / D .S 0 ; S /; and in the same way we will write  D .1; /. At time t D 0, an investor will choose a portfolio  D . 0 ; / D . 0 ;  1 ; : : : ;  d / 2 Rd C1 ; where  i represents the number of shares of the i th asset. The price for buying the portfolio  equals d X   D i i : iD0

At time t D 1, the portfolio will have the value   S.!/ D

d X

 i S i .!/ D  0 .1 C r/ C   S.!/;

iD0

depending on the scenario ! 2 . Here we assume implicitly that buying and selling assets does not create extra costs, an assumption which may not be valid for a small investor but which becomes more realistic for a large financial institution. Note our convention of writing x  y for the inner product of two vectors x and y in Euclidean space.

Section 1.1 Assets, portfolios, and arbitrage opportunities

5

Our definition of a portfolio allows the components  i to be negative. If  0 < 0, this corresponds to taking out a loan such that we receive the amount j 0 j at t D 0 and pay back the amount .1 C r/j 0 j at time t D 1. If  i < 0 for i  1, a quantity of j i j shares of the i th asset is sold without actually owning them. This corresponds to a short sale of the asset. In particular, an investor is allowed to take a short position  i < 0, and to use up the received amount  i j i j for buying quantities  j  0, j ¤ i , of the other assets. In this case, the price of the portfolio  D . 0 ; / is given by    D 0. Remark 1.1. So far we have not assumed that anything is known about probabilities that might govern the realization of the various scenarios ! 2 . Such a situation is often referred to as Knightian uncertainty, in honor of F. Knight [176], who introduced the distinction between “risk” which refers to an economic situation in which the probabilistic structure is assumed to be known, and “uncertainty” where no such assumption is made. } Let us now assume that a probability measure P is given on .; F /. The asset prices S 1 ; : : : ; S d and the portfolio values   S can thus be regarded as random variables on .; F ; P /. Definition 1.2. A portfolio  2 Rd C1 is called an arbitrage opportunity if     0 but   S  0 P -a.s. and P Œ   S > 0  > 0. Intuitively, an arbitrage opportunity is an investment strategy that yields with positive probability a positive profit and is not exposed to any downside risk. The existence of such an arbitrage opportunity may be regarded as a market inefficiency in the sense that certain assets are not priced in a reasonable way. In real-world markets, arbitrage opportunities are rather hard to find. If such an opportunity would show up, it would generate a large demand, prices would adjust, and the opportunity would disappear. Later on, the absence of such arbitrage opportunities will be our key assumption. Absence of arbitrage implies that S i vanishes P -a.s. once  i D 0. Hence, there is no loss of generality if we assume from now on that i > 0

for i D 1; : : : ; d .

Remark 1.3. Note that the probability measure P enters the definition of an arbitrage opportunity only through the null sets of P . In particular, the definition can be formulated without any explicit use of probabilities if  is countable. In this case, we can simply apply Definition 1.2 with an arbitrary probability measure P such that P Œ¹!º > 0 for every ! 2 . Then an arbitrage opportunity is a portfolio  with     0, with   S.!/  0 for all ! 2 , and such that   S .!0 / > 0 for at least one !0 2 . }

6

Chapter 1 Arbitrage theory

The following lemma shows that absence of arbitrage is equivalent to the following property of the market: Any investment in risky assets which yields with positive probability a better result than investing the same amount in the risk-free asset must be exposed to some downside risk. Lemma 1.4. The following statements are equivalent. (a) The market model admits an arbitrage opportunity. (b) There is a vector  2 Rd such that   S  .1 C r/   P -a.s.

and

P Π  S > .1 C r/    > 0:

Proof. To see that (a) implies (b), let  be an arbitrage opportunity. Then 0     D  0 C   . Hence,   S  .1 C r/      S C .1 C r/ 0 D   S : Since   S is P -a.s. non-negative and strictly positive with non-vanishing probability, the same must be true of   S  .1 C r/  . Next let  be as in (b). We claim that the portfolio . 0 ; / with  0 WD    is an arbitrage opportunity. Indeed,    D  0 C    D 0 by definition. Moreover,   S D .1 C r/   C   S, which is P -a.s. non-negative and strictly positive with non-vanishing probability. Exercise 1.1.1. On  D ¹!1 ; !2 ; !3 º we fix a probability measure P with P Œ !i  > 0 for i D 1; 2; 3. Suppose that we have three assets with prices 0 1 1  D@2A 7 at time 0 and 0 1 1 S.!1 / D @ 3 A; 9

0 1 1 S.!2 / D @ 1 A; 5

0

1 1 S .!3 / D @ 5 A 10

at time 1. Show that this market model admits arbitrage.

}

Exercise 1.1.2. We consider a market model with a single risky asset defined on a probability space with a finite sample space  and a probability measure P that assigns strictly positive probability to each ! 2 . We let a WD min S.!/ !2

and

b WD max S.!/: !2

Show that the model does not admit arbitrage if and only if a < .1 C r/ < b.

}

Section 1.2 Absence of arbitrage and martingale measures

7

Exercise 1.1.3. Show that the existence of an arbitrage opportunity implies the following seemingly stronger condition. (a) There exists an arbitrage opportunity  such that    D 0. Show furthermore that the following condition implies the existence of an arbitrage opportunity. (b) There exists  2 Rd C1 such that    < 0 and   S  0 P -a.s. What can you say about the implication (a))(b)?

1.2

}

Absence of arbitrage and martingale measures

In this section, we are going to characterize those market models which do not admit any arbitrage opportunities. Such models will be called arbitrage-free. Definition 1.5. A probability measure P  is called a risk-neutral measure, or a martingale measure, if   Si i D E ; i D 0; 1; : : : ; d: (1.1) 1Cr Remark 1.6. In (1.1), the price of the i th asset is identified as the expectation of the discounted payoff under the measure P  . Thus, the pricing formula (1.1) can be seen as a classical valuation formula which does not take into account any risk aversion, in contrast to valuations in terms of expected utility which will be discussed in Section 2.3. This is why a measure P  satisfying (1.1) is called risk-neutral. The connection to martingales will be made explicit in Section 1.6. } The following basic result is sometimes called the “fundamental theorem of asset pricing” or, in short, FTAP. It characterizes arbitrage-free market models in terms of the set ® ¯ P WD P  j P  is a risk-neutral measure with P   P of risk-neutral measures which are equivalent to P . Recall that two probability measures P  and P are said to be equivalent (P   P ) if, for A 2 F , P  Œ A  D 0 if and only if P Œ A  D 0. This holds if and only if P  has a strictly positive density dP  =dP with respect to P ; see Appendix A.2. An equivalent risk-neutral measure is also called a pricing measure or an equivalent martingale measure. Theorem 1.7. A market model is arbitrage-free if and only if P ¤ ;. In this case, there exists a P  2 P which has a bounded density dP  =dP . We show first that the existence of a risk-neutral measure implies the absence of arbitrage.

8

Chapter 1 Arbitrage theory

Proof of the implication “(” of Theorem 1:7. Suppose that there exists a risk-neutral measure P  2 P . Take a portfolio  2 Rd C1 such that   S  0 P -a.s. and EŒ   S  > 0. Both properties remain valid if we replace P by the equivalent measure P  . Hence,   D

d X

i i

  D

iD0

d X iD0

E





i S i 1Cr

 DE





 S 1Cr

 > 0:

Thus,  cannot be an arbitrage opportunity. For the proof of the implication ) of Theorem 1.7, it will be convenient to introduce the random vector Y D .Y 1 ; : : : ; Y d / of discounted net gains: Y i WD

Si  i ; 1Cr

i D 1; : : : ; d:

(1.2)

With this notation, Lemma 1.4 implies that the absence of arbitrage is equivalent to the following condition: For  2 Rd :

  Y  0 P -a.s. H)   Y D 0 P -a.s.

(1.3)

Since Y i is bounded from below by  i , the expectation E  ΠY i  of Y i under any measure P  is well-defined, and so P  is a risk-neutral measure if and only if E  ΠY  D 0:

(1.4)

Here, E  Œ Y  is a shorthand notation for the d -dimensional vector with components E  Œ Y i , i D 1; : : : ; d . The assertion of Theorem 1.7 can now be read as follows: Condition (1.3) holds if and only if there exists some P   P such that E  Œ Y  D 0, and in this case, P  can be chosen such that the density dP  =dP is bounded. Proof of the implication “)” of Theorem 1:7. We have to show that (1.3) implies the existence of some P   P such that (1.4) holds and such that the density dP  =dP is bounded. We will do this first in the case in which EŒ jY j  < 1: Let Q denote the convex set of all probability measures Q  P with bounded densities dQ=dP , and denote by EQ Œ Y  the d -dimensional vector with components EQ Œ Y i , i D 1; : : : ; d . Due to our assumption jY j 2 L1 .P /, all these expectations are finite. Let C WD ¹EQ Œ Y  j Q 2 Qº;

9

Section 1.2 Absence of arbitrage and martingale measures

and note that C is a convex set in Rd : If Q1 , Q0 2 Q and 0  ˛  1, then Q˛ WD ˛Q1 C .1  ˛/Q0 2 Q and ˛EQ1 Œ Y  C .1  ˛/EQ0 Œ Y  D EQ˛ Œ Y ; which lies in C . Our aim is to show that C contains the origin. To this end, we suppose by way of contradiction that 0 … C . Using the “separating hyperplane theorem” in the elementary form of Proposition A.1, we obtain a vector  2 Rd such that   x  0 for all x 2 C , and such that   x0 > 0 for some x0 2 C . Thus,  satisfies EQ Œ   Y   0 for all Q 2 Q and EQ0 Œ   Y  > 0 for some Q0 2 Q. Clearly, the latter condition yields that P Œ   Y > 0  > 0. We claim that the first condition implies that   Y is P -a.s. non-negative. This fact will be a contradiction to our assumption (1.3) and thus will prove that 0 2 C . To prove the claim that   Y  0 P -a.s., let A WD ¹  Y < 0º, and define functions   1 1 'n WD 1   IA C  IAc : n n We take 'n as densities for new probability measures Qn : dQn 1 WD  'n ; dP EŒ 'n 

n D 2; 3; : : : :

Since 0 < 'n  1, it follows that Qn 2 Q, and thus that 0    EQn ΠY  D

1 EΠ  Y 'n : EΠ'n 

Hence, Lebesgue’s dominated convergence theorem yields that EŒ   Y I¹Y 0I d d see (5.43) in Chapter 5. The law m; of Ym; satisfies m. m; / D m for all > 0. Condition (c) of Corollary 2.61 implies that m; is decreasing in > 0 with respect to 0. Then Yn WD

1 Y C 2 B; .Y / n

and Yn ! Y =.Y / uniformly, whence EQX ΠY  D lim EQX ΠYn   1: .Y / n"1 On the other hand, X=.X / 2 C2  C yields the inequality EQX ΠX   c D 1: .X / We are now ready to complete the proof of the first main result in this section. Proof of Theorem 2:78. (a): By Remark 2.80, it suffices to consider the induced preference relation on X once the function u has been determined. According to Lemma 2.81 and the two Propositions 2.83 and 2.84, there exists a convex set Q  M1;f such that U.X / D min EQ Πu.X /  Q2Q

is a numerical representation of on X. This proves the first part of the assertion. (b): The assumption (2.34) applied to X  1 and Y  b < 1 gives that any sequence with Xn % 1 is such that Xn b for large enough n. We claim that this implies that U.Xn / % u.1/ D 1. Otherwise, U.Xn / would increase to some number a < 1. Since u is continuous and strictly increasing, we may take b such that a < u.b/ < 1. But then U.Xn / > U.b/ D u.b/ > a for large enough n, which is a contradiction.

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Chapter 2 Preferences

S In particular, we obtain that for any increasing sequence of events An 2 F with n An D  lim min QΠAn  D lim U.IAn / D 1: n"1 Q2Q

n"1

But this means that each Q 2 Q satisfies limn QΠAn  D 1, which is equivalent to the -additivity of Q. The continuity assumption (2.34), required for all Xn 2 X, is actually quite strong. In a topological setting, our discussion of risk measures in Chapter 4 will imply the following version of the representation theorem. Proposition 2.85. Consider a preference order as in Theorem 2:78. Suppose that  is a Polish space with Borel field F and that (2.34) holds if Xn and X are continuous. Then there exists a class of probability measures Q  M1 .; F / such that the induced preference order on X has the robust Savage representation U.X / D min EQ Πu.X /  for continuous X 2 X. Q2Q

Proof. As in the proof of Theorem 2.78, the continuity property of implies the corresponding continuity property of U , and hence of the functional  in (2.42). The result follows by combining Proposition 2.83, which reduces the representation of U to a representation of , with Proposition 4.27 applied to the coherent risk measure  WD . Now we consider an alternative setting where we fix in advance a reference measure P on .; F /. In this context, X will be identified with the space L1 .; F ; P /, and the representation of preferences will involve measures which are absolutely continuous with respect to P . Note, however, that this passage from measurable functions to equivalence classes of random variables in L1 .; F ; P /, and from arbitrary probability measures to absolutely continuous measures, involves a certain loss of robustness in the face of model uncertainty. Theorem 2.86. Let be a preference relation as in Theorem 2:78, and assume that X Y

whenever X D Y P -a.s.

(a) There exists a robust Savage representation of the form U.X / D inf EQ Πu.X / ; Q2Q

X 2 X;

where Q consists of probability measures on .; F / which are absolutely continuous with respect to P , if and only if satisfies the following condition of continuity from above: Y X and Xn & X P -a.s.

H)

Y Xn

P -a.s. for all large n.

109

Section 2.5 Robust preferences on asset profiles

(b) There exists a representation of the form U.X / D min EQ Πu.X / ; Q2Q

X 2 X;

where Q consists of probability measures on .; F / which are absolutely continuous with respect to P , if and only if satisfies the following condition of continuity from below: X Y and Xn % X P -a.s.

H)

Xn Y

P -a.s. for all large n.

Proof. As in the proof of Theorem 2.78, the continuity property of implies the corresponding continuity property of U , and hence of the functional  in (2.42). The results follow by combining Proposition 2.83, which reduces the representation of U to a representation of , with Corollary 4.37 and Corollary 4.38 applied to the coherent risk measure  WD . In the following two exercises we explore the impact of replacing the axiom of certainty independence by stronger requirements as discussed in Remark 2.77. Exercise 2.5.3. Show that the following conditions are equivalent: (a) The preference relation satisfies the following unrestricted independence axiom on XQ , Independence: For XQ ; YQ ; ZQ 2 XQ and ˛ 2 .0; 1 we have XQ YQ

”

Q ˛ XQ C .1  ˛/ZQ ˛ YQ C .1  ˛/Z:

(b) The functional  is additive: .X C Y / D .X / C .Y / for X 2 X. (c) The set Q has exactly one element Q 2 M1;f , and so U.X / D .u.X // admits the Savage representation U.X / D EQ Πu.X / ;

X 2 X:

}

Exercise 2.5.4. Two random variables X; Y 2 X will be called comonotone when ŒX.!/  X.! 0 /ŒY .!/  Y .! 0 /  0 for all pairs .!; ! 0 / 2   . Show that the following conditions are equivalent: (a) The preference relation satisfies the axiom of comonotonic independence (2.33). (b) The functional  is comonotonic in the sense that .X C Y / D .X / C .Y / whenever X; Y 2 X are comonotone. }

110

Chapter 2 Preferences

Now we discuss what happens if we replace the assumption of certainty independence by weak certainty independence as introduced in Remark 2.77. We thus assume from now on that is a preference relation on XQ satisfying the following conditions. 

Weak certainty independence.



Uncertainty aversion.



Monotonicity.



Continuity.

Exercise 2.5.5. Show that the restriction of to Mb .R/ satisfies the independence axiom of von Neumann–Morgenstern theory and hence admits a von Neumann–Morgenstern representation Z (2.45) u. / Q WD u.x/ .dx/; 2 Mb .R/; with a continuous and strictly increasing function u W R ! R.

}

For simplicity we will assume for the rest of this section that the function u in (2.45) has an unbounded range u.R/ containing zero. The assumption of an unbounded range is satisfied automatically if the restriction of to Mb .R/ is risk averse and u hence concave. Lemma 2.81 and Remark 2.82 imply that the numerical representation uQ in (2.45) admits a unique extension UQ W XQ ! R that is a numerical representation Q We define again of on all of X. U.X / WD UQ .ıX / for X 2 X. As in Remark 2.80, we see that XQ  ıX when X 2 X is defined as X.!/ WD c.XQ .!// with Z  1 c. / D u u d

denoting the certainty equivalent of a lottery 2 Mb .R/. In particular, the preference relation on XQ is uniquely determined by its restriction to X. We now analyze the structure of U in analogy to Proposition 2.83. Our next result shows that U is again of the form U.X / D .u.X // for a functional  W X ! R. However,  may no longer be positively homogeneous, since we have replaced certainty independence with weak certainty independence. Proposition 2.87. Under the above assumptions, there exists a unique functional  W X ! R such that UQ .X / D .u. Q XQ //

Q for all XQ 2 X,

and such that the following three properties are satisfied:

(2.46)

111

Section 2.5 Robust preferences on asset profiles 

Monotonicity: If Y .!/  X.!/ for all !, then .Y /  .X /.



Concavity: If 2 Œ0; 1 then . X C .1  /Y /  .X / C .1  /.Y /.



Cash invariance: .X C z/ D .X / C z for all z 2 R.

Proof. Let Xu denote the set of all X 2 X that take values in a compact subset of the range u.R/ of u. Since u is strictly increasing, we can define  on Xu via .X / WD UQ .ıu1 .X/ /;

X 2 Xu :

Then we have Q Q X//; UQ .XQ / D UQ .ıc.XQ / / D .u.c.XQ /// D .u.

(2.47)

and so (2.46) follows. Moreover,  is monotone on Xu due to our monotonicity assumption. We now prove that  is cash invariant on Xu . To this end, we assume first that u.R/ D R and take X 2 X and some z 2 R. We then let X0 WD u1 .2X /, z0 WD u1 .2z/, and y WD u1 .0/. Taking a > 0 such that a  X0 .!/  a for each !, we see as in (2.37) that there exists ˇ 2 Œ0; 1 such that 1 1 1 ˇ 1ˇ 1 ıX0 C ıy  .ıa C ıy / C .ıa C ıy / D C ıy ; 2 2 2 2 2 2 where D ˇıa C .1  ˇ/ıa . Using weak certainty independence, we may replace ıy by ız0 and obtain 12 .ıX0 C ız0 /  12 . C ız0 /. Hence, from (2.47), 1

  1 1 1  u.X0 / C u.z0 / D  uQ ıX0 C ız0 2 2 2 2 1  1  1 1 D UQ ıX0 C ız0 D UQ C ız0 2 2 2 2 1 1  1 1 D uQ C ız0 D u. / Q C u.z0 /: 2 2 2 2

.X C z/ D 

Cash invariance now follows from u.z0 / D 2z and the fact that 1 1 1 1 1  1  u. / Q D .u. / Q C u.ıy // D uQ C ıy D UQ C ıy 2 2 2 2 2 2  1  1 1 1 D UQ ıX0 C ıy D  u.X0 / C u.y/ D .X /: 2 2 2 2 Here we have again applied (2.47). If u.R/ is not equal to R it is sufficient to consider the cases in which u.R/ contains Œ0; 1/ or .1; 0 and to work with positive or negative quantities X and z, respectively. Then the preceding argument establishes the cash invariance of  on the spaces of positive or negative bounded measurable functions, and  can be extended by translation to the entire space of bounded measurable functions.

112

Chapter 2 Preferences

Now we prove the concavity of  by showing . 12 .X C Y //  12 .X / C 12 .Y /. This is enough due to Exercise 2.5.2. Let X0 WD u1 .X / and Y0 WD u1 .Y / and suppose first that .X / D .Y /. Then ıX0  ıY0 and uncertainty aversion implies that ZQ WD 12 ıX0 C 12 ıY0 ıX0 . Hence, by using (2.47), 

1

 Q  UQ .ıX0 / D .X / D 1 .X / C 1 .Y /: .X C Y / D UQ .Z/ 2 2 2

When .X / ¤ .Y / we let z WD .X /  .Y / so that Yz WD Y C z satisfies .Yz / D .X /. Hence, 

 1 1  1 1 1 1 1 .X CY / C z D  .X CYz /  .X /C .Yz / D .X /C .Y /C z: 2 2 2 2 2 2 2 2

1

A functional  W X ! R satisfying the properties of monotonicity, concavity, and cash invariance is sometimes called a concave monetary utility functional, and  WD  is called a convex risk measure. In Chapter 4 we will derive various representation results for convex risk measures. In particular it will follow from Theorem 4.16 that every concave monetary utility functional  W X ! R is of the form .X / D

min .EQ Œ X  C ˛.Q//;

Q2M1;f

X 2 X;

(2.48)

where the penalty function ˛ W M1;f ! R [ ¹C1º is bounded from below. Exercise 2.5.6. Prove the representation (2.48) for the case in which X is the set of all functions X W  ! R on a finite set . To this end, one can either use biduality (Theorem A.62) or, more directly, a separation argument as given in Proposition A.1. } Combining Proposition 2.87 with the representation (2.48) yields the final result of this section: Theorem 2.88. Consider a preference order on XQ satisfying the four properties of uncertainty aversion, weak certainty independence, monotonicity, and continuity. Assume moreover that the function u in (2.45) has an unbounded range u.R/. Then there exists a penalty function ˛ W M1;f ! R [ ¹C1º that is bounded from below such that  Z   Q UQ .XQ / D min EQ u.x/ XQ .; dx/ C ˛.Q/ ; XQ 2 X; Q2M1;f

is a numerical representation of .

113

Section 2.6 Probability measures with given marginals

2.6

Probability measures with given marginals

In this section, we study the construction of probability measures with given marginals. In particular, this will yield the missing implication in the characterization of uniform preference in Theorem 2.57, but the results in this section are of independent interest. We focus on the following basic question: Suppose 1 and 2 are two probability measures on S, and ƒ is a convex set of probability measures on S  S ; when does ƒ contain some which has 1 and 2 as marginals? The answer to this question will be given in a general topological setting. Let S be a Polish space, and let us fix a continuous function on S with values in Œ1; 1/. As in Section 2.2 and in Appendix A.6, we use as a gauge function in order to define the space of measures Z ° ± ˇ ˇ M1 .S/ WD 2 M1 .S/ .x/ .dx/ < 1 and the space of continuous test functions C .S/ WD ¹f 2 C.S/ j 9 c W jf .x/j  c  The

.x/ for all x 2 S º:

-weak topology on M1 .S/ is the coarsest topology such that Z M1 .S/ 3 7! f d

is a continuous mapping for all f 2 C .S/; see Appendix A.6 for details. On the product space S  S, we take the gauge function .x; y/ WD

.x/ C

.y/;

and define the corresponding set M1 .S S/, which will be endowed with the -weak topology. Theorem 2.89. Suppose that ƒ  M1 .S  S/ is convex and closed in the -weak topology, and that 1 , 2 are probability measures in M1 .S /. Then there exists some

2 ƒ with marginal distributions 1 and 2 if and only if Z Z Z f1 d 1 C f2 d 2  sup .f1 .x/ C f2 .y// .dx; dy/ for all f1 ; f2 2 C .S /.

2ƒ

Theorem 2.89 is due to V. Strassen [255]. Its proof boils down to an application of the Hahn–Banach theorem; the difficult part consists in specifying the right topological setting. First, let us investigate the relations between M1 .S  S / and M1 .S /. To this end, we define mappings i W M1 .S  S/ ! M1 .S /;

i D 1; 2,

114

Chapter 2 Preferences

that yield the i th marginal distribution of a measure 2 M1 .S  S /: Z Z Z Z f d.1 / D f .x/ .dx; dy/ and f d.2 / D f .y/ .dx; dy/; for all f 2 C .S/. Lemma 2.90. 1 and 2 are continuous and affine mappings from M1 .S  S / to M1 .S/. Proof. Suppose that n converges to in M1 .S S /. For f 2 C .S / let f .x; y/ WD f .x/. Clearly, f 2 C .S  S/, and thus Z Z Z Z f d.1 n / D f d n ! f d D f d.1 /: Therefore, 1 is continuous, and the same is true of 2 . Affinity is obvious. Now, let us consider the linear space E WD ¹˛  ˇ j ;  2 M1 .S /; ˛; ˇ 2 R º R spanned by M1 .S/. For  D ˛  ˇ 2 E the integral f d against a function f 2 C .S/ is well-defined and given by Z Z Z f d D ˛ f d  ˇ f d: R In particular,  7! f d is linear functional Ron E, so we R can regard C .S / as a subset of the algebraic dual E  of E. Note that f d D f d Q for all f 2 C .S / implies  D , Q i.e., C .S/ separates the points of E. We endow E with the coarsest topology .E; C .S// for which all maps Z E 3  7! f d; f 2 C .S /; are continuous; see Definition A.58. With this topology, E becomes a locally convex topological vector space. Lemma 2.91. Under the above assumptions, M1 .S / is a closed convex subset of E, and the relative topology of the embedding coincides with the -weak topology. Proof. The sets of the form U" .I f1 ; : : : ; fn / WD

Z Z n ° ˇ ± \ ˇ ˇˇ ˇ Q 2 E ˇ ˇ fi d  fi d Qˇ < " iD1

Section 2.6 Probability measures with given marginals

115

with  2 E, n 2 N, fi 2 C .S/, and " > 0 form a base of the topology .E; C .S //. Thus, if U  E is open, then every point 2 U \ M1 .S / possesses some neighborhood U" . I f1 ; : : : ; fn /  U . But U" . I f1 ; : : : ; fn / \ M1 .S / is an open neighborhood of in the -weak topology. Hence, U \ M1 .S / is open in the -weak topology. Similarly, one shows that every open set V  M1 .S / is of the form V D U \M1 .S/ for some open subset U of E. This shows that the relative topology M1 .S/ \ .E; C .S// coincides with the -weak topology. Moreover, M1 .S/ is an intersection of closed subsets of E Z Z ° ± ± \ ° ˇ ˇ M1 .S / D  2 E ˇ 2E ˇ 1 d D 1 \ f d  0 : f 2C .S/ f 0

Therefore, M1 .S/ is closed in E. Next, let E 2 denote the product space E  E. We endow E 2 with the product topology for which the sets U  V with U; V 2 .E; C .S // form a neighborhood base. Clearly, E 2 is a locally convex topological vector space. Lemma 2.92. Every continuous linear functional ` on E 2 is of the form Z Z `.1 ; 2 / D f1 d1 C f2 d2 for some f1 ; f2 2 C .S/. Proof. By linearity, ` is of the form `.1 ; 2 / D `1 .1 / C `2 .2 /, where `1 .1 / WD `.1 ; 0/ and `2 .2 / WD `.0; 2 /. By continuity of `, the set V WD `1 ..1; 1// is open in E 2 and contains the point .0; 0/. Hence, there are two open neighborhoods U1 ; U2  E such that .0; 0/ 2 U1  U2  V . Therefore, 0 2 Ui  `1 i ..1; 1//

for i D 1; 2,

i.e., 0 is an interior point of `1 i ..1; 1//. It follows that the `i are continuous at 0, which in view of their linearity implies continuity everywhere on E.R Finally, we may conclude from Proposition A.59 that each `i is of the form `i ./ D fi di for some fi 2 C .S /. The proof of the following lemma uses the characterization of compact sets for the -weak topology that is stated in Corollary A.47. It is here that we need our assumption that S is Polish.

116

Chapter 2 Preferences N

Lemma 2.93. If ƒ is a closed convex subset of M1 .S  S /, then Hƒ WD ¹.1 ; 2 / j 2 ƒº is a closed convex subset of E 2 . Proof. It is enough to show that Hƒ is closed in M1 .S /2 WD M1 .S /  M1 .S /, because Lemma 2.91 implies that the relative topology induced by E 2 on M1 .S /2 coincides with the product topology for the -weak topology. This is a metric topology by Corollary A.45. So let . n ; n / 2 Hƒ , n 2 N, be a sequence converging to some . ; / 2 M1 .S/2 in the product topology. Since both sequences . n /n2N and .n /n2N are relatively compact for the -weak topology, Corollary A.47 yields functions i W S ! Œ1; 1, i D 1; 2, such that sets of the form Kki WD ¹i  k º, k 2 N, are relatively compact in S and such that Z Z sup 1 d n C sup 2 dn < 1 : n2N

n2N

For each n, there exists n 2 ƒ such that 1 n D n and 2 n D n . Hence, if we let .x; y/ WD 1 .x/ C 2 .y/, then Z Z Z  sup  d n D sup 1 d n C 2 dn < 1 : n2N

n2N

Moreover, we claim that each set ¹  k º is relatively compact in S  S . To prove this claim, let li 2 N be such that li  sup

.x/ :

x2Kki

Then, since

 1, 2 1 ¹  k º  Kk1  Kk.1Cl [ Kk.1Cl  Kk2 ; 1/ 2/

and the right-hand side is a relatively compact set in S  S . It follows from Corollary A.47 that the sequence . n /n2N is relatively compact for the -weak topology. Any accumulation point of this sequence belongs to the closed set ƒ. Moreover, has marginal distributions and , since the projections i are continuous according to Lemma 2.90. Hence . ; / 2 Hƒ . Proof of Theorem 2:89. Let 1 , 2 2 M1 .S / be given. Since Hƒ is closed and convex in E 2 by Lemma 2.93, we may apply Theorem A.57 with B WD ¹. 1 ; 2 /º and C WD Hƒ : We conclude that . 1 ; 2 / … Hƒ if and only if there exists a linear functional ` on E 2 such that `. 1 ; 2 / >

sup

`.1 ; 2 / D sup `.1 ; 2 /:

.1 ;2 /2Hƒ

Applying Lemma 2.92 to ` completes the assertion.

2ƒ

117

Section 2.6 Probability measures with given marginals

We will now use Theorem 2.89 to deduce the remaining implication of Theorem 2.57. We consider here a more general, d -dimensional setting. To this end, let x D .x 1 ; : : : ; x d / and y D .y 1 ; : : : ; y d / be two d -dimensional vectors. We will say that x  y if x i  y i for all i . A function on Rd is called increasing, if it is increasing with respect to the partial order . Theorem 2.94. Suppose 1 and 2 are Borel probability measures on Rd with R jxj i .dx/ < 1 for i D 1; 2. Then the following assertions are equivalent: R R (a) f d 1  f d 2 for all increasing concave functions f on Rd . (b) There exists a probability space .; F ; P / with random variables X1 and X2 having distributions 1 and 2 , respectively, such that EΠX2 j X1   X1

P -a.s.

(c) There exists a kernel Q.x; dy/ on Rd such that Z y Q.x; dy/  x for all x 2 Rd Rd

and such that 2 D 1 Q. Proof. (a) ) (b): We will apply Theorem 2.89 with S WD Rd and with the gauge functions .x/ WD 1 C jxj and .x; y/ WD .x/ C .y/. We denote by Cb .Rd / the set of bounded and continuous functions on Rd . Let Z Z ± \ ° ˇ ƒ WD

2 M1 .Rd Rd / ˇ yf .x/ .dx; dy/  xf .x/ .dx; dy/ : f 2Cb .Rd /

Each single set of the intersection is convex and closed in M1 .Rd  Rd /, because the functions g.x; y/ WD yf .x/ and g.x; Q y/ WD xf .x/ belong to C .Rd  Rd / for f 2 Cb .S/. Therefore, ƒ itself is convex and closed. Suppose we can show that ƒ contains an element P that has 1 and 2 as marginal distributions. Then we can take  WD Rd  Rd with its Borel -algebra F , and let X1 and X2 denote the canonical projections on the first and the second components, respectively. By definition, Xi will have the distribution i , and EŒ EŒ X2 j X1 f .X1 /  D EŒ X2 f .X1 /   EŒ X1 f .X1 /  By monotone class arguments, we may thus conclude that EŒ X2 j X1   X1 so that the assertion will follow.

P -a.s.

for all f 2 Cb .Rd /.

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Chapter 2 Preferences

It remains to prove the existence of P . To this end, we will apply Theorem 2.89 with the set ƒ defined above. Take a pair f1 ; f2 2 C .Rd /, and let fQ2 .x/ WD inf¹g.x/ j g is concave, increasing, and dominates f2 º: Then fQ2 is concave, increasing, and dominates f2 . In fact, fQ2 is the smallest function with these properties. We have Z Z Z Z f1 d 1 C f2 d 2  f1 d 1 C fQ2 d 2 Z  .f1 C fQ2 / d 1  sup .f1 .x/ C fQ2 .x// DW r0 : x2Rd

We will establish the condition in Theorem 2.89 for our set ƒ by showing that for r < r0 we have Z r < sup .f1 .x/ C f2 .y// .dx; dy/:

2ƒ

To this end, let for z 2 Rd Z ° ± ˇ d ˇ ƒz WD  2 M1 .R / x .dx/  z and

²Z g2 .z/ WD sup

³ ˇ ˇ f2 d ˇ  2 ƒz :

Then g2 is increasing and g2 .z/  f2 .z/, because ız 2 ƒz . Moreover, if 1 2 ƒz1 and 2 2 ƒz2 , then ˛1 C .1  ˛/2 2 ƒ˛z1 C.1˛/z2 for ˛ 2 Œ0; 1. Therefore, g2 is concave, and we conclude that g2  fQ2 (recall that fQ2 is the smallest increasing and concave function dominating f2 ). Hence, r < f1 .z/ C g2 .z/ for some z 2 Rd , i.e., there exists some  2 ƒz such that the product measure WD ız ˝  satisfies Z Z r < f1 .z/ C f2 d D .f1 .x/ C f2 .y// .dx; dy/: But D ız ˝  2 ƒ. (b) ) (c): This follows as in the proof of the implication (f) ) (g) of Theorem 2.57 by using regular conditional distributions. (c) ) (a): As in the proof of (g) ) (a) of Theorem 2.57, this follows by an application of Jensen’s inequality.

119

Section 2.6 Probability measures with given marginals

By the same arguments as for Corollary 2.61, we obtain the following result from Theorem 2.94. Corollary 2.95. Suppose 1 and 2 are Borel probability measures on Rd such that R jxj i .dx/ < 1, for i D 1; 2. Then the following conditions are equivalent: R R (a) f d 1  f d 2 for all concave functions f on Rd . (b) There exists a probability space .; F ; P / with random variables X1 and X2 having distributions 1 and 2 , respectively, such that EΠX2 j X1  D X1

P -a.s.

(c) There exists a kernel Q.x; dy/ on Rd such that Z y Q.x; dy/ D x for all x 2 Rd .i.e., Q is a mean-preserving spread/ and such that 2 D 1 Q. We conclude this section with a generalization of Theorem 2.68. Let S be a Polish space which is endowed with a preference order . We will assume that is continuous in the sense of Definition 2.8. A function on S will be called increasing if it is increasing with respect to . Theorem 2.96. For two Borel probability measures 1 and 2 on S , the following conditions are equivalent. R R (a) f d 1  f d 2 for all bounded, increasing, and measurable functions f on S. (b) There exists a probability space .; F ; P / with random variables X1 and X2 having distributions 1 and 2 , respectively, such that X1 X2 P -a.s. (c) There exists a kernel Q on S such that 2 D 1 Q and Q.x; ¹y j x yº / D 1 for all x 2 S . Proof. (a) ) (b): We will apply Theorem 2.89 with the gauge function  1, so that M1 .S/ is just the space M1 .S/ of all Borel probability measures on S with the usual weak topology. Then  2 which is equivalent to taking WD 1. Let M WD ¹.x; y/ 2 S  S j x yº: This set M is closed in S  S by Proposition 2.11. Hence, the portmanteau theorem in the form of Theorem A.39 implies that the convex set ƒ WD ¹ 2 M1 .S  S/ j .M / D 1º

120

Chapter 2 Preferences

is closed in M1 .S  S/. For f2 2 Cb .S/, let fQ2 .x/ WD sup¹f2 .y/ j x yº: Then fQ2 is bounded, increasing, and dominates f2 . Therefore, if f1 2 Cb .S /, Z Z Z Z f1 d 1 C f2 d 2  f1 d 1 C fQ2 d 2 Z  .f1 C fQ2 / d 1  sup .f1 .x/ C fQ2 .x// x2S

D sup .f1 .x/ C f2 .y//: xy

If x y, then the product measure WD ıx ˝ ıy is contained in ƒ, and so Z sup .f1 .x/ C f2 .y// D sup .f1 .x/ C f2 .y// .dx; dy/: xy

2ƒ

Hence, all assumptions of Theorem 2.89 are satisfied, and we conclude that there exists a probability measure P 2 ƒ with marginals 1 and 2 . Taking  WD S  S and Xi as the projection on the i th coordinate finishes the proof of (a) ) (b). (b) ) (c) follows as in the proof of Theorem 2.57 by using regular conditional distributions. (c) ) (a) is proved as the corresponding implication of Theorem 2.68.

Chapter 3

Optimality and equilibrium

Consider an investor whose preferences can be expressed in terms of expected utility. In Section 3.1, we discuss the problem of constructing a portfolio which maximizes the expected utility of the resulting payoff. The existence of an optimal solution is equivalent to the absence of arbitrage opportunities. This leads to an alternative proof of the “fundamental theorem of asset pricing”, and to a specific choice of an equivalent martingale measure defined in terms of marginal utility. Section 3.2 contains a detailed case study describing the interplay between exponential utility and relative entropy. In Section 3.3, the optimization problem is formulated for general contingent claims. Typically, optimal profiles will be non-linear functions of a given market portfolio, and this is one source of the demand for financial derivatives. Section 3.6 introduces the idea of market equilibrium. Prices of risky assets will no longer be given in advance; they will be derived as equilibrium prices in a microeconomic setting, where different agents demand contingent claims in accordance with their preferences and with their budget constraints.

3.1

Portfolio optimization and the absence of arbitrage

Let us consider the one-period market model of Section 1.1 in which d C 1 assets are priced at time 0 and at time 1. Prices at time 0 are given by the price system  D . 0 ; / D . 0 ;  1 ; : : : ;  d / 2 RdCC1 ; prices at time 1 are modeled by the price vector S D .S 0 ; S/ D .S 0 ; S 1 ; : : : ; S d / consisting of non-negative random variables S i defined on some probability space .; F ; P /. The 0th asset models a riskless bond, and so we assume that  0 D 1 and

S0  1 C r

for some constant r > 1. At time t D 0, an investor chooses a portfolio  D . 0 ; / D . 0 ;  1 ; : : : ;  d / 2 Rd C1

122

Chapter 3 Optimality and equilibrium

where  i represents the amount of shares of the i th asset. Such a portfolio  requires an initial investment    and yields at time 1 the random payoff   S . Consider a risk-averse economic agent whose preferences are described in terms of a utility function u, Q and who wishes to invest a given amount w into the financial market. Recall from Definition 2.35 that a real-valued function uQ is called a utility function if it is continuous, strictly increasing, and strictly concave. A rational choice of the investor’s portfolio  D . 0 ; / will be based on the expected utility EŒ u. Q  S/ 

(3.1)

of the payoff   S at time 1, where the portfolio  satisfies the budget constraint     w:

(3.2)

Thus, the problem is to maximize the expected utility (3.1) among all portfolios  2 Rd C1 which satisfy the budget constraint (3.2). Here we make the implicit assumption that the payoff   S is P -a.s. contained in the domain of definition of the utility function u. Q In a first step, we remove the constraint (3.2) by considering instead of (3.1) the expected utility of the discounted net gain  S   D Y 1Cr earned by a portfolio  D . 0 ; /. Here Y is the d -dimensional random vector with components Si   i ; i D 1; : : : ; d: Yi D 1Cr For any portfolio  with    < w, adding the risk-free investment w     would lead to the strictly better portfolio . 0 Cw  ; /. Thus, we can focus on portfolios  which satisfy    D w, and then the payoff is an affine function of the discounted net gain   S D .1 C r/.  Y C w/: Moreover, for any  2 Rd there exists a unique numéraire component  0 2 R such that the portfolio  WD . 0 ; / satisfies    D w. Let u denote the following transformation of our original utility function u: Q u.y/ WD u..1 Q C r/.y C w//: Note that u is again a utility function, and that CARA and (shifted) HARA utility functions are transformed into utility functions in the same class.

Section 3.1 Portfolio optimization and the absence of arbitrage

123

Clearly, the original constrained utility maximization problem is equivalent to the unconstrained problem of maximizing the expected utility EŒ u.  Y /  among all  2 Rd such that   Y is contained in the domain D of u. Assumption 3.1. We assume one of the following two cases: (a) D D R. In this case, we will admit all portfolios  2 Rd , but we assume that u is bounded from above. (b) D D Œa; 1/ for some a < 0. In this case, we only consider portfolios which satisfy the constraint   Y  a P -a.s., and we assume that the expected utility generated by such portfolios is finite, i.e., EŒ u.  Y /  < 1 for all  2 Rd with   Y  a P -a.s. Remark 3.2. Part (a) of this assumption is clearly satisfied in the case of an exponential utility function u.x/ D 1  e ˛x . Domains of the form D D Œa; 1/ appear, for example, in the case of (shifted) HARA utility functions u.x/ D log.x  b/ for b < a and u.x/ D 1 .x  c/ for c  a and 0 <  < 1. The integrability assumption in (b) holds if EŒ jY j  < 1, because any concave function is bounded from above by an affine function. } In order to simplify notations, let us denote by S.D/ WD ¹ 2 Rd j   Y 2 D P -a.s.º the set of admissible portfolios for D. Clearly, S.D/ D Rd if D D R. Our aim is to find some   2 S.D/ which is optimal in the sense that it maximizes the expected utility EŒ u. Y /  among all  2 S.D/. In this case,   will be an optimal investment strategy into the risky assets. Complementing   with a suitable numéraire component   0 yields a portfolio  D . 0 ;   / which maximizes the expected utility EŒ u. Q  S/  under the budget constraint    D w. Our first result in this section will relate the existence of such an optimal portfolio to the absence of arbitrage opportunities. Theorem 3.3. Suppose that the utility function u W D ! R satisfies Assumption 3:1. Then there exists a maximizer of the expected utility EŒ u.  Y / ;

 2 S.D/;

if and only if the market model is arbitrage-free. Moreover, there exists at most one maximizer if the market model is non-redundant in the sense of Definition 1:15.

124

Chapter 3 Optimality and equilibrium

Proof. The uniqueness part of the assertion follows immediately from the strict concavity of the function  7! EŒ u.  Y /  for non-redundant market models. As to existence, we may assume without loss of generality that our model is non-redundant. If the non-redundance condition (1.8) does not hold, then we define a linear space N  Rd by N WD ¹ 2 Rd j   Y D 0 P -a.s.º: Clearly, Y takes P -a.s. values in the orthogonal complement N ? of N . Moreover, the no-arbitrage condition (1.3) holds for all  2 Rd if and only if it is satisfied for all  2 N ? . By identifying N ? with some Rn , we arrive at a situation in which the non-redundance condition (1.8) is satisfied and where we may apply our result for non-redundant market models. If the model admits arbitrage opportunities, then a maximizer   of the expected utility EŒ u.  Y /  cannot exist: Adding to   some non-zero  2 Rd for which   Y  0 P -a.s., which exists by Lemma 1.4, would yield a contradiction to the optimality of   , because then EŒ u.   Y /  < EŒ u..  C /  Y / : From now on, we assume that the market model is arbitrage-free. Let us first consider the case in which D D Œa; 1/ for some a 2 .1; 0/. Then S.D/ is compact. In order to prove this claim, suppose by way of contradiction that .n / is a sequence in S.D/ such that jn j ! 1. By choosing a subsequence if necessary, we may assume that n WD n =jn j converges to some unit vector  2 Rd . Clearly, n  Y a  lim D0 n"1 jn j n"1 jn j

  Y D lim

P -a.s.,

and so non-redundance implies that  WD .  ; / is an arbitrage-opportunity. In the next step, we show that our assumptions guarantee the continuity of the function S.D/ 3  7! EΠu.  Y / ; which, in view of the compactness of S.D/, will imply the existence of a maximizer of the expected utility. To this end, it suffices to construct an integrable random variable which dominates u.  Y / for all  2 S.D/. Define  2 Rd by i WD 0 _ max  i < 1: 2S.D/

Then,   S    S for  2 S.D/, and hence  Y D

S  S     0 ^ min    0 : 1Cr 1Cr  0 2S.D/

Note that   Y is bounded from below by    and that there exists some ˛ 2 .0; 1 such that ˛   < jaj. Hence ˛ 2 S.D/, and so EŒ u.˛  Y /  < 1. Applying

Section 3.1 Portfolio optimization and the absence of arbitrage

125

Lemma 3.4 below first with b WD ˛   and then with b WD 0 ^ min 0 2S.D/    0 shows that    S 0 E u  0 ^ min    < 1: 1Cr  0 2S.D/ This concludes the proof of the theorem in case D D Œa; 1/. Let us now turn to the case of a utility function on D D R which is bounded from above. We will reduce the assertion to a general existence criterion for minimizers of lower semicontinuous convex functions on Rd , given in Lemma 3.5 below. It will be applied to the convex function h./ WD EŒ u.  Y / . We must show that h is lower semicontinuous. Take a sequence .n /n2N in Rd converging to some . By part (a) of Assumption 3.1, the random variables u.n  Y / are uniformly bounded from below, and so we may apply Fatou’s lemma: lim inf h.n / D lim inf EŒ u.n  Y /   EŒ u.  Y /  D h./: n"1

n"1

Thus, h is lower semicontinuous. By our non-redundance assumption, h is strictly convex and admits at most one minimizer. We claim that the absence of arbitrage opportunities is equivalent to the following condition: lim h.˛ / D C1

˛"1

for all non-zero  2 Rd .

(3.3)

This is just the condition (3.4) required in Lemma 3.5. It follows from (1.3) and (1.8) that a non-redundant market model is arbitrage-free if and only if each non-zero  2 Rd satisfies P Œ  Y < 0 > 0. Since the utility function u is strictly increasing and concave, the set ¹  Y < 0º can be described as ¹  Y < 0º D

®

lim u.˛   Y / D 1

¯

˛"1

for  2 Rd .

The probability of the right-hand set is strictly positive if and only if lim EŒ u.˛  Y /  D 1;

˛"1

because u is bounded from above. This observation proves that the absence of arbitrage opportunities is equivalent to the condition (3.3) and completes the proof. Lemma 3.4. If D D Œa; 1/, b < jaj, 0 < ˛  1, and X is a non-negative random variable, then EŒ u.˛X  b/  < 1

H)

EΠu.X /  < 1:

126

Chapter 3 Optimality and equilibrium

Proof. As in (A.1) in the proof of Proposition A.4 we obtain that u.X /  u.0/ u.˛X /  u.0/ u.˛X  b/  u.b/   : X 0 ˛X  0 ˛X  b  .b/ Multiplying by X shows that u.X / can be dominated by a multiple of u.˛X  b/ plus some constant. Lemma 3.5. Suppose h W Rd ! R [ ¹C1º is a convex and lower semicontinuous function with h.0/ < 1. Then h attains its infimum provided that lim h.˛ / D C1

˛"1

for all non-zero  2 Rd .

(3.4)

Moreover, if h is strictly convex on ¹h < 1º, then also the converse implication holds: the existence of a minimizer implies (3.4). Proof. First suppose that (3.4) holds. We will show below that for c > inf h the level sets ¹x j h.x/  cº of h are bounded and hence compact. Once the compactness of the level sets is established, it follows that the set \ ¹x 2 Rd j h.x/ D inf hº D ¹x 2 Rd j h.x/  cº c>inf h

of minimizers of h is non-empty as an intersection of decreasing and non-empty compact sets. Suppose c > inf h is such that the level set ¹h  cº is not compact, and take a sequence .xn / in ¹h  cº such that jxn j ! 1. By passing to a subsequence if necessary, we may assume that xn =jxn j converges to some non-zero . For any ˛ > 0,       ˛ xn ˛ h.˛/  lim inf h ˛ xn C 1  D lim inf h 0 jxn j jxn j jxn j n"1 n"1     ˛ ˛ cC 1 h.0/  lim inf jxn j jxn j n"1 D h.0/: Thus, we arrive at a contradiction to condition (3.4). This completes the proof of the existence of a minimizer under assumption (3.4). In order to prove the converse implication, suppose that the strictly convex function h has a minimizer x  but that there exists a non-zero  2 Rd violating (3.4), i.e., there exists a sequence .˛n /n2N and some c < 1 such that ˛n " 1 but h.˛n /  c for all n. Let xn WD n x  C .1  n /˛n 

Section 3.1 Portfolio optimization and the absence of arbitrage

127

where n is such that jx   xn j D 1, which is possible for all large enough n. By the compactness of the Euclidean unit sphere centered in x  , we may assume that xn converges to some x. Then necessarily jx  x  j D 1. As ˛n  diverges, we must have that n ! 1. By using our assumption that h.˛n / is bounded, we obtain h.x/  lim inf h.xn /  lim . n h.x  / C .1  n /h.˛n // D h.x  /: n"1

n"1

Hence, x is another minimizer of h besides x  , contradicting the strict convexity of h. Thus, (3.4) must hold if the strictly convex function h takes on its infimum. Remark 3.6. Note that the proof of Theorem 3.3 under Assumption 3.1 (a) did not use the fact that the components of Y are bounded from below. The result remains true for arbitrary Y . } We turn now to a characterization of the solution   of our utility maximization problem for continuously differentiable utility functions. Proposition 3.7. Let u be a continuously differentiable utility function on D such that EΠu.  Y /  is finite for all  2 S.D/. Suppose that   is a solution of the utility maximization problem, and that one of the following two sets of conditions is satisfied: 

u is defined on D D R and is bounded from above.



u is defined on D D Œa; 1/, and   is an interior point of S.D/.

Then u0 .   Y / jY j 2 L1 .P /; and the following “first-order condition” holds: EŒ u0 .   Y / Y  D 0:

(3.5)

Proof. For  2 S.D/ and " 2 .0; 1 let " WD " C .1  "/  , and define " WD

u."  Y /  u.   Y / : "

The concavity of u implies that "  ı for "  ı, and so " % u0 .   Y / .    /  Y

as " # 0.

Note that our assumptions imply that u.Y / 2 L1 .P / for all  2 S.D/. In particular, we have 1 2 L1 .P /, so that monotone convergence and the optimality of   yield that 0  EΠ"  % EΠu0 .   Y / .    /  Y  as " # 0. (3.6) In particular, the expectation on the right-hand side of (3.6) is finite.

128

Chapter 3 Optimality and equilibrium

Both sets of assumptions imply that   is an interior point of S.D/. Hence, we deduce from (3.6) by letting  WD     that EŒ u0 .   Y /   Y   0 for all  in a small ball centered in the origin of Rd . Replacing  by  shows that the expectation must vanish. Remark 3.8. Let us comment on the assumption that the optimal   is an interior point of S.D/: (a) If the non-redundance condition (1.8) is not satisfied, then either each or none of the solutions to the utility maximization problem is contained in the interior of S.D/. This can be seen by using the reduction argument given at the beginning of the proof of Theorem 3.3. (b) Note that   Y is bounded from below by    in case  has only non-negative components. Thus, the interior of S.D/ is always non-empty. (c) As shown by the following example, the optimal   need not be contained in the interior of S.D/ and, in this case, the first-order condition (3.5) will generally fail. } Example 3.9. Take r D 0, and let S 1 be integrable but unbounded. We choose D D Œa; 1/ with a WD  1 , and we assume that P Œ S 1  "  > 0 for all " > 0. Then S.D/ D Œ0; 1. If 0 < EŒ S 1  <  1 then Example 2.40 shows that the optimal investment is given by   D 0, and so   lies in the boundary of S.D/. Thus, if u is sufficiently smooth, EŒ u0 .   Y / Y  D u0 .0/.EŒ S 1    1 / < 0: The intuitive reason for this failure of the first-order condition is that taking a short position in the asset would be optimal as soon as EŒ S 1  <  1 . This choice, however, is ruled out by the constraint  2 S.D/. } Proposition 3.7 yields a formula for the density of a particular equivalent riskneutral measure. Recall that P  is risk-neutral if and only if E  Œ Y  D 0. Corollary 3.10. Suppose that the market model is arbitrage-free and that the assumptions of Proposition 3:7 are satisfied for a utility function u W D ! R and an associated maximizer   of the expected utility EŒu.  Y /. Then dP  u0 .   Y / D dP EŒu0 .   Y / defines an equivalent risk-neutral measure.

(3.7)

Section 3.1 Portfolio optimization and the absence of arbitrage

129

Proof. Proposition 3.7 states that u0 .   Y /Y is integrable with respect to P and that its expectation vanishes. Hence, we may conclude that P  is an equivalent riskneutral measure if we can show that P  is well-defined by (3.7), i.e., if u0 .   Y / 2 L1 .P /. Let ´ for D D Œa; 1/, u0 .a/ c WD sup¹u0 .x/ j x 2 D and jxj  j  jº  0  u .j j/ for D D R, which is finite by our assumption that u is continuously differentiable on all of D. Thus, 0  u0 .   Y /  c C u0 .   Y /jY j  I¹jY j1º ; and the right-hand side has a finite expectation. Remark 3.11. Corollary 3.10 yields an independent and constructive proof of the “fundamental theorem of asset pricing” in the form of Theorem 1.7: Suppose that the model is arbitrage-free. If Y is P -a.s. bounded, then so is u.   Y /, and the measure P  of (3.7) is an equivalent risk-neutral measure with a bounded density dP  =dP . If Y is unbounded, then we may consider the bounded random vector YQ WD

Y ; 1 C jY j

which also satisfies the no-arbitrage condition (1.3). Let Q  be a maximizer of the expected utility EΠu.  YQ / . Then an equivalent risk-neutral measure P  is defined through the bounded density dP  u0 . Q   YQ / WD c  ; dP 1 C jY j where c is an appropriate normalizing constant.

}

Example 3.12. Consider the exponential utility function u.x/ D 1  e ˛x with constant absolute risk aversion ˛ > 0. The requirement that EŒ u.  Y /  is finite is equivalent to the condition EŒ e Y  < 1

for all  2 Rd .

If   is a maximizer of the expected utility, then the density of the equivalent riskneutral measure P  in (3.7) takes the particular form 

dP  e ˛ Y D : dP EŒ e ˛  Y 

130

Chapter 3 Optimality and equilibrium

In fact, P  is independent of ˛ since   maximizes the expected utility 1EŒ e ˛Y  if and only if  WD ˛  is a minimizer of the moment generating function Z. / WD EŒ e Y ;

2 Rd ;

of Y . In Corollary 3.25 below, the measure P  will be characterized by the fact that it minimizes the relative entropy with respect to P among the risk-neutral measures in P ; see Definition 3.20 below. }

3.2

Exponential utility and relative entropy

In this section we give a more detailed study of the problem of portfolio optimization with respect to a CARA utility function u.x/ D 1  e ˛x for ˛ > 0. As in the previous Section 3.1, the problem is to maximize the expected utility EŒ u.  Y /  of the discounted net gain   Y earned by an investment into risky assets. The key assumption for this problem is that for all  2 Rd .

EΠu.  Y /  > 1

(3.8)

Recall from Example 3.12 that the maximization of EΠu.  Y /  is reduced to the minimization of the moment generating function Z. / WD EΠe Y ;

2 Rd ;

which does not depend on the risk aversion ˛. The key assumption (3.8) is equivalent to the condition that Z. / < 1 for all 2 Rd . (3.9) Throughout this section, we will always assume that (3.9) holds. But we will not need the assumption that Y is bounded from below (which in our financial market model follows from assuming that asset prices are non-negative); all results remain true for general random vectors Y ; see also Remarks 1.9 and 3.6. Lemma 3.13. The condition (3.9) is equivalent to EŒ e ˛jY j  < 1

for all ˛ > 0.

131

Section 3.2 Exponential utility and relative entropy

Proof. Clearly, the condition in the statement of the lemma implies (3.9). To prove P the converse assertion, take a constant c > 0 such that jxj  c diD1 jx i j for x 2 Rd . By Hölder’s inequality, d d h  X i Y i EŒ e ˛jY j   E exp ˛c jY i j  EŒ e ˛cd jY j 1=d : iD1

iD1

In order to show that the i th factor on the right is finite, take 2 Rd such that

i D ˛cd and j D 0 for j ¤ i . With this choice, i

EŒ e ˛cd jY j   EŒ e Y  C EŒ e  Y ; which is finite by (3.9). Definition 3.14. The exponential family of P with respect to Y is the set of measures ¹P j 2 Rd º defined via

dP

e Y D : dP Z. /

Example 3.15. Suppose that the risky asset S 1 has under P a Poisson distribution with parameter ˛ > 0, i.e., S 1 takes values in ¹0; 1; : : : º and satisfies P Œ S 1 D k  D e ˛

˛k ; kŠ

k D 0; 1; : : : :

Then (3.9) is satisfied for Y WD S 1   1 , and S 1 has under P a Poisson distribution with parameter e ˛. Hence, the exponential family of P generates the family of all Poisson distributions. } Example 3.16. Let Y have a standard normal distribution N.0; 1/. Then (3.9) is satisfied, and the distribution of Y under P is equal to the normal distribution N. ; 1/ with mean and variance 1. } Remark 3.17. Two parameters and 0 in Rd determine the same element in the exponential family of P if and only if .  0 /  Y D 0 P -almost surely. It follows that the mapping

7! P

is injective provided that the non-redundance condition holds in the form   Y D 0 P -a.s.

H)

 D 0:

(3.10) }

132

Chapter 3 Optimality and equilibrium

In the sequel, we will be interested in the barycenters of the members of the exponential family of P with respect to Y . We denote m. / WD E ΠY  D

1 EΠY e Y ; Z. /

2 Rd :

The next lemma shows that m. / can be obtained as the gradient of the logarithmic moment generating function. Lemma 3.18. Z is a smooth function on Rd , and the gradient of log Z at is the expectation of Y under P

.r log Z/. / D E ΠY  D m. /: Moreover, the Hessian of log Z at equals the covariance matrix .covP .Y i ; Y j //i;j of Y under the measure P

@2 log Z. / D cov.Y i ; Y j / D E Œ Y i Y j   E Œ Y i E Œ Y j : @ i @ j P In particular, log Z is convex. Proof. Observe that ˇ x ˇ ˇ @e ˇ i x ˇ ˇ  expŒ.1 C j j/  jxj: ˇ @ i ˇ D jx j e Hence, Lemma 3.13 and Lebesgue’s dominated convergence theorem justify the interchanging of differentiation and integration (see the “differentiation lemma” in [21], § 16, for details). The following corollary summarizes the results we have obtained so far. Recall from Section 1.5 the notion of the convex hull ./ of the support of a measure  on Rd and the definition of the relative interior ri C of a convex set C . Corollary 3.19. Denote by WD P ı Y 1 the distribution of Y under P . Then the function

7!  m0  log Z. / takes on its maximum if and only if m0 is contained in the relative interior of the convex hull of the support of , i.e., if and only if m0 2 ri . /: In this case, any maximizer  satisfies m0 D m.  / D E  Œ Y : In particular, the set ¹m. / j 2 Rd º coincides with ri . /. Moreover, if the non-redundance condition (3.10) holds, then there exists at most one maximizer  .

Section 3.2 Exponential utility and relative entropy

133

Proof. Taking YQ WD Y  m0 reduces the problem to the situation where m0 D 0. Applying Theorem 3.3 with the utility function u.z/ D 1  e z shows that the existence of a maximizer  of  logZ is equivalent to the absence of arbitrage opportunities. Corollary 3.10 states that m.  / D 0 and that 0 belongs to Mb . /, where Mb . / was defined in Lemma 1.44. An application of Theorem 1.49 completes the proof. It will turn out that the maximization problem of the previous corollary is closely related to the following concept. Definition 3.20. The relative entropy of a probability measure Q with respect to P is defined as ´  dQ E dQ if Q P , log dP dP H.QjP / WD C1 otherwise. Remark 3.21. Jensen’s inequality applied to the strictly convex function h.x/ D x log x yields    dQ H.QjP / D E h  h.1/ D 0; (3.11) dP with equality if and only if Q D P . } Example 3.22. Let  be a finite set and F be its power set. Every probability Q on .; F / is absolutely continuous with respect to the uniform distribution P . Let us denote Q.!/ WD QŒ ¹!º . Clearly, X X Q.!/ D Q.!/ log Q.!/ log Q.!/ C log jj: H.QjP / D P .!/ !2

!2

The quantity H.Q/ WD 

X

Q.!/ log Q.!/

!2

is usually called the entropy of Q. Observe that H.P / D log jj, so that H.QjP / D H.P /  H.Q/: Since the left-hand side is non-negative by (3.11), the uniform distribution P has maximal entropy among all probability distributions on .; F /. } Example 3.23. Let D N.m; 2 / denote the normal distribution with mean m and Q Q 2 / variance 2 on R. Then, for Q D N.m;   d Q .x  m/ Q 2 .x  m/2 .x/ D exp  C ; d

Q 2 Q 2 2 2 and hence

    1 Q 2 Q 2 Q 2 1 mm H. j / Q D  log 2  1 C : 2 2 2

}

134

Chapter 3 Optimality and equilibrium

The following result shows that P is the unique minimizer of the relative entropy H.QjP / among all probability measures Q with EQ ΠY  D E ΠY . Theorem 3.24. Let m0 WD m.P 0 / for some given 0 2 Rd . Then, for any probability measure Q on .; F / such that EQ ΠY  D m0 , H.QjP /  H.P 0 jP / D 0  m0  log Z. 0 /; and equality holds if and only if Q D P 0 . Moreover, 0 maximizes the function

 m0  log Z. / over all 2 Rd . Proof. Let Q be a probability measure on .; F / such that EQ ΠY  D m0 . We show first that for all 2 Rd H.QjP / D H.QjP / C  m0  log Z. /:

(3.12)

To this end, note that both sides of (3.12) are infinite if Q 6 P . Otherwise dQ dP

dQ e Y dQ D D ;   dP dP dP dP Z. / and taking logarithms and integrating with respect to Q yields (3.12). Since H.QjP /  0 according to (3.11), we get from (3.12) that H.QjP /   m0  log Z. /

(3.13)

for all 2 Rd and all measures Q such that EQ ΠY  D m0 . Moreover, equality holds in (3.13) if and only if H.QjP / D 0, which is equivalent to Q D P . In this case,

must be such that m. / D m0 . In particular, for any such H.P jP / D  m0  log Z. /: Thus, 0 maximizes the right-hand side of (3.13), and P 0 minimizes the relative entropy on the set M0 WD ¹Q j EQ Œ Y  D m0 º: But the relative entropy H.QjP / is a strictly convex functional of Q, and so it can have at most one minimizer in the convex set M0 . Thus, any with m. / D m0 induces the same measure P 0 . Taking m0 D 0 in the preceding theorem yields a special equivalent risk-neutral measure in our financial market model, namely the entropy-minimizing risk-neutral measure. Sometimes it is also called the Esscher transform of P . Recall our assumption (3.9).

135

Section 3.2 Exponential utility and relative entropy

Corollary 3.25. Suppose the market model is arbitrage-free. Then there exists a unique equivalent risk-neutral measure P  2 P which minimizes the relative entropy H.PO jP / over all PO 2 P . The density of P  is of the form 

e Y dP  ; D dP EŒ e  Y  where  denotes a minimizer of the moment generating function EŒ e Y  of Y . Proof. This follows immediately from Corollary 3.19 and Theorem 3.24. By combining Theorem 3.24 with Remark 3.17, we obtain the following corollary. It clarifies the question of uniqueness in the representation of points in the relative interior of .P ı Y 1 / as barycenters of the exponential family. Corollary 3.26. If the non-redundance condition (3.10) holds, then

7! m. / is a bijective mapping from Rd to ri .P ı Y 1 /. Remark 3.27. It follows from Corollary 3.19 and Theorem 3.24 that for all m 2 ri .P ı Y 1 / min

EQ ΠY Dm

H.QjP / D max Π m  log Z. /:

2Rd

(3.14)

Here, the right-hand side is the Fenchel–Legendre transform of the convex function log Z evaluated at m 2 Rd . } The following theorem shows that the variational principle (3.14) remains true for all m 2 Rd , if we replace “min” and “max” by “inf” and “sup”. Theorem 3.28. For m 2 Rd H.QjP / D sup Œ  m  log Z. /:

inf EQ ΠY Dm

2Rd

The proof of this theorem relies on the following two general lemmas. Lemma 3.29. For any probability measure Q, H.QjP / D

.EQ ΠZ   log EΠe Z /

sup Z2L1 .;F

;P /

D sup¹EQ Œ Z   log EŒ e Z  j e Z 2 L1 .P /º: The second supremum is attained by Z WD log dQ if Q P . dP

(3.15)

136

Chapter 3 Optimality and equilibrium

Proof. We first show  in (3.15). To this end, we may assume that H.QjP / < 1. For Z with e Z 2 L1 .P / let P Z be defined by dP Z eZ : D dP EΠe Z  Then P Z is equivalent to P and log

dQ dQ dP Z D log : C log dP dP dP Z

Integrating with respect to Q gives H.QjP / D H.QjP Z / C EQ Œ Z   log EŒ e Z : Since H.QjP Z /  0 by (3.11), we have proved that H.QjP / is larger than or equal to both suprema on the right of (3.15). To prove the reverse inequality, consider first the case Q 6 P . Take Zn WD nIA where A is such that QŒ A  > 0 and P Œ A  D 0. Then, as n " 1, EQ Œ Zn   log EŒ e Zn  D n  QŒ A  ! 1 D H.QjP /: Now suppose that Q P with density ' D dQ=dP . Then Z WD log ' satisfies e Z 2 L1 .P / and H.QjP / D EQ Œ Z   log EŒ e Z : For the first identity we use an approximation argument. Let Zn D .n/_.log '/^n. We split the expectation EŒ e Zn  according to the two sets ¹'  1º and ¹' < 1º. Using monotone convergence for the first integral and dominated convergence for the second yields EŒ e Zn  ! EŒ e log '  D 1: Since x log x  1=e, we have 'Zn  1=e uniformly in n, and Fatou’s lemma yields lim inf EQ Œ Zn  D lim inf EŒ 'Zn   EŒ ' log '  D H.QjP /: n"1

n"1

Putting both facts together shows lim inf.EQ ΠZn   log EΠe Zn /  H.QjP /; n"1

and the inequality  in (3.15) follows.

Section 3.2 Exponential utility and relative entropy

137

Remark 3.30. The preceding lemma shows that the relative entropy is monotone with respect to an increase of the underlying -algebra: Let P and Q be two probability measures on a measurable space .; F /, and denote by H.QjP / their relative entropy. Suppose that F0 is a -field such that F0  F and denote by H0 .QjP / the relative entropy of Q with respect to P considered as probability measures on the smaller space .; F0 /. Then the relation L1 .; F0 ; P /  L1 .; F ; P / implies H0 .QjP /  H.QjP /I }

in general this inequality is strict. Lemma 3.31. For all ˛  0, the set ˆ˛ WD ¹' 2 L1 .; F ; P / j '  0; EŒ '  D 1; EŒ ' log '   ˛º is weakly sequentially compact in L1 .; F ; P /. Proof. Let Lp WD Lp .; F ; P /. The set of all P -densities, D WD ¹' 2 L1 j '  0; EŒ '  D 1 º;

is clearly convex and closed in L1 . Hence, this set is also weakly closed in L1 by Theorem A.60. Moreover, Lemma 3.29 states that for ' 2 D EΠ' log '  D sup .EΠZ '   log EΠe Z /: Z2L1

In particular, ' 7! EŒ ' log '  is a weakly lower semicontinuous functional on D, and so ˆ˛ is weakly closed. In addition, ˆ˛ is bounded in L1 and uniformly integrable, due to the criterion of de la Vallée Poussin; see, e.g., Lemma 3 in § 6 of Chapter II of [251]. Applying the Dunford–Pettis theorem and the Eberlein–Šmulian theorem as stated in Appendix A.7 concludes the proof. Proof of Theorem 3:28. In view of Theorem 3.24 and inequality (3.13) (whose proof extends to all m 2 Rd ), it remains to prove that inf EQ Œ Y Dm

H.QjP /  sup Π m  log Z. /

(3.16)

2Rd

for those m which do not belong to ri . /, where WD P ıY 1 . The right-hand side of (3.16) is just the Fenchel–Legendre transform at m of the convex function log Z and, thus, denoted .log Z/ .m/.

138

Chapter 3 Optimality and equilibrium

First, we consider the case in which m is not contained in the closure . / of the convex hull of the support of . Proposition A.1, the separating hyperplane theorem, yields some  2 Rd such that   m > sup¹  x j x 2 . /º  sup¹  x j x 2 supp º: By taking n WD n, it follows that

n  m  log Z. n /  n   m 

 sup   y ! C1

as n " 1.

y2supp 

Hence, the right-hand side of (3.16) is infinite if m … . /. It remains to prove (3.16) for m 2 . /n ri . / with .log Z/ .m/ < 1. Recall from (1.25) that ri . / D ri . /. Pick some m1 2 ri . / and let   1 1 mn WD m1 C 1  m: n n Then mn 2 ri . / by (1.24). By the convexity of .log Z/ , we have   1 n1 .log Z/ .m1 / C .log Z/ .m/ lim sup.log Z/ .mn /  lim sup n n n"1 n"1

(3.17)



D .log Z/ .m/: We also know that to each mn there corresponds a n 2 Rd such that mn D E n ΠY 

and

H.P n jP / D .log Z/ .mn /:

(3.18)

From (3.17) and (3.18) we conclude that lim sup H.P n jP / D lim sup.log Z/ .mn /  .log Z/ .m/ < 1: n"1

n"1

In particular, H.P n jP / is uniformly bounded in n, and Lemma 3.31 implies that – after passing to a suitable subsequence if necessary – the densities dP n =dP converge weakly in L1 .; F ; P / to a density '. Let dP1 D ' dP . By the weak lower semicontinuity of dQ 7! H.QjP /; dP which follows from Lemma 3.29, we may conclude that H.P1 jP /  .log Z/ .m/. The theorem will be proved once we can show that E1 Œ Y  D m. To this end, let  WD supn .log Z/ .mn /, which is a finite non-negative number by (3.17). Taking Z WD ˛ I¹jY jcº jY j

139

Section 3.3 Optimal contingent claims

on the right-hand side of (3.15) yields   ˛E n Œ jY j  I¹jY jcº   log EŒ exp.˛jY jI¹jY jcº / 

for all n  1.

Note that the rightmost expectation is finite due to condition (3.9) and Lemma 3.13. By taking ˛ large so that =˛ < "=2 for some given " > 0, and by choosing c such that ˛" log EŒ exp.˛jY jI¹jY jcº /  < ; 2 we obtain that sup E n Œ jY j  I¹jY jcº   ": n1

But E n Œ jY j  I¹jY j 0, the random variable XQ WD X C c IA would satisfy E  Œ XQ  D E  Œ X  and EŒ u.XQ /  > EŒ u.X / . Similarly, if P  Œ A  > 0 and P Œ A  D 0 then XO WD X C c 

c I P ΠA  A

would have the same price as X but higher expected utility. In particular, the expectations in (3.20) would be unbounded in both cases if X is the class of all measurable functions on .; F / and if the function u is not bounded from above. } Remark 3.33. If a solution X  with EŒ u.X  /  < 1 exists then it is unique, since B is convex and u is strictly concave. Moreover, if X D L0 .; F ; P / or X D L0C .; F ; P / then X  satisfies E Œ X   D w since E  Œ X   < w would imply that X WD X  C w  E  Œ X   is a strictly better choice, due to the strict monotonicity of u. } Let us first consider the unrestricted case X D L0 .; F; P / where any finite random variable on .; F; P / is admissible. The following heuristic argument identifies a candidate X  for the maximization of the expected utility. Suppose that a solution X  exists. For any X 2 L1 .P / and any 2 R, X WD X  C .X  E  Œ X / satisfies the budget constraint E  Œ X  D w. A formal computation yields d ˇˇ EŒ u.X /  d D0 D EŒ u0 .X  /.X  E  Œ X /

0D

D EΠu0 .X  /X   EΠXEΠu0 .X  /  ' D EΠX.u0 .X  /  c '/  where c WD EΠu0 .X  / . The identity EΠX u0 .X  /  D c EΠX '  for all bounded measurable X implies u0 .X  / D c ' P -almost surely. Thus, if we denote by I WD .u0 /1

Section 3.3 Optimal contingent claims

141

the inverse function of the strictly decreasing function u0 , then X  should be of the form X  D I.c '/: We will now formulate a set of assumptions on our utility function u which guarantee that X  WD I.c '/ is indeed a maximizer of the expected utility, as suggested by the preceding argument. Theorem 3.34. Suppose u W R ! R is a continuously differentiable utility function which is bounded from above, and whose derivative satisfies lim u0 .x/ D C1:

x#1

(3.21)

Assume moreover that c > 0 is a constant such that X  WD I.c '/ 2 L1 .P  /: Then X  is the unique maximizer of the expected utility EΠu.X /  among all those X 2 L1 .P  / for which E  ΠX   E  ΠX  . In particular, X  solves our optimization problem (3.20) for X D L0 .; F ; P / if c can be chosen such that E  ΠX   D w. Proof. Uniqueness follows from Remark 3.33. Since u is bounded from above, its derivative satisfies lim u0 .x/ D 0; x"1

in addition to (3.21). Hence, .0; 1/ is contained in the range of u0 , and it follows that I.c '/ is P -a.s. well-defined for all c > 0. To show the optimality of X  D I.c '/, note that the concavity of u implies that for any X 2 L1 .P  / u.X /  u.X  / C u0 .X  /.X  X  / D u.X  / C c '.X  X  /: Taking expectations with respect to P yields EŒ u.X /   EŒ u.X  /  C c E  Œ X  X  : Hence, X  is indeed a maximizer in the class ¹X 2 L1 .P  / j E  Œ X   E  Œ X  º.

Example 3.35. Let u.x/ D 1  e ˛x be an exponential utility function with constant absolute risk aversion ˛ > 0. In this case, 1 y I.y/ D  log : ˛ ˛

142

Chapter 3 Optimality and equilibrium

It follows that c 1 1 E  Œ I.c '/  D  log   EŒ ' log '  ˛ ˛ ˛ c 1 1 D  log   H.P  jP /; ˛ ˛ ˛ where H.P  jP / denotes the relative entropy of P  with respect to P ; see Definition 3.20. Hence, the utility maximization problem can be solved for any w 2 R if and only if the relative entropy H.P  jP / is finite. In this case, the optimal profile is given by 1 1 X  D  log ' C w C H.P  jP /; ˛ ˛ and the maximal value of expected utility is EŒ u.X  /  D 1  exp.˛w  H.P  jP //; corresponding to the certainty equivalent wC

1 H.P  jP /: ˛

Let us now return to the financial market model considered in Section 3.1, and let P  be the entropy-minimizing risk-neutral measure constructed in Corollary 3.25. The density of P  is of the form  e ˛ Y 'D ; EŒ e ˛  Y  where   2 Rd denotes a maximizer of the expected utility EŒ u.  Y / ; see Example 3.12. In this case, the optimal profile takes the form 

X  D   Y C w D

 S ; 1Cr 

i.e., X  is the discounted payoff of the portfolio  D . 0 ;   /, where  0 D w    is determined by the budget constraint    D w. Thus, the optimal profile is given by a linear profile in the given primary assets S 0 ; : : : ; S d : No derivatives are needed at this point. } In most situations it will be natural to restrict the discussion to payoff profiles which are non-negative. For the rest of this section we will make this restriction, and so the utility function u may be defined only on Œ0; 1/. In several applications we will also use an upper bound on payoff profiles given by an F -measurable random variable W W  ! Œ0; 1 . We include the case W  C1 and define the convex class of admissible payoff profiles as X WD ¹X 2 L0 .P / j 0  X  W P -a.s.º:

143

Section 3.3 Optimal contingent claims

Thus, our goal is to maximize the expected utility EŒ u.X /  among all X 2 B where the budget set B is defined in terms of X and P  as in (3.19), i.e., B D ¹X 2 L1 .P  / j 0  X  W P -a.s. and E  Œ X   w º: We first formulate a general existence result: Proposition 3.36. Let u be any utility function on Œ0; 1/, and suppose that W is P -a.s. finite and satisfies EŒ u.W /  < 1. Then there exists a unique X  2 B which maximizes the expected utility EŒ u.X /  among all X 2 B. Proof. Take a sequence .Xn / in B with E  Œ Xn   w and such that EŒ u.Xn /  converges to the supremum of the expected utility. Since supn jXn j  W < 1 P -almost surely, we obtain from Lemma 1.70 a sequence XQ n 2 conv¹Xn ; XnC1 ; : : : º of convex combinations which converge almost-surely to some XQ . Clearly, every XQ n is contained in B. Fatou’s lemma implies E  Œ XQ   lim inf E  Œ XQ n   w; n"1

and so XQ 2 B. Each XQ n can be written as coefficients ˛in  0 summing up to 1. Hence, u.XQn / 

m X

Pm

n iD1 ˛i Xni

for indices ni  n and

˛in u.Xni /;

iD1

and it follows that EΠu.XQn /   inf EΠu.Xm / : mn

By dominated convergence, EΠu.XQ /  D lim EΠu.XQ n / ; n"1

and the right-hand side is equal to the supremum of the expected utility. Remark 3.37. The argument used to prove the preceding proposition works just as well in the following general setting. Let U W B ! R be a concave functional on a set B of random variables defined on a probability space .; F ; P / and with values in Rn . Assume that

144  

Chapter 3 Optimality and equilibrium

B is convex and closed under P -a.s. convergence, there exists a random variable W 2 L0C .; F ; P / with jX i j  W < 1 P -a.s. for each X D .X 1 ; : : : ; X n / 2 B,



supX2B U.X / < 1,



U is upper semicontinuous with respect to P -a.s. convergence.

Then there exists an X  2 B which maximizes U on B, and X  is unique if U is strictly concave. As a special case, this includes the utility functionals U.X / D inf EQ Πu.X / ; Q2Q

appearing in a robust Savage representation of preferences on n-dimensional asset profiles, where u is a utility function on Rn and Q is a set of probability measures equivalent to P ; see Section 2.5. } We turn now to a characterization of the optimal profile X  in terms of the inverse of the derivative u0 of u in case where u is continuously differentiable on .0; 1/. Let a WD lim u0 .x/  0 and x"1

b WD u0 .0C/ D lim u0 .x/  C1: x#0

We define I C W .a; b/ ! .0; 1/ as the continuous, bijective, and strictly decreasing inverse function of u0 on .a; b/, and we extend I C to the full half axis Œ0; 1 by setting ´ 0 for y  b, C I .y/ WD (3.22) C1 for y  a. With this convention, I C W Œ0; 1 ! Œ0; 1 is continuous. Remark 3.38. If u is a utility function defined on all of R, the function I C is the inverse of the restriction of u0 to Œ0; 1/. Thus, I C is simply the positive part of the function I D .u0 /1 . For instance, in the case of an exponential utility function u.x/ D 1  e ˛x , we have a D 0, b D ˛, and   1 y  I C .y/ D log D .I.y//C ; y  0: (3.23) ˛ ˛ } Theorem 3.39. Assume that X  2 B is of the form X  D I C .c '/ ^ W for some constant c > 0 such that E  Œ X   D w. If EŒ u.X  /  < 1 then X  is the unique maximizer of the expected utility EŒ u.X /  among all X 2 B.

145

Section 3.3 Optimal contingent claims

Proof. In a first step, we consider the function v.y; !/ WD

.u.x/  xy/

sup

(3.24)

0xW .!/

defined for y 2 R and ! 2 . Clearly, for each ! with W .!/ < 1 the supremum above is attained in a unique point x  .y/ 2 Œ0; W .!/, which satisfies x  .y/ D 0 

x .y/ D W .!/

” ”

u0 .x/ < y 0

u .x/ > y

for all x 2 .0; W .!//, for all x 2 .0; W .!//.

Moreover, y D u0 .x  .y// if x  .y/ is an interior point of the interval Œ0; W .!/. It follows that x  .y/ D I C .y/ ^ W .!/; or X  D x  .c '/

on ¹W < 1º.

(3.25)

If W .!/ D C1, then the supremum in (3.24) is not attained if and only if u0 .x/ > y for all x 2 .0; 1/. By our convention (3.22), this holds if and only if y  a and hence I C .y/ D C1. But our assumptions on X  imply that I C .c '/ < 1 P -a.s. on ¹W D 1º, and hence that X  D x  .c '/

P -a.s. on ¹W D 1º.

(3.26)

Putting (3.24), (3.25), and (3.26) together yields u.X  /  X  c ' D v.c'; /

P -a.s.

Applied to an arbitrary X 2 B, this shows that u.X  /  c 'X   u.X /  c 'X

P -a.s.

Taking expectations gives EΠu.X  /   EΠu.X /  C c  E  ΠX   X   EΠu.X / : Hence, X  maximizes the expected utility on B. Uniqueness follows from Remark 3.33. In the following examples, we study the application of the preceding theorem to CARA and HARA utility functions. For simplicity we consider only the case W  1. The extension to a non-trivial bound W is straightforward.

146

Chapter 3 Optimality and equilibrium

Example 3.40. For an exponential utility function u.x/ D 1  e ˛x we have by (3.23)  y '  1  1 y '  'I C .y '/ D ' log D h ; ˛ ˛ y ˛ where h.x/ D .x log x/ . Since h is bounded by e 1 , it follows that 'I C .y '/ belongs to L1 .P / for all y > 0. Thus, 1 h y '  i g.y/ WD E  Œ I C .y '/  D E h y ˛ decreases continuously from C1 to 0 as y increases from 0 to 1, and there exists a unique c with g.c/ D w. The corresponding profile X  WD I C .c '/ maximizes the expected utility EŒ u.X /  among all X  0. Let us now return to the special situation of the financial market model of Section 3.1, and take P  as the entropy-minimizing risk-neutral measure of Corollary 3.25. Then the optimal profile X  takes the form X  D .   Y  K/C ; where   is the maximizer of the expected utility EŒ u.  Y / , and where K is given by 1 c 1 c 1 1  K D log  log EŒ e ˛ Y  D log C H.P  jP /: ˛ ˛ ˛ ˛ ˛ ˛ Note that X  is a linear combination of the primary assets only in the case where    Y  K P -almost surely. In general, X  is a basket call option on the attainable asset w C .1 C r/   Y 2 V with strike price w C .1 C r/K. Thus, a demand for derivatives appears. } Example 3.41. If u is a HARA utility function of index  2 Œ0; 1/ then u0 .x/ D x 1 , hence 1 I C .y/ D y  1 and

1

1

I C .y '/ D y  1  '  1 : In the logarithmic case  D 0, we assume that the relative entropy H.P jP  / of P with respect to P  is finite. Then X D

dP w Dw ' dP 

is the unique maximizer, and the maximal value of expected utility is EΠlog X   D log w C H.P jP  /:

147

Section 3.3 Optimal contingent claims

If  2 .0; 1/ and

1

EΠ'  1  D E  Π'  1  < 1; then the unique optimal profile is given by

1

X  D w .EΠ'  1 /1 '  1 ; and the maximal value of expected utility is equal to EΠu.X  /  D

1 w .EΠ'  1 /1 : 

}

Exercise 3.3.1. In the context of Example 3.41, compute the maximal value of expected utility when ' has a log-normal distribution. } The following corollary gives a simple condition on W which guarantees the existence of the maximizer X  in Theorem 3.39. Corollary 3.42. If EŒ u.W /  < 1 and if 0 < w < E  Œ W  < 1, then there exists a unique constant c > 0 such that X  D I C .c '/ ^ W satisfies E  Œ X   D w. In particular, X  is the unique maximizer of the expected utility EŒ u.X /  among all X 2 B. Proof. For any ˇ 2 .0; 1/, y 7! I C .y/ ^ ˇ is a continuous decreasing function with limy"b I C .y/ ^ ˇ D 0 and I C .y/ ^ ˇ D ˇ for all y  u0 .ˇ/. Hence, dominated convergence implies that the function g.y/ WD E  Œ I C .y '/ ^ W ; is continuous and decreasing with lim g.y/ D 0 < w < E  Œ W  D lim g.y/:

y"1

y#0

Moreover, g is even strictly decreasing on ¹y j 0 < g.y/ < E  Œ W º. Hence, there exists a unique c with g.c/ D w, and Theorem 3.39 yields the optimality of the corresponding X  . Let us now extend the discussion to the case where preferences themselves are uncertain. This additional uncertainty can be modelled by incorporating the choice of a utility function into the description of possible scenarios; for an axiomatic discussion see [172]. More precisely, we assume that preferences are described by a measurable

148

Chapter 3 Optimality and equilibrium

function u on Œ0; 1/   such that u.; !/ is a utility function on Œ0; 1/ which is continuously differentiable on .0; 1/. For each ! 2 , the inverse of u0 .; !/ is extended as above to a function I C .; !/ W Œ0; 1 ! Œ0; 1: Using exactly the same arguments as above, we obtain the following extension of Corollary 3.42 to the case of random preferences: Corollary 3.43. If EŒ u.W; /  < 1 and if 0 < w < E  Œ W  < 1, then there exists a unique constant c > 0 such that X  .!/ W D I C .c '.!/; !/ ^ W .!/ is the unique maximizer of the expected utility Z EŒ u.X; /  D u.X.!/; !/ P .d!/ among all X 2 B.

3.4

Optimal payoff profiles for uniform preferences

So far, we have discussed the structure of asset profiles which are optimal with respect to a fixed utility function u. Let us now introduce an optimization problem with respect to the uniform order 1  > 0 and admits a continuous and strictly increasing quantile function q' with respect to P , and t is the solution of a certain nonlinear equation. Let us also assume for the sake of concreteness that u.x/ D 1 x is a HARA utility function with risk aversion  2 .0; 1/ and that EΠ'  =.1 /  < 1.

159

Section 3.6 Microeconomic equilibrium

It was shown in Example 3.41 that the standard utility maximization problem under P has the solution 1 X  D w.EΠ'  1 /1 '  1 : Note that X  is large for small values of ', that is, for low-price scenarios. Now let us replace the single probability measure P by the entire set Q . By Theorem 3.56, the corresponding robust utility maximization problem will be solved by 1 XQ  D w.EΠ  1 /1   1 : where D

dP  D .' _ q' .t //: dQ0

It follows that XQ  D c.X  ^ r/ where r > 0 and c > 1 are certain constants. Thus, the effect of robustness is here that the optimal payoff profile X  is cut off at a certain threshold. That is, one gives up the opportunity for very high profits in low-price scenarios in favor of enhanced returns in all other scenarios. }

3.6

Microeconomic equilibrium

The aim of this section is to provide a brief introduction to the theory of market equilibrium. Prices of assets will no longer be given in advance. Instead, they will be derived from “first principles” in a microeconomic setting where different agents demand asset profiles in accordance with their preferences and with their budget constraints. These budget constraints are determined by a given price system. The role of equilibrium prices consists in adjusting the constraints in such a way that the resulting overall demand is matched by the overall supply of assets. Consider a finite set A of economic agents and a convex set X  L0 .; F ; P / of admissible claims. At time t D 0, each agent a 2 A has an initial endowment whose discounted payoff at time t D 1 is described by an admissible claim Wa 2 X; The aggregated claim W WD

a 2 A: X

Wa

a2A

is also called the market portfolio. Agents may want to exchange their initial endowment Wa against some other admissible claim Xa 2 X. This could lead to a new allocation .Xa /a2A if the resulting total demand matches the overall supply.

160

Chapter 3 Optimality and equilibrium

Definition 3.58. A collection .Xa /a2A  X is called a feasible allocation if it satisfies the market clearing condition X Xa D W P -a.s. (3.41) a2A

The budget constraints will be determined by a linear pricing rule of the form ˆ.X / WD EŒ ' X ;

X 2 X;

where ' is a price density, i.e., an integrable function on .; F / such that ' > 0 P a.s. and EŒ jWa j '  < 1 for all a 2 A. To any such ' we can associate a normalized price measure P '  P with density 'EŒ ' 1 . Remark 3.59. In the context of our one-period model of a financial market with d risky assets S 1 ; : : : ; S d and a risk-free asset S 0  1 C r, P ' is a risk-neutral measure if the pricing rule ˆ is consistent with the given price vector  D . 0 ; /, where  0 D 1. In this section, the pricing rule will be derived as an equilibrium price measure, given the agents’ preferences and endowments. In particular, this will amount to an endogenous derivation of the price vector . In a situation where the structure of the equilibrium is already partially known in the sense that it is consistent with the given price vector , the construction of a microeconomic equilibrium yields a specific choice of a martingale measure P  , i.e., of a specific extension of  from the space V of attainable payoffs to a larger space of admissible claims. } The preferences of agent a 2 A are described by a utility function ua . Given the price density ', an agent a 2 A may want to exchange the endowment Wa for an ' admissible claim Xa which maximizes the expected utility EŒ ua .X /  among all X in the agent’s budget set Ba .'/ WD ¹X 2 X j EŒ 'jX j  < 1 and EŒ ' X   EŒ ' Wa º D ¹X 2 X j X 2 L1 .P ' / and E ' Œ X   E ' Œ Wa º: '

In this case, we will say that Xa solves the utility maximization problem of agent a 2 A with respect to the price density '. The key problem is whether ' can be ' chosen in such a way that the requested profiles Xa , a 2 A, form a feasible allocation. Definition 3.60. A price density '  together with a feasible allocation .Xa /a2A is called an Arrow–Debreu equilibrium if each Xa solves the utility maximization problem of agent a 2 A with respect to '  .

161

Section 3.6 Microeconomic equilibrium

Thus, the price density '  appearing in an Arrow–Debreu equilibrium decentralizes the crucial problem of implementing the global feasibility constraint (3.41). This is achieved by adjusting the budget sets in such a way that the resulting demands respect the market clearing condition, even though the individual demand is determined without any regard to this global constraint. Example 3.61. Assume that each agent a 2 A has an exponential utility function with parameter ˛a > 0, and let us consider the unconstrained case X D L0 .; F ; P /: In this case, there is a unique equilibrium, and it is easy to describe it explicitly. For a given pricing measure P   P such that Wa 2 L1 .P  / for all a 2 A, the utility maximization problem for agent a 2 A can be solved if and only if H.P  jP / < 1, and in this case the optimal demand is given by Xa D 

1 1 log '  C wa C H.P  jP / ˛a ˛a

where wa WD E  Œ Wa I see Example 3.35. The market clearing condition (3.41) takes the form X 1 1 W D  log '  C wa C H.P  jP / ˛ ˛ a2A

where ˛ is defined via

X 1 1 D : ˛ ˛a

(3.42)

a2A

Thus, a normalized equilibrium price density must have the form ' D

e ˛W ; EŒ e ˛W 

(3.43)

and this shows uniqueness. As to existence, let us assume that EŒ jWa je ˛W  < 1;

a 2 AI

this condition is satisfied if, e.g., the random variables Wa are bounded from below. Define P   P via (3.43). Then H.P  jP / D ˛ E  Œ W   log EŒ e ˛W  < 1;

162

Chapter 3 Optimality and equilibrium

and the optimal profile for agent a 2 A with respect to the pricing measure P  takes the form ˛ Xa D wa C .W  E  Œ W /: (3.44) ˛a Since X wa D E  Œ W ; a2A

.Xa /a2A

is feasible, and so we have constructed an Arrow–Debreu the allocation equilibrium. Thus, the agents share the market portfolio in a linear way, and in inverse proportion to their risk aversion. Let us now return to our financial market model of Section 3.1. We assume that the initial endowment of agent a 2 A is given by a portfolio a 2 Rd C1 so that the discounted payoff at time t D 1 is Wa D

a  S ; 1Cr

a 2 A:

In this case, the market portfolio is given by W D   S =.1 C r/ with  WD .0 ; /. The optimal claim for agent a 2 A in (3.44) takes the form   ˛ S  Xa D a   C  ;  ˛a 1Cr

P

a

a D

where  D .1; / and i

 DE





Si 1Cr

 for i D 1; : : : ; d .

Thus, we could have formulated the equilibrium problem within the smaller space X D V of attainable payoffs, and the resulting equilibrium allocation would have been the same. In particular, the extension of X from V to the general space L0 .; F ; P / of admissible claims does not create a demand for derivatives in our present example. } From now on we assume that the set of admissible claims is given by X D L0C .; F ; P /; and that the preferences of agent a 2 A are described by a utility function ua W Œ0; 1/ ! R which is continuously differentiable on .0; 1/. In particular, the initial endowments Wa are assumed to be non-negative. Moreover, we assume P Œ Wa > 0  ¤ 0

for all a 2 A

and EΠW  < 1:

(3.45)

163

Section 3.6 Microeconomic equilibrium

A function ' 2 L1 .; F ; P / such that ' > 0 P -a.s. is a price density if EΠ' W  < 1I note that this condition is satisfied as soon as ' is bounded, due to our assumption (3.45). Given a price density ', each agent faces exactly the optimization problem discussed in Section 3.3 in terms of the price measure P '  P . Thus, if .Xa /a2A is an equilibrium allocation with respect to the price density '  , feasibility implies 0  Xa  W , and so it follows as in the proof of Corollary 3.42 that Xa D IaC .ca '  /;

a 2 A;

(3.46)

with positive constants ca > 0. Note that the market clearing condition X X W D Xa D IaC .ca '  / a2A

a2A

will determine '  as a decreasing function of W , and thus the optimal profiles Xa will be increasing functions of W . Before we discuss the existence of an Arrow–Debreu equilibrium, let us first illustrate the structure of such equilibria by the following simple examples. In particular, they show that an equilibrium allocation will typically involve non-linear derivatives of the market portfolio W . Example 3.62. Let us consider the constrained version of the preceding example where agents a 2 A have exponential utility functions with parameters ˛a > 0. Define w WD sup¹c j W  c P -a.s.º  0; and let P  be the measure defined via (3.43). For any agent a 2 A such that wa WD E  Œ Wa  

˛ .E  Œ W   w/; ˛a

(3.47)

the unrestricted optimal profile Xa D wa C

˛ .W  E  Œ W / ˛a

satisfies Xa  0 P -a.s. Thus, if all agents satisfy the requirement (3.47) then the unrestricted equilibrium computed in Example 3.61 is a forteriori an Arrow–Debreu equilibrium in our present context. In this case, there is no need for non-linear derivatives of the market portfolio. If some agents do not satisfy the requirement (3.47) then the situation becomes more involved, and the equilibrium allocation will need derivatives such as call options. Let us illustrate this effect in the simple setting where there are only two agents

164

Chapter 3 Optimality and equilibrium

a 2 A D ¹1; 2º. Suppose that agent 1 satisfies condition (3.47), while agent 2 does not. For c  0, we define the measure P c  P in terms of the density ´ 1 ˛1 W e on ¹W  cº; c ' WD Z11 ˛W on ¹W  cº; Z2 e where ˛ is given by (3.42), and where the constants Z1 and Z2 are determined by the continuity condition log Z2  log Z1 D c.˛1  ˛/ and by the normalization EŒ ' c  D 1. Note that P 0 D P  with P  as in (3.43). Consider the equation ˛ c E Œ .W  c/C  D w2c WD E c Œ W2 : ˛2

(3.48)

Both sides are continuous in c. As c increases from 0 to C1, the left-hand side decreases from ˛˛2 E  Œ W  to 0, while w2c goes from w20 < ˛˛2 E  Œ W  to E 1 Œ W2  > 0. Thus, there exists a solution c of (3.48). Let us now check that X2c WD

˛ .W  c/C ; ˛2

X1c WD W  X2c

defines an equilibrium allocation with respect to the pricing measure P c . Clearly, X1c and X2c are non-negative and satisfy X1c CX2c D W . The budget condition for agent 2 is satisfied due to (3.48), and this implies the budget condition E c ΠX1c  D E c ΠW   w2c D w1c for agent 1. Both are optimal since Xac D IaC . a ' c / with

1 WD ˛1 Z1

and

2 WD ˛2 Z2 e ˛c :

Thus, agent 2 demands ˛˛2 shares of a call option on the market portfolio W with strike c, agent 1 demands the remaining part of W , and so the market is cleared. In the general case of a finite set A of agents, the equilibrium price measure PO has the following structure. There are levels 0 WD c0 <    < cN D 1 with 1  N  jAj such that the price density 'O is given by 'O D

1 ˇi W e Zi

for i D 1; : : : ; N , where ˇi WD

on ¹W 2 Œci1 ; ci º

 X 1 1 ; ˛a ˛2Ai

165

Section 3.6 Microeconomic equilibrium

and where Ai .i D 1; : : : ; N / are the increasing sets of agents which are active at the i th layer in the sense that Xa > 0 on ¹W 2 .ci1 ; ci º. At each layer .ci1 ; ci , the active agents are sharing the market portfolio in inverse proportions to their risk aversion. Thus, the optimal profile XO a of any agent a 2 A is given by an increasing piecewise linear function in W , and thus it can be implemented by a linear combination of call options with strikes ci . More precisely, an agent a 2 Ai takes ˇi =˛a shares of the spread .W  ci1 /C  .W  ci /C ; i.e., the agent goes long on a call option with strike ci1 and short on a call option } with strike ci . Example 3.63. Assume that all agents a 2 A have preferences described by HARA utility functions so that 1 IaC .y/ D y  1 a ; a 2 A with 0  a < 1. For a given price density ', the optimal claims take the form 1

Xa D IaC .ca '/ D ba '  1 a

(3.49)

with constants ba > 0. If a D  for all a 2 A, then the market clearing condition (3.41) implies X  X 1 Xa D ba '  1 ; W D a2A

i.e., the equilibrium price density

a2A

'

takes the form

' D

1 1 W ; Z

where Z is the normalizing constant, and so the agents demand linear shares of the market portfolio W . If risk aversion varies among the agents then the structure of the equilibrium becomes more complex, and it will involve non-linear derivatives of the market portfolio. Let us number the agents so that A D ¹1; : : : ; nº and 1      n . Condition (3.49) implies Xi D di Xnˇi with some constants di , and where ˇi WD

1  n 1  i

satisfies ˇ1      ˇn D 1 with at least one strict inequality. Thus, each Xi is a convex increasing function of Xn . In equilibrium, Xn is a concave function of W determined by the condition n X di Xnˇi D W; (3.50) iD1

166

Chapter 3 Optimality and equilibrium

and the price density '  takes the form ' D

1 n 1 X : Z n

As an illustration, we consider the special case “Bernoulli vs. Cramer”, where p A D ¹1; 2º with u1 .x/ D x and u2 .x/ D log x, i.e., 1 D 12 and 2 D 0; see Example 2.38. The solutions of (3.50) can be parameterized with c  0 such that p p p  X2c D 2 c W C c  c 2 Œ0; W  and X1c D W  X2c : The corresponding price density takes the form 'c D

1 1 p p ; Z.c/ W C c  c

where Z.c/ is the normalizing constant. Now assume that W 1 2 L1 .P /, and let P 1 denote the measure with density W 1 .EŒ W 1 /1 . As c increases from 0 to 1, E c Œ X2c  increases continuously from 0 to E 1 Œ W , while E c Œ W2  goes continuously from E 0 Œ W2  > 0 to E 1 Œ W2  < E 1 Œ W ; here we use our assumption that P Œ Wa > 0  ¤ 0 for all a 2 A. Thus, there is a c 2 .0; 1/ such that E c Œ X2c  D E c Œ W2 ; and this implies that the budget constraint is satisfied for both agents. With this choice of the parameter c, .X1c ; X2c / is an equilibrium allocation with respect to the pricing measure P c : Agent 2 demands the concave profile X2c , agent 1 demands the convex profile X1c , both in accordance with their budget constraints, and the market is cleared. } Let us now return to our general setting, and let us prove the existence of an Arrow– Debreu equilibrium. Consider the following condition:    W 0 0 lim sup x ua .x/ < 1 and E ua < 1; a 2 A: (3.51) jAj x#0 Remark 3.64. Condition (3.51) is clearly satisfied if u0a .0/ WD lim u0a .x/ < 1; x#0

a 2 A:

(3.52)

But it also includes HARA utility functions ua with parameter a 2 Œ0; 1/ if we assume EŒ W a 1  < 1; a 2 A; in addition to our assumption EŒ W  < 1.

}

167

Section 3.6 Microeconomic equilibrium

Theorem 3.65. Under assumptions (3.45) and (3.51), there exists an Arrow–Debreu equilibrium. In a first step, we are going to show that an equilibrium allocation maximizes a suitable weighted average U .X / WD

X

a EΠua .Xa / 

a2A

of the individual utility functionals over all feasible allocations X D .Xa /a2A . The weights are non-negative, and without loss of generality we can assume that they are normalized so that the vector WD . a /a2A belongs to the convex compact set ± ° ˇ X

a D 1 : ƒ D 2 Œ0; 1A ˇ a2A

In a second step, we will use a fixed-point argument to obtain a weight vector and a corresponding price density such that the maximizing allocation satisfies the individual budget constraints. Definition 3.66. A feasible allocation .Xa /a2A is called -efficient for 2 ƒ if it maximizes U over all feasible allocations. In view of (3.46), part (b) of the following lemma shows that the equilibrium allocation .Xa /a2A in an Arrow–Debreu P 1 equilibrium is -efficient for the vector 1 1

D .c  ca /a2A , where c WD a ca . Thus, the existence proof for an Arrow– Debreu equilibrium is reduced to the construction of a suitable vector  2 ƒ. Lemma 3.67. (a) For any 2 ƒ there exists a unique -efficient allocation .Xa /a2A . (b) A feasible allocation .Xa /a2A is -efficient if and only if it satisfies the first order conditions

a u0a .Xa /  ';

with equality on ¹Xa > 0º

(3.53)

with respect to some price density '. In this case, .Xa /a2A coincides with .Xa /a2A , and the price density can be chosen as ' WD max a u0a .Xa /: a2A

(c) For each a 2 A, Xa maximizes EΠua .X /  over all X 2 X such that EΠ' X   EΠ' Xa :

(3.54)

168

Chapter 3 Optimality and equilibrium

Proof. (a): Existence and uniqueness follow from the general argument in Remark 3.37 applied to the set B of all feasible allocations and to the functional U . Note that U .X /  max EΠua .W /  a2A

for any feasible allocation, and that the right-hand side is finite due to our assumption (3.45). Moreover, by dominated convergence, U is indeed continuous on B with respect to P -a.s. convergence. (b): Let us first show sufficiency. If X D .Xa /a2A is a feasible allocation satisfying the first order conditions, and Y D .Ya /a2A is another feasible allocation then U .X /  U .Y / D

X

a EΠua .Xa /  ua .Ya / 

a2A



X

a EΠu0a .Xa /.Xa  Ya / 

a2A

h X X i E ' Xa  Ya D 0; a2A

a2A

using concavity of ua in the second step and the first order conditions in the third. This shows that X is -efficient. Turning to necessity, consider the -efficient allocation .Xa /a2A for 2 ƒ and another feasible allocation .Xa /a2A . For " 2 .0; 1, let Ya" WD "Xa C .1  "/Xa . Since .Ya" /a2A is feasible, -efficiency of .Xa /a2A yields 0

1X

a EΠua .Ya" /  ua .Xa /  " a2A

1X

a EΠu0a .Ya" /.Ya"  Xa /  " a2A X D

a EΠu0a .Ya" /.Xa  Xa / :



(3.55)

a2A

Let us first assume (3.52); in part (d) of the proof we show how to modify the argument under condition (3.51). Using dominated convergence and (3.52), we may let " # 0 in the above inequality to conclude X a2A

EΠ'a Xa  

X

EΠ'a Xa   EΠ' W ;

a2A

where 'a WD a u0a .Xa /:

(3.56)

169

Section 3.6 Microeconomic equilibrium

Note that ' is a price density since by (3.52) 0 < '  max¹ a u0a .0/ j a 2 Aº < 1: Take a feasible allocation .Xa /a2A such that X 'a Xa D ' W I

(3.57)

a2A

for example, we can enumerate A WD ¹1; : : : ; jAjº and take Xa WD W I¹T Daº where T .!/ WD min¹a j 'a .!/ D ' .!/º: In view of (3.56), we see that X

EΠ'a Xa  D EΠ' W :

(3.58)

a2A

This implies 'a D ' on ¹Xa > 0º, which is equivalent to the first order condition (3.53) with respect to ' . (c): In order to show optimality of Xa , we may assume without loss of generality that P Œ Xa > 0  > 0, and hence a > 0. Thus, the first order condition with respect to ' takes the form

Xa D IaC . 1 a ' /; due to our convention (3.22). By Corollary 3.42, Xa solves the optimization problem for agent a 2 A under the constraint EŒ ' X   EŒ ' Xa : (d): If (3.52) is replaced by (3.51), then we first need an additional argument in order to pass from (3.55) to (3.56). Note first that by Fatou’s lemma, X X

a EΠu0a .Ya" / Xa  

a lim inf EΠu0a .Ya" / Xa  lim inf "#0

a2A

a2A



X

"#0

a EΠu0a .Xa / Xa :

a2A

On the other hand, since  WD max sup x u0a .x/ < 1 a2A 0 .0/, the coherent risk measures c .X / WD sup EQ Œ X  Q2ƒc

are continuous from below, where ƒc D ¹Q 2 M1;f j ˛ min .Q/  cº.

}

Example 4.26. Let us consider a utility function u on R, a probability measure Q 2 M1 .; F /, and fix some threshold c 2 R. As in Example 4.10, we suppose that a position X is acceptable if its expected utility EQ Œ u.X /  is bounded from below by u.c/. Alternatively, we can introduce the convex increasing loss function `.x/ D u.x/ and define the convex set of acceptable positions A WD ¹X 2 X j EQ Œ `.X /   x0 º; where x0 WD u.c/. Let  WD A denote the convex risk measure induced by A. In Section 4.9, we will show that  is continuous from below, and we will derive a formula for its minimal penalty function. } Let us now continue the discussion in a topological setting. More precisely, we will assume for the rest of this section that  is a separable metric space and that F is the -field of Borel sets. As before, X is the linear space of all bounded measurable functions on .; F /. We denote by Cb ./ the subspace of bounded continuous functions on , and we focus on the representation of convex risk measures viewed as functionals on Cb ./. Proposition 4.27. Let  be a convex risk measure on X such that .Xn / & . / for any sequence .Xn / in Cb ./ that increases to a constant > 0. (4.28)

196

Chapter 4 Monetary measures of risk

Then there exists a penalty function ˛ on M1 such that .X / D max .EQ Œ X   ˛.Q//

for X 2 Cb ./.

(4.29)

Q j E Q Œ   D EQ Œ   on Cb ./º: ˛.Q/ WD inf¹˛ min .Q/ Q

(4.30)

Q2M1

In fact, one can take

Proof. Let ˛ min be the minimal penalty function of  on M1;f . We show that for any Q < 1 there exists Q 2 M1 such that E Q Œ X  D EQ Œ X  for all QQ with ˛ min .Q/ Q X 2 Cb ./. Take a sequence .Yn / in Cb ./ which increases to some Y 2 Cb ./, and choose ı > 0 such that Xn WD 1 C ı.Yn  Y /  0 for all n. Clearly, .Xn / satisfies condition (a) of Lemma 4.23, and so EQQ Œ Xn  ! 1, i.e., EQQ Œ Yn  % EQQ Œ Y : This continuity property of the linear functional EQQ Œ   on Cb ./ implies, via the Daniell–Stone representation theorem as stated in Appendix A.6, that it coincides on Cb ./ with the integral with respect to a -additive measure Q. Taking ˛ as in (4.30) gives the result. Remark 4.28. If  is compact then any convex risk measure admits a representation (4.29) on the space Cb ./ D C./. Indeed, if .Xn / is a sequence in Cb ./ that increases to a constant , then this convergence is even uniform by Lemma 4.24. Since  is Lipschitz continuous on C./ by Lemma 4.3, it satisfies condition (4.28). Alternatively, we could argue as in Remark 4.18 and apply the general duality theorem for the Fenchel–Legendre transform to the convex functional  on the Banach space C./. Just note that any continuous functional ` on C./ which is positive and normalized is of the form `.X / D EQ Œ X  for some probability measure Q 2 M1 ; see Theorem A.48. } Definition 4.29. A convex risk measure  on X is called tight if there exists an increasing sequence K1  K2     of compact subsets of  such that . IKn / ! . /

for all  1.

Note that every convex risk measure is tight if  is compact. Proposition 4.30. Suppose that the convex risk measure  on X is tight. Then (4.28) holds and the conclusion of Proposition 4:27 is valid. Moreover, if  is a Polish space and ˛ is a penalty function on M1 such that .X / D sup .EQ Œ X   ˛.Q//

for X 2 Cb ./,

Q2M1

then the level sets ƒc D ¹Q 2 M1 j ˛.Q/  cº are relatively compact for the weak topology on M1 .

Section 4.2 Robust representation of convex risk measures

197

Proof. First we show (4.28). Suppose Xn 2 Cb ./ are such that Xn % > 0. We may assume without loss of generality that  is normalized. Convexity and normalization guarantee that condition (4.28) holds for all > 0 as soon as it holds for all

 c where c is an arbitrary constant larger than 1. Hence, the cash invariance of  implies that there is no loss of generality in assuming Xn  0 for all n. We must show that .Xn /  . / C 2" eventually, where we take " 2 .0;  1/. By assumption, there exists a compact set KN such that ..  "/IKN /  .  "/ C " D . / C 2": By Dini’s lemma as recalled in Lemma 4.24, there exists some n0 2 N such that

 "  Xn on KN for all n  n0 . Finally, monotonicity implies .Xn /  ..  "/IKN /  . / C 2": To prove the relative compactness of ƒc , we will show that for any " > 0 there exists a compact set K"   such that for all c > .0/ inf QŒ K"   1  ".c C .0/ C 1/:

Qc Q2ƒ

The relative compactness of ƒc will then be an immediate consequence of Prohorov’s characterization of weakly compact sets in M1 , as stated in Theorem A.42. We fix a countable dense set ¹!1 ; !2 ; : : : º   and a complete metric ı which generates the topology of . For r > 0 we define continuous functions ri on  by ri .!/ WD 1 

ı.!; !i / ^ r : r

The function ri is dominated by the indicator function of the closed metric ball B r .!i / WD ¹! 2  j ı.!; !i /  rº: Let Xnr .!/ WD max ri .!/: in

Xnr

is continuous and satisfies 0  Xnr  1 as well as Xnr % 1 for n " 1. Clearly, According to (4.25), we have for all > 0 inf Q

Q2ƒc

n h [ iD1

i c C . Xnr / : B r .!i /  inf EQ Œ Xnr    Q2ƒc

Now we take k WD 2k =" and rk WD 1=k. The first part of this proof and (4.28) yield the existence of nk 2 N such that . k Xnrkk /  . k / C 1 D  k C 1;

198

Chapter 4 Monetary measures of risk

and thus sup Q

nk h \

Q2ƒc

iD1

i cC1 nB rk .!i /  D "2k .c C 1/:

k

We let K" WD

nk 1 [ \

B rk .!i /:

kD1 iD1

Then, for each Q 2 ƒc QŒ K"  D 1  Q

nk 1 \ h [

nB rk .!i /

i

kD1 iD1

1

1 X

"2k .c C 1/

kD1

D 1  ".c C 1/: The reader may notice that K" is closed, totally bounded and, hence, compact. A short proof of this fact goes as follows: Let .xj / be a sequence in K" . We must show that .xj / has a convergent subsequence. Since K" is covered by B rk .!1 /; : : : ; B rk .!nk / for each k, there exists some ik  nk such that infinitely many xj are contained in B rk .!ik /. A diagonalization argument yields a single subsequence .xj 0 / which for each k is contained in some B rk .!ik /. Thus, .xj 0 / is a Cauchy sequence with respect to the complete metric ı and, hence, converging to some element ! 2 . Remark 4.31. Note that the representation (4.29) does not necessarily extend from Cb ./ to the space X of all bounded measurable functions. Suppose in fact that  is compact but not finite, so that condition (4.28) holds as explained in Remark 4.28. There is a finitely additive Q0 2 M1;f which does not belong to M1 ; see Example A.53. The proof of Proposition 4.27 shows that there is some QQ 2 M1 such that the coherent risk measure  defined by .X / WD EQ0 Œ X  coincides with EQQ Œ X  for X 2 Cb ./. But  does not admit a representation of the form .X / D sup .EQ Œ X   ˛.Q//

for all X 2 X.

Q2M1

In fact, this would imply ˛.Q/  EQ0 Œ X   EQ Œ X  for Q 2 M1 and any X 2 X, hence ˛.Q/ D 1 for any Q 2 M1 .

}

Section 4.3 Convex risk measures on L1

4.3

199

Convex risk measures on L1

In the sequel, we fix a probability measure P on .; F / and consider risk measures  such that .X / D .Y / if X D Y P -a.s. (4.31) Note that only the nullsets of P will matter in this section. Lemma 4.32. Let  be a convex risk measure that satisfies (4.31) and which is represented by a penalty function ˛ as in (4.15). Then ˛.Q/ D C1 for any Q 2 M1;f .; F / which is not absolutely continuous with respect to P . Proof. If Q 2 M1;f .; F / is not absolutely continuous with respect to P , then there exists A 2 F such that QŒ A  > 0 but P Œ A  D 0. Take any X 2 A , and define Xn WD X  n IA . Then .Xn / D .X /, i.e., Xn is again contained in A . Hence, ˛.Q/  ˛ min .Q/  EQ Œ Xn  D EQ Œ X  C n QŒ A  ! 1 as n " 1. In view of (4.31), we can identify X with the Banach space L1 WD L1 .; F ; P /. Let us denote by M1 .P / WD M1 .; F ; P / the set of all probability measures on .; F / which are absolutely continuous with respect to P . The following theorem characterizes those convex risk measures on L1 that can be represented by a penalty function concentrated on probability measures, and hence on M1 .P /, due to Lemma 4.32. Theorem 4.33. Suppose  W L1 ! R is a convex risk measure. Then the following conditions are equivalent: (a)  can be represented by some penalty function on M1 .P /. (b)  can be represented by the restriction of the minimal penalty function ˛ min to M1 .P / .X / D

sup

.EQ Œ X   ˛ min .Q//;

X 2 L1 :

(4.32)

Q2M1 .P /

(c)  is continuous from above: If Xn & X P -a.s. then .Xn / % .X /. (d)  has the following Fatou property: for any bounded sequence .Xn / which converges P -a.s. to some X , .X /  lim inf .Xn /: n"1

200

Chapter 4 Monetary measures of risk

(e)  is lower semicontinuous for the weak topology .L1 ; L1 /. (f) The acceptance set A of  is weak closed in L1 , i.e., A is closed with respect to the topology .L1 ; L1 /. Proof. The implication (b) ) (a) is obvious, and (a) ) (c) , (d) follows as in Lemma 4.21, replacing pointwise convergence by P -a.s. convergence. (c) ) (e): We have to show that C WD ¹  cº is weak closed for c 2 R. To this end, let Cr WD C \ ¹X 2 L1 j kX k1  rº for r > 0. If .Xn / is a sequence in Cr converging in L1 to some random variable X , then there is a subsequence that converges P -a.s., and the Fatou property of  implies that X 2 Cr . Hence, Cr is closed in L1 , and Lemma A.65 implies that C WD ¹  cº is weak closed. (e) ) (f) is obvious. (f) ) (b): We fix some X 2 L1 and let mD

sup

. EQ Œ X   ˛ min .Q/ /:

(4.33)

Q2M1 .P /

In view of Theorem 4.16, we need to show that m  .X / or, equivalently, that m C X 2 A . Suppose by way of contradiction that m C X … A . Since the nonempty convex set A is weak closed by assumption, we may apply Theorem A.57 in the locally convex space .L1 ; .L1 ; L1 // with C WD A and B WD ¹m C X º. We obtain a continuous linear functional ` on .L1 ; .L1 ; L1 // such that ˇ WD inf `.Y / > `.m C X / DW  > 1: Y 2A

(4.34)

By Proposition A.59, ` is of the form `.Y / D EŒ Y Z  for some Z 2 L1 . In fact, Z  0. To show this, fix Y  0 and note that . Y /  .0/ for  0, by monotonicity. Hence Y C .0/ 2 A for all  0. It follows that 1 <  < `. Y C .0// D `.Y / C `..0//: Taking " 1 yields that `.Y /  0 and in turn that Z  0. Moreover, P Œ Z > 0  > 0 since ` is non-zero. Thus, Z dQ0 WD dP EŒ Z  defines a probability measure Q0 2 M1 .P /. By (4.34), we see that ˛ min .Q0 / D sup EQ0 Œ Y  D  Y 2A

ˇ : EŒ Z 

However, EQ0 ΠX  C m D

 ˇ `.m C X / D < D ˛ min .Q0 /; EŒ Z  EŒ Z  EŒ Z 

in contradiction to (4.33). Hence, m C X must be contained in A , and thus m  .X /.

Section 4.3 Convex risk measures on L1

201

The theorem shows that any convex risk measure of L1 that is continuous from above arises in the following manner. We consider any probabilistic model Q 2 M1 .P /, but these models are taken more or less seriously as described by the penalty function. Thus, the value .X / is computed as the worst case, over all models Q 2 M1 .P /, of the expected loss EQ Œ X , but reduced by ˛.Q/. In the following example, the given model P is the one which is taken most seriously, and the penalty function ˛.Q/ is proportional to the deviation of Q from P , measured by the relative entropy. Example 4.34. Consider the penalty function ˛ W M1 .P / ! .0; 1 defined by ˛.Q/ WD

1 H.QjP /; ˇ

where ˇ > 0 is a given constant and h dQ i H.QjP / D EQ log dP is the relative entropy of Q 2 M1 .P / with respect to P ; see Definition 3.20. The corresponding entropic risk measure ˇ is given by ˇ .X / D

  1 EQ Œ X   H.QjP / : ˇ Q2M1 .P / sup

The variational principle for the relative entropy as stated in Lemma 3.29 shows that EQ ΠX  

1 1 H.QjP /  log EŒ e ˇX ; ˇ ˇ

and the upper bound is attained by the measure with the density e ˇX =EŒ e ˇX . Thus, the entropic risk measure takes the form ˇ .X / D

1 log EŒ e ˇX : ˇ

Note that ˛ is in fact the minimal penalty function representing ˇ , since Lemma 3.29 implies   1 1 ˛ min .Q/ D sup EQ Œ X   log EŒ e ˇX  D H.QjP /: ˇ ˇ X2L1 A financial interpretation of the entropic risk measure in terms of shortfall risk will be discussed in Example 4.114. } Exercise 4.3.1. Show that the entropic risk measure ˇ converges to the worst-case risk measure for ˇ " 1 and to the expected loss under P for ˇ # 0. }

202

Chapter 4 Monetary measures of risk

The following corollary characterizes those convex risk measures on L1 that satisfy the property of continuity from below, which is stronger than continuity from above. Corollary 4.35. For a convex risk measure  on L1 , the following conditions are equivalent: (a)  is continuous from below: Xn % X H) .Xn / & .X /. (b)  satisfies the Lebesgue property: .Xn / ! .X / whenever .Xn / is a bounded sequence in L1 which converges P -a.s. to X . (c) The minimal penalty function ˛ min is concentrated on M1 .P /, i.e., ˛ min .Q/ < 1 implies Q 2 M1 .P /. In particular we have .X / D

max

Q2M1 .P /

.EQ Œ X   ˛ min .Q//;

X 2 L1 ;

whenever one of these three equivalent conditions is satisfied. Proof. The equivalence of conditions (a) and (b) was shown in Exercise 4.2.2. The equivalence of conditions (a) and (c) follows from Theorem 4.22 and Lemma 4.32. Exercise 4.3.2. Show that the three conditions in Corollary 4.35 are equivalent to the following fourth condition: (d) For each c 2 R, the level set ƒc WD ¹Q j ˛ min .Q/  cº is contained in M1 .P /, and the corresponding set of densities, ³ ² dQ ˇˇ ˇ Q 2 ƒc ; dP is weakly compact in L1 .; F ; P /. Hint: Use Lemma 4.23 and the Dunford–Pettis theorem (Theorem A.67).

}

Example 4.36. Let g W Œ0; 1Œ! R [ ¹C1º be a lower semicontinuous convex function satisfying g.1/ < 1 and the superlinear growth condition g.x/=x ! C1 as x " 1. Associated with it is the g-divergence h  dQ  i Ig .QjP / WD E g ; dP

Q 2 M1 .P /:

(4.35)

The g-divergence Ig .QjP / quantifies the deviation of the hypothetical model Q from the reference measure P . Thus, ˛g .Q/ WD Ig .QjP /;

Q 2 M1 .P /;

Section 4.3 Convex risk measures on L1

203

is a natural choice for a penalty function. The resulting risk measure g .X / WD sup .EQ ΠX   Ig .QjP //

(4.36)

QP

is sometimes called divergence risk measure. Note that, for g.x/ D ˇ1 x log x, g is just the entropic risk measure discussed in Example 4.34. Divergence risk measures will be discussed in more detail in Section 4.9. } Exercise 4.3.3. Show that the risk measure in (4.36) is continuous from below and that ˛g .Q/ D Ig .QjP /, Q 2 M1 .P /, is its minimal penalty function. In particular, the supremum in (4.36) is in fact a maximum. Hint: Using Exercise 4.3.2 can be helpful. } Theorem 4.33 takes the following form for coherent risk measures; the proof is the same as the one for Corollary 4.19. Corollary 4.37. A coherent risk measure on L1 can be represented by a set Q  M1 .P / if and only if the equivalent conditions of Theorem 4:33 are satisfied. In this case, the maximal representing subset of M1 .P / is given by Qmax WD ¹Q 2 M1 .P / j ˛ min .Q/ D 0º: Let us also state a characterization of those coherent risk measures on L1 which are continuous from below. Corollary 4.38. For a coherent risk measure  on L1 the following properties are equivalent: (a)  is continuous from below: Xn % X H) .Xn / & .X /. (b)  satisfies the Lebesgue property: .Xn / ! .X / whenever .Xn / is a bounded sequence in L1 which converges P -a.s. to X . (c) We have Qmax  M1 .P /. (d) The set of densities

²

³ dQ ˇˇ ˇ Q 2 Qmax dP

is weakly compact in L1 .; F ; P /. In this case, the representation .X / D max EQ ΠX ; Q2Qmax

involves only -additive probability measures.

X 2 L1 ;

204

Chapter 4 Monetary measures of risk

We now give three examples of coherent risk measures which will be studied in more detail in Section 4.4. Example 4.39. In our present context, where we require condition (4.31), the worstcase risk measure takes the form max .X / WD  ess inf X D inf¹m 2 R j X C m  0 P -a.s.º: One can easily check that max is coherent and satisfies the Fatou property. Moreover, 1 the acceptance set of max is equal to the positive cone L1 C in L , and this implies min ˛ .Q/ D 0 for any Q 2 M1 .P /. Thus, max .X / D

sup

EQ ΠX :

Q2M1 .P /

Note however that the supremum on the right cannot be replaced by a maximum as soon as .; F ; P / cannot be reduced to a finite model. Indeed, in that case there exists X 2 L1 such that X does not attain its essential infimum, and so there can be no Q 2 M1 .P / such that EQ ΠX  D ess inf X D max .X /. In this case, the preceding corollary shows that max is not continuous from below. } Example 4.40. Let Q be the class of all Q 2 M1 .P / whose density dQ=dP is bounded by 1= for some fixed parameter 2 .0; 1/. The corresponding coherent risk measure AV@R .X / WD sup EQ ΠX  (4.37) Q2Q

will be called the Average Value at Risk at level . This terminology will become clear in Section 4.4, which contains a detailed study of AV@R . By taking g.x/ WD 0 for x  1 and g.x/ WD C1 for x > 1 , one sees that AV@R falls into the class of divergence risk measures as introduced in Example 4.36. It follows from Exercise 4.3.3 that Q is equal to the maximal representing subset of AV@R , that AV@R is continuous from below, and that the supremum in (4.37) is actually attained. An explicit construction of the maximizing measure will be given in the proof of Theorem 4.52. } Example 4.41. We take for Q the class of all conditional distributions P Œ  j A  such that A 2 F has P Œ A  > for some fixed level 2 .0; 1/. The coherent risk measure induced by Q, WCE .X / WD sup¹ EŒ X j A  j A 2 F ; P Œ A  > º;

(4.38)

is called the worst conditional expectation at level . We will show in Section 4.4 that it coincides with the Average Value at Risk of Example 4.40 if the underlying probability space is rich enough. }

Section 4.3 Convex risk measures on L1

205

Let  be a convex risk measure with the Fatou property. We now consider the situation in which  admits a representation in terms of equivalent probability measures Q  P , i.e., .X / D sup .EQ Œ X   ˛ min .Q//;

X 2 L1 :

(4.39)

Q P

We will show in the next theorem that this property can be characterized by the following concept of sensitivity, which is sometimes also called relevance. It formalizes the idea that  should react to every nontrivial loss at a sufficiently high level. Definition 4.42. A convex risk measure  on L1 is called sensitive with respect to P if for every nonconstant X 2 L1 with X  0 there exists > 0 such that . X / > .0/. Theorem 4.43. For a convex risk measure with the Fatou property, the following conditions are equivalent: (a)  admits the representation (4.39) in terms of equivalent probability measures. (b)  is sensitive with respect to P . (c) For every A 2 F with P Œ A  > 0 there exists > 0 such that . IA / > .0/. (d) For every A 2 F with P Œ A  > 0 there exists Q 2 M1 .P / with QŒ A  > 0 and ˛ min .Q/ < 1. (e) There exists Q  P with ˛ min .Q/ < 1. Proof. Throughout the proof we will assume for simplicity that  is normalized in the sense that .0/ D 0. This can be done without loss of generality. (a) ) (b): Take X 2 L1 C with EŒ X  > 0. By (4.39) there exists Q  P with min ˛ .Q/ < 1 and . X /  EQ Œ X   ˛ min .Q/: The right-hand side is strictly positive as soon as

>

˛ min .Q/ ; EQ Œ X 

which is a finite number since EQ Œ X  > 0 due to Q  P . The implications “(b) ) (c)” and “(c) ) (d)” are both obvious. (d) ) (e): For every c > 0 the level set ƒc WD ¹Q 2 M1 .P / j ˛ min .Q/  cº is nonempty due to our assumption .0/ D 0. We will show that ƒc contains some Q  P . We show first the following auxiliary claim: For any A 2 F with P Œ A  > 0 there exists Q 2 ƒc with QŒ A  > 0.

(4.40)

206

Chapter 4 Monetary measures of risk

Indeed, (d) implies that there is Q0 2 M1 .P / with Q0 Œ A  > 0 and ˛ min .Q0 / < 1. Now we take Q1 2 ƒc=2 and let Q" WD "Q0 C .1  "/Q1 for 0 < " < 1. We clearly have Q" Œ A  > 0 and c ˛ min .Q" /  "˛ min .Q1 / C .1  "/ < c 2 for sufficiently small " > 0. This implies (4.40). We now apply the Halmos–Savage theorem in the form of Theorem 1.61. It yields the existence of Q 2 ƒc with Q  P if we can show that ƒc is countably convex. To show countable convexity, let .ˇk /k2N be a sequence of nonnegative real numbers P k summing up to 1 and take Qk 2 ƒc for k 2 N. We define Q WD 1 kD1 ˇk Q . Then Q 2 M1 .P / and ˛ min .Q/ D sup EQ Œ X  D sup X2A



1 X kD1

1 X

X2A kD1

ˇk sup EQk Œ X  D X2A

ˇk EQk Œ X 

1 X

ˇk ˛ min .Qk /  c:

kD1

Thus, Q belongs to ƒc , and (e) follows. (e) ) (a): By the representation (4.32) in Theorem 4.33 it is sufficient to show that .X / D

sup Q2M1 .P /

.EQ Œ X   ˛ min .Q//  sup .EQ Œ X   ˛ min .Q// Q P

for any given X 2 L1 . To this end, we take ı > 0 and choose Q1 2 M1 .P / such that EQ1 Œ X   ˛ min .Q1 / > .X /  ı: Then we take Q0  P with ˛ min .Q0 / < 1, which exists due to (e). When letting Q" WD "Q0 C .1  "/Q1 we have Q"  P for all " 2 .0; 1/ and ˛ min .Q" /  "˛ min .Q1 / C .1  "/˛ min .Q0 /. Hence, EQ" Œ X   ˛ min .Q" /  ".EQ0 Œ X   ˛ min .Q0 // C .1  "/.EQ1 Œ X   ˛ min .Q1 //; and this is larger that .X /  ı when " is sufficiently small. Condition (e) in Theorem 4.43 is of course satisfied when ˛ min .P / < 1. This is the case for the worst conditional expectation WCE (Example 4.41), Average Value at Risk AV@R (Example 4.40), the divergence risk measures (Example 4.36), and in particular the entropic risk measure (Example 4.34). These, and many other risk measures, are therefore sensitive and admit a representation (4.39) in terms of equivalent risk measures.

207

Section 4.4 Value at Risk

Remark 4.44. In analogy to Remark 4.18, the implication (e) ) (a) in the Representation Theorem 4.33 can be viewed as a special case of the general duality in Theorem A.62 for the Fenchel–Legendre transform of the convex function  on L1 , combined with the properties of a monetary risk measure. From this general point of view, it is now clear how to state representation theorems for convex risk measures on the Banach spaces Lp .; F ; P / for 1  p < 1. More precisely, let q 2 .1; 1 be such that p1 C q1 D 1, and define ° ± ˇ dQ q M1 .P / WD Q 2 M1 .P / ˇ 2 Lq : dP A convex risk measure  on Lp is of the form .X / D

sup q

.EQ Œ X   ˛.Q//

Q2M1 .P /

if and only if it is lower semicontinuous on Lp , i.e., the Fatou property holds in the form Xn ! X in Lp H) .X /  lim inf .Xn /: } n"1

4.4

Value at Risk

A common approach to the problem of measuring the risk of a financial position X consists in specifying a quantile of the distribution of X under the given probability measure P . For 2 .0; 1/, a -quantile of a random variable X on .; F ; P / is any real number q with the property P ΠX  q   and

P ΠX < q   ;

and the set of all -quantiles of X is an interval ŒqX . /; qXC . /, where qX .t / D sup¹x j P Œ X < x  < t º D inf¹x j P Œ X  x   t º is the lower and qXC .t / D inf¹x j P Œ X  x  > t º D sup¹x j P Œ X < x   t º is the upper quantile function of X ; see Appendix A.3. In this section, we will focus on the properties of qXC . /, viewed as a functional on a space of financial positions X . Definition 4.45. Fix some level 2 .0; 1/. For a financial position X , we define its Value at Risk at level as  V@R .X / WD qXC . / D qX .1  / D inf¹m j P Œ X C m < 0   º:

(4.41)

208

Chapter 4 Monetary measures of risk

In financial terms, V@R .X / is the smallest amount of capital which, if added to X and invested in the risk-free asset, keeps the probability of a negative outcome below the level . However, Value at Risk only controls the probability of a loss; it does not capture the size of such a loss if it occurs. Clearly, V@R is a monetary risk measure on X D L0 , which is positively homogeneous; see also Example 4.11. The following example shows that the acceptance set of V@R is typically not convex, and so V@R is not a convex risk measure. Thus, V@R may penalize diversification instead of encouraging it. Example 4.46. Consider an investment into two defaultable corporate bonds, each with return rQ > r, where r  0 is the return on a riskless investment. The discounted net gain of an investment w > 0 in the i th bond is given by ´ w in case of default, Xi D w. rQ r/ otherwise. 1Cr If a default of the first bond occurs with probability p  , then i h w. rQ  r/ < 0 D P Œ 1st bond defaults  D p  : P X1  1Cr Hence, w. rQ  r/ < 0: 1Cr This means that the position X1 is acceptable in the sense that is does not carry a positive Value at Risk, regardless of the possible loss of the entire investment w. Diversifying the portfolio by investing the amount w=2 into each of the two bonds leads to the position Y WD .X1 C X2 /=2. Let us assume that the two bonds default independently of each other, each of them with probability p. For realistic r, Q the probability that Y is negative is equal to the probability that at least one of the two bonds defaults: P Œ Y < 0  D p.2  p/. If, for instance, p D 0:009 and D 0:01 then we have p < < p.2  p/, hence   rQ  r w V@R .Y / D  1  : 2 1Cr

V@R .X1 / D 

Typically, this value is close to one half of the invested capital w. In particular, the acceptance set of V@R is not convex. This example also shows that V@R may strongly discourage diversification: It penalizes quite drastically the increase of the probability that something goes wrong, without rewarding the significant reduction of the expected loss conditional on the event of default. Thus, optimizing a portfolio with respect to V@R may lead to a concentration of the portfolio in one single asset with a sufficiently small default probability, but with an exposure to large losses. }

209

Section 4.4 Value at Risk

Exercise 4.4.1. Let .Yn / be a sequence of independent and identical distributed random variables in L1 .; F ; P /. Show that

V@R

n 1 X

n

 Yi ! EΠY1 

as n " 1

iD1

for any 2 .0; 1/. Choose the common distribution in such a way that convexity is violated for large n, i.e.,

V@R

n 1 X

n

 Yi > V@R .Y1 /:

}

iD1

In the remainder of this section, we will focus on monetary measures of risk which, in contrast to V@R , are convex or even coherent on X WD L1 . In particular, we are looking for convex risk measures which come close to V@R . A first guess might be that one should take the smallest convex measure of risk, continuous from above, which dominates V@R . However, since V@R itself is not convex, the following proposition shows that such a smallest V@R -dominating convex risk measure does not exist. Proposition 4.47. For each X 2 X and each 2 .0; 1/,

V@R .X / D min¹.X / j  is convex, continuous from above, and  V@R º: Proof. Let q WD V@R .X / D qXC . / so that P Œ X < q   . If A 2 F satisfies P Œ A  > , then P Œ A \ ¹X  qº  > 0. Thus, we may define a measure QA by QA WD P Œ  j A \ ¹X  qº : It follows that EQA Œ X   q D V@R .X /. Let Q WD ¹QA j P Œ A  > º, and use this set to define a coherent risk measure  via .Y / WD sup EQ Œ Y : Q2Q

Then .X /  V@R .X /. Hence, the assertion will follow if we can show that .Y /  V@R .Y / for each Y 2 X. Let " > 0 and A WD ¹Y  V@R .Y / C "º. Clearly P Œ A  > , and so QA 2 Q. Moreover, QA Œ A  D 1, and we obtain .Y /  EQA Œ Y   V@R .Y /  ": Since " > 0 is arbitrary, the result follows. For the rest of this section, we concentrate on the following risk measure which is defined in terms of Value at Risk, but does satisfy the axioms of a coherent risk measure.

210

Chapter 4 Monetary measures of risk

Definition 4.48. The Average Value at Risk at level 2 .0; 1 of a position X 2 X is given by Z 1

AV@R .X / D V@R .X / d:

0 Sometimes, the Average Value at Risk is also called the “Conditional Value at Risk”or the “expected shortfall”, and one writes CV@R .X / or ES .X /. These terms are motivated by formulas (4.44) and (4.42) below, but they are potentially misleading: “Conditional Value at Risk” might also be used to denote the Value at Risk with respect to a conditional distribution, and “expected shortfall” might be understood as the expectation of the shortfall X  . For these reasons, we prefer the term Average Value at Risk. Note that Z 1

AV@R .X / D  qX .t / dt

0 by (4.41). In particular, the definition of AV@R .X / makes sense for any X 2 L1 .; F ; P / and we have, in view of Lemma A.19, Z 1 AV@R1 .X / D  qXC .t / dt D EΠX : 0

Exercise 4.4.2. Compute AV@R .X / when X is (a) uniform, (b) normally distributed, (c) log-normally distributed, i.e., X D e ZCm with Z  N.0; 1/ and m; 2 R. Recalling the results from Example 4.11 and Exercise 4.1.5, compare the behavior of V@R .X / and AV@R .X / as the parameter increases from 0 to 1. } Remark 4.49. Theorem 2.57 shows that the partial order 0 define X .c/ 2 X by X .c/ WD c.' ^ k/I¹'1=0 º : Since PŒX

.c/



1 < 0 D P '  0



 0 < ;

we have V@R .X .c/ / D 0, and (4.42) yields that

AV@R .X

.c/

  1 1 c .c/ / D EΠX  D E ' ^ kI '  0 :



On the other hand, EQ ΠX

.c/



1  D c  E '  ' ^ kI '  0



  1 c  0 E ' ^ kI '  0 :

Thus, the difference between EQ Œ X .c/  and AV@R .X .c/ / becomes arbitrarily large as c " 1. Remark 4.53. The proof shows that for 2 .0; 1/ the maximum in (4.43) is attained by the measure Q0 2 Q , whose density is given by 1 dQ0 D .I¹X 0, and let A WD ¹X  V@R .X /  "º and Y WD EŒ X j X IAc  D X  IAc C EŒ X j A   IA : Since Y > qXC . / C "  EŒ X j A  on Ac , we get P Œ Y < EŒ X j A   D 0. On the other hand, P Œ Y  EŒ X j A    P Œ A  > , and this implies that V@R .Y / D EŒ X j A . Since  dominates V@R , we have .Y /  EŒ X j A . Thus, .X /  .Y / D EŒ X j X  V@R .X /  " ; by Corollary 4.65. Taking " # 0 yields .X /  EŒ X j X  V@R .X / : If the distribution of X is continuous, Corollary 4.54 states that the conditional expectation on the right equals AV@R .X /, and we obtain (4.50). When the distribution of X is not continuous, we denote by D the set of all points x such that P Œ X D x  > 0 and take any bounded random variable Z  0 with a continuous distribution. Such a random variable exists due to our assumption that .; F ; P / is atomless. Note that Xn WD X C n1 ZI¹X 2Dº has a continuous distribution. Indeed, for any x, X P Œ Xn D x  D P Œ X D x; X … D  C P Œ X D y; Z D n.x  y/  D 0: y2D

Moreover, Xn decreases to X . The inequality (4.50) holds for each Xn and extends to X by continuity from above. Corollary 4.68. AV@R and WCE coincide under our assumption that the probability space is atomless. Proof. We know from Corollary 4.54 that WCE .X / D AV@R .X / if X has a continuous distribution. Repeating the approximation argument at the end of the preceding proof yields WCE .X / D AV@R .X / for each X 2 X.

4.6

Concave distortions

Let us now have a closer look at the coherent risk measures Z  .X / WD AV@R .X / .d /;

(4.51)

which appear in the Representation Theorem 4.62 for law-invariant convex risk measures. We are going to characterize these risk measures  in two ways, first as Choquet integrals with respect to some concave distortion of the underlying probability measure P , and then, in the next section, by a property of comonotonicity.

220

Chapter 4 Monetary measures of risk

Again, we will assume throughout this section that the underlying probability space .; F ; P / is atomless. Since AV@R is coherent, continuous from below, and lawinvariant, any mixture  for some probability measure on .0; 1 has the same properties. According to Remark 4.50, we may set AV@R0 .X / D  ess inf X so that we can extend the definition (4.51) to probability measures on the closed interval Œ0; 1. However,  will only be continuous from above and not from below if .¹0º/ > 0, because AV@R0 is not continuous from below. Our first goal is to show that  .X / can be identified with the Choquet integral of the loss X with respect to the set function c .A/ WD .P Œ A /, where is the concave function defined in the following lemma. Choquet integrals were introduced in Example 4.14, and the risk measure MINVAR of Exercise 4.1.7 provides a first example for a risk measure arising as the Choquet integral of a set function c . Recall 0 ; see that every concave function admits a right-continuous right-hand derivative C Proposition A.4. Lemma 4.69. The identity 0 C .t /

Z

s 1 .ds/;

D

0 < t < 1;

(4.52)

.t;1

defines a one-to-one correspondence between probability measures on Œ0; 1 and increasing concave functions W Œ0; 1 ! Œ0; 1 with .0/ D 0 and .1/ D 1. Moreover, we have .0C/ D .¹0º/. Proof. Suppose first that is given and is defined by is concave and increasing on .0; 1. Moreover, Z 1

1

.0C/ D

0

Z .t / dt D .0;1

0

1 s

Z

.1/ D 1 and (4.52). Then

1 0

I¹t x / dx: 0

Proof. Using the fact that V@R .X / D qX .1  /, we get as in (4.49) that Z 1 Z AV@R .X / .d / D qX .t / 0 .1  t / dt: .0;1

0

Hence, we obtain the first identity. For the second one, we will first assume X  0. Then Z 1 qXC .t / D sup¹x  0 j FX .x/  t º D

0

I¹FX .x/t º dx;

where FX is the distribution function of X . Using Fubini’s theorem, we obtain Z 1 Z 1Z 1 0 qX .t / .1  t / dt D I¹FX .x/1t º 0 .t / dt dx 0

0

Z

0

1

.1  FX .x// dx 

D

.0C/ ess sup X;

0

Ry since 0 0 .t / dt D . .y/  .0C//I¹y>0º . This proves the second identity for X  0, since .0C/ D .¹0º/ and ess sup X D AV@R0 .X /. If X 2 L1 is arbitrary, we consider X C C , where C WD  ess inf X . The cash invariance of  yields Z 1 C C  .X / D .P Œ X > x  C / dx Z

0

Z

0

D

.P ΠX > x / dx C C

Z

1

.P ΠX > x / dx 0

0

DC C

Z

. .P ΠX > x /  1/ dx C 1

1

.P ΠX > x / dx: 0

Example 4.71. Clearly, the risk measure AV@R is itself of the form  where D ı . For > 0, the corresponding concave distortion function is given by t  1 .t / D ^ 1 D .t ^ /:

Thus, we obtain yet another representation of AV@R : Z 1 1 AV@R .X / D P ΠX > x  ^ dx for X 2 L1 } C.

0

222

Chapter 4 Monetary measures of risk

Exercise 4.6.1. Find the probability measure on Œ0; 1 such that the coherent risk measure MINVAR introduced in Exercise 4.1.7 is of the form (4.51). } Exercise 4.6.2. Suppose that there exists a set A 2 F with 0 < P Œ A  < 1 such that  .IA / D  .IA /. Use the representation (4.53) to deduce that D ı1 , i.e.,  .X / D EŒ X  for all X 2 L1 . More generally, show that D ı1 if there exists a nonconstant X 2 L1 such that  .X / D  .X /. } Corollary 4.72. If .¹0º/ D 0 in Theorem 4.70, then Z 1 qX .'.t // dt;  .X / D  0

where ' is an inverse function of

, taken in the sense of Definition A.14.

Proof. Due to Lemma A.15, the distribution of ' under the Lebesgue measure has the distribution function and hence the density 0 . Therefore Z 1 Z 1 Z 1 0 qX .'.t // dt D qX .t / .t / dt D  qX .1  t / 0 .t / dt; 0

0

0

where we have used Lemma A.23 in the last step. An application of Theorem 4.70 concludes the proof. Let us continue with a brief discussion of the set function c .A/ D

.P ΠA /.

Definition 4.73. Let W Œ0; 1 ! Œ0; 1 be an increasing function such that and .1/ D 1. The set function c .A/ WD

.P ΠA /;

.0/ D 0

A2F;

is called the distortion of the probability measure P with respect to the distortion function . Definition 4.74. A set function c W F ! Œ0; 1 is called monotone if c.A/  c.B/

for A  B

and normalized if c.;/ D 0

and

c./ D 1:

A monotone set function is called submodular or 2-alternating if c.A [ B/ C c.A \ B/  c.A/ C c.B/: Clearly, any distortion c is normalized and monotone.

223

Section 4.6 Concave distortions

Proposition 4.75. Let c be the distortion of P with respect to the distortion function . If is concave, then c is submodular. Moreover, if the underlying probability space is atomless, then also the converse implication holds. Proof. Suppose first that is concave. Take A; B 2 F with P ΠA   P ΠB . We must show that c WD c satisfies c.A/  c.A \ B/  c.A [ B/  c.B/: This is trivial if r D 0, where r WD P ΠA   P ΠA \ B  D P ΠA [ B   P ΠB : For r > 0 the concavity of

yields via (A.1) that

c.A/  c.A \ B/ c.A [ B/  c.B/  : PŒA  PŒA \ B  PŒA [ B   PŒB  Multiplying both sides with r gives the result. Now suppose that c D c is submodular and assume that .; F ; P / is atomless. By Exercise A.1.1 below it is sufficient to show that .y/  . .x/ C .z//=2 whenever 0  x  z  1 and y D .x C z/=2. To this end, we will construct two sets A; B  F such that P Œ A  D P Œ B  D y, P Œ A \ B  D x, and P Œ A [ B  D z. Submodularity then gives .x/ C .z/  2 .y/ and in turn the concavity of . In order to construct the two sets A and B, take a random variable U with a uniform distribution on Œ0; 1, which exists by Proposition A.27. Then A WD ¹0  U  yº

and

B WD ¹z  y  U  zº

are as desired. Let us now recall the notion of a Choquet integral, which was introduced in Example 4.14. Definition 4.76. Let c W F ! Œ0; 1 be any set function which is normalized and monotone. The Choquet integral of a bounded measurable function X on .; F / with respect to c is defined as Z

Z

Z

0

X dc WD

1

.c.X > x/  1/ dx C 1

c.X > x/ dx: 0

Note that the Choquet integral coincides with the usual integral as soon as c is a -additive probability measure; see also Lemma 4.97 below. With this definition, Theorem 4.70 allows us to identify the risk measure  as the Choquet integral of the loss with respect to a concave distortion c of the underlying probability measure P .

224

Chapter 4 Monetary measures of risk

Corollary 4.77. For a probability measure on Œ0; 1, let be the concave distortion function defined in Lemma 4:69, and let c denote the distortion of P with respect to . Then, for X 2 L1 , Z  .X / D

.X / dc :

Combining Corollary 4.77 with Theorem 4.62, we obtain the following characterization of law-invariant convex risk measures in terms of concave distortions: Corollary 4.78. A convex risk measure  is law-invariant and continuous from above if and only if  Z .X / D sup .X / dc   min . / ; where the supremum is taken over the class of all concave distortion functions Z min .X / dc :  . / WD sup

and

X2A

The following series of exercises should be compared with Exercises 4.1.7 and 4.6.1. Exercise 4.6.3. For ˇ  1 consider the concave distortion function Show that for ˇ 2 N the corresponding risk measure Z MAXVARˇ .X / WD .X / dc ˇ

ˇ .x/

1

WD x ˇ .

has the property that MAXVARˇ .X / D EŒ Y1 , if Y1 ; : : : ; Yˇ are independent and identically distributed (i.i.d.) random variables for which max.Y1 ; : : : ; Yˇ /  X . } 1

Exercise 4.6.4. For ˇ  1 consider the distortion function Q ˇ .x/ WD .1.1x/ˇ / ˇ . Show that for ˇ 2 N the corresponding risk measure Z MAXMINVARˇ .X / WD .X / dc Q ˇ has the property that MAXMINVARˇ .X / D EŒ Y1 , if Y1 ; : : : ; Yˇ are i.i.d. random variables for which max.Y1 ; : : : ; Yˇ /  min.X1 ; : : : ; Xˇ /, when X1 ; : : : ; Xˇ are independent copies of X . } 1

Exercise 4.6.5. For ˇ  1 consider the distortion function O ˇ .x/ WD .1.1x/ ˇ /ˇ . Show that for ˇ 2 N the corresponding risk measure Z MINMAXVARˇ .X / WD .X / dc O ˇ has the property that MINMAXVARˇ .X / D EŒ min.Y1 ; : : : ; Yˇ / , if Y1 ; : : : ; Yˇ are i.i.d. random variables for which max.Y1 ; : : : ; Yˇ /  X . }

225

Section 4.6 Concave distortions

As another consequence of Theorem 4.70, we obtain an explicit description of the maximal representing set Q  M1 .P / for the coherent risk measure  . Theorem 4.79. Let be a probability measure on Œ0; 1, and let be the corresponding concave function defined in Lemma 4:69. Then  can be represented as  .X / D sup EQ Œ X ; Q2Q

where the set Q is given by Z 1 ° ˇ dQ ˇ Q WD Q 2 M1 .P / Z WD satisfies qZ .s/ ds  dP t

± .1  t / for t 2 .0; 1/ :

Moreover, Q is the maximal subset of M1 .P / that represents  . Proof. The risk measure  is coherent and continuous from above. By Corollary 4.37, it can be represented by taking the supremum of expectations over the set Qmax D ¹Q 2 M1 .P / j ˛ min .Q/ D 0º. Using (4.47) and Theorem 4.70, we see that a measure Q 2 M1 .P / with density Z D dQ=dP belongs to Qmax if and only if Z 1 qX .s/qZ .s/ ds   .X / 0 (4.54) Z 1

D

.0C/AV@R0 .X / C

qX .s/

0

.1  s/ ds

0

for all X 2 L1 . For constant random variables X  t , we have qX D IŒt;1 a.e., and so we obtain Z 1 Z 1 0 qZ .s/ ds  .0C/ C .1  s/ ds D .1  t / t

t

Qmax

 Q . For the proof of the converse inclusion, we for all t 2 .0; 1/. Hence show that the density Z of a fixed measure Q 2 Q satisfies (4.54) for any given X 2 L1 . We may assume without loss of generality that X  0. Let  be the positive finite measure on Œ0; 1 such that qXC .s/ D .Œ0; s/. Using Fubini’s theorem and the definition of Q , we get Z 1 Z 1 Z qX .s/qZ .s/ ds D qZ .s/ ds .dt / 0

Z

Œ0;1

t

.1  t / .dt /

 Œ0;1

D

Z

1

.0C/.Œ0; 1/ C 0

which coincides with the right-hand side of (4.54).

0

Z .1  s/

.dt / ds; Œ0;s

226

Chapter 4 Monetary measures of risk

Corollary 4.80. In the context of Theorem 4.79, the following conditions are equivalent: (a)  is continuous from below. (b) .¹0º/ D 0. (c)  .X / D maxQ2Q EQ Œ X  for all X 2 L1 . If these equivalent conditions are satisfied, then the maximum in (c) is attained by the measure QX 2 Q with density dQX =dP D f .X /, where f is the decreasing function defined by f .x/ WD 0 .FX .x// if x is a continuity point of FX , and by f .x/ WD

1 FX .x/  FX .x/

Z

FX .x/

0

.t / dt

FX .x/

otherwise. Moreover, with denoting the Lebesgue measure on .0; 1/, °  dQ 1 ˇ 0. In this case, we can write  D ı AV@R0 C .1  ı/0 ; where 0 WD .  j.0; 1/. Then 0 is continuous from below since 0 .¹0º/ D 0, but AV@R0 is not, and so  does not satisfy (a); see Remark 4.50. Let us now prove the remaining assertions. Since .0C/ RD .¹0º/ D 0, Ra measure 1 1 Q with density Z D dQ=dP belongs to Q if and only if t qZ .s/ ds  t 0 .1  0 0 s/ ds for all t . Since .1  t / is a quantile function for the law of under , part (e) of Theorem 2.57 implies (4.55). The problem of identifying the maximizing measure QX is hence equivalent to minimizing EŒ ZX  under the constraint that Z is a density function such that P ı Z 1 x/  1/ dx C c.X > x/ dx: 1

0

The proof of the following proposition was already given in Example 4.14. Proposition 4.84. The Choquet integral of the loss, Z .X / WD .X / dc; is a monetary risk measure on X which is positively homogeneous.

229

Section 4.7 Comonotonic risk measures

Definition 4.85. Let X be a measurable function on .; F /. An inverse function rX W .0; 1/ ! R of the increasing function GX .x/ WD 1  c.X > x/, taken in the sense of Definition A.14, is called a quantile function for X with respect to c. If c is a probability measure, then GX .x/ D c.X  x/. Hence, the preceding definition extends the notion of a quantile function given in Definition A.20. The following proposition yields an alternative representation of the Choquet integral in terms of quantile functions with respect to c. Proposition 4.86. Let rX be a quantile function with respect to c for X 2 X. Then Z Z 1 X dc D rX .t / dt: 0

R

R

Proof. We have .X C m/ dc D X dc C m, and one easily checks that rXCm D rX C m a.e. for all m 2 R and each quantile function rXCm of X C m. Thus, we may assume without loss of generality that X  0. In this case, Remark A.16 and Lemma A.15 imply that the largest quantile function rXC is given by Z 1 C rX .t / D sup¹x  0 j GX .x/  t º D I¹GX .x/t º dx: 0

Since rX D rXC a.e. on .0; 1/, Fubini’s theorem implies Z

1Z 1

Z

1

rX .t / dt D 0

Z

0

0

I¹GX .x/t º dx dt

1

.1  GX .x// dx

D Z

0

D

X dc:

The preceding proposition yields the following generalization of Corollary 4.72 when applied to a continuous distortion of a probability measure as defined in Definition 4.73. Corollary 4.87. Let c .A/ D .P ΠA / be the distortion of the probability measure P with respect to the continuous distortion function . If ' is an inverse function for the increasing function in the sense of Definition A.14, then the Choquet integral with respect to c satisfies Z

Z X dc D

1

qX .1  '.t // dt; 0

where qX is a quantile function for X 2 X, taken with respect to P .

230

Chapter 4 Monetary measures of risk

Proof. Due to the continuity of , we have .a/  t if and only if a  ' C .t / D inf¹x j .x/ > t º. Thus, we can compute the lower quantile function of X with respect to c rX .t / D inf¹x 2 R j 1  c .X > x/  t º D inf¹x 2 R j .P Œ X > x /  1  t º D inf¹x 2 R j P Œ X > x   ' C .1  t /º D qX .1  ' C .1  t //: Next note that ' C .t / D '.t / for a.e. t . Moreover, ' has the continuous distribution function under the Lebesgue measure, and so we can replace qX by the arbitrary quantile function qX . Theorem 4.88. A monetary risk measure  on X is comonotonic if and only if there exists a normalized monotone set function c on .; F / such that Z .X / D .X / dc; X 2 X: In this case, c is given by c.A/ D .IA /. The preceding theorem implies in view of Corollary 4.77 that all mixtures Z AV@R .d /  D Œ0;1

are comonotonic. We will see in Theorem 4.93 below that these are in fact all convex risk measures that are law-invariant and comonotonic. The proof of Theorem 4.88 requires a further analysis of comonotone random variables. Lemma 4.89. Two measurable functions X and Y on .; F / are comonotone if and only if there exists a third measurable function Z on .; F / and increasing functions f and g on R such that X D f .Z/ and Y D g.Z/. Proof. Clearly, X WD f .Z/ and Y WD g.Z/ are comonotone for given Z, f , and g. Conversely, suppose that X and Y are comonotone and define Z by Z WD X C Y . We show that z WD Z.!/ has a unique decomposition as z D x C y, where .x; y/ D .X.! 0 /; Y .! 0 // for some ! 0 2 . Having established this, we can put f .z/ WD x and g.z/ WD y. The existence of the decomposition as z D x C y follows by taking x WD X.!/ and y WD Y .!/, so it remains to show that these are the only possible values x and y. To this end, let us suppose that X.!/ C Y .!/ D z D X.! 0 / C Y .! 0 / for some ! 0 2 . Then X.!/  X.! 0 / D .Y .!/  Y .! 0 //;

231

Section 4.7 Comonotonic risk measures

and comonotonicity implies that this expression vanishes. Hence x D X.! 0 / and y D Y .! 0 /. Next, we check that both f and g are increasing functions on Z./. So let us suppose that X.!1 / C Y .!1 / D z1  z2 D X.!2 / C Y .!2 /: This implies X.!1 /  X.!2 /  .Y .!1 /  Y .!2 //: Comonotonicity thus yields that X.!1 /X.!2 /  0 and Y .!1 /Y .!2 /  0, whence f .z1 /  f .z2 / and g.z1 /  g.z2 /. Thus, f and g are increasing on Z./, and it is straightforward to extend them to increasing functions defined on R. Lemma 4.90. If X; Y 2 X is a pair of comonotone functions, and rX , rY , rXCY are quantile functions with respect to c, then rXCY .t / D rX .t / C rY .t /

for a.e. t .

Proof. Write X D f .Z/ and Y D g.Z/ as in Lemma 4.89. The same argument as in the proof of Lemma A.23 shows that f .rZ / and g.rZ / are quantile functions for X and Y under c if rZ is a quantile function for Z. An identical argument applied to the increasing function h WD f C g shows that h.rZ / D f .rZ / C g.rZ / is a quantile function for X CY . The assertion now follows from the fact that all quantile functions of a random variable coincide almost everywhere, due to Lemma A.15. Remark 4.91. Applied to the special case of quantile function with respect to a probability measure, the preceding lemma yields that V@R and AV@R are comonotonic. } Proof of Theorem 4:88. We already know from Proposition 4.84 that the Choquet integral of the loss is a monetary risk measure. Comonotonicity follows by combining Proposition 4.86 with Lemma 4.90. Conversely, suppose now that  is comonotonic. Then  is positively homogeneous according to Lemma 4.83. In particular we have .m/ D m for m  0. Thus, we obtain a normalized monotone set function by letting c.A/ WD .IA /. Moreover, R c .X / WD .X / dc is a comonotonic monetary risk measure on X that coincides with  on indicator functions: .IA / D c.A/ D c .IA /. Let us now show that  and c coincide on simple random variables of the form XD

n X

xi IAi ;

xi 2 R; Ai 2 F :

iD1

Since these random variables are dense in L1 , Lemma 4.3 will then imply that  D c . In order to show that c .X / D .X / for X as above, we may assume without

232

Chapter 4 Monetary measures of risk

loss of generality that x1  x2      xn and that the sets Ai are disjoint. By cash invariance, we may also assume X  0, i.e., xn  0. Thus, weS can write P X D niD1 bi IBi , where bi WD xi  xiC1  0, xnC1 WD 0, and Bi WD ikD1 Ak . Pk1 bi IBi Note that bi IBi and bk IBk is a pair of comonotone functions. Hence, also iD1 and bk IBk are comonotone, and we get inductively .X / D

n X

bi .IBi / D

iD1

n X

bi c .IBi / D c .X /:

iD1

Remark 4.92. The argument at the end of the preceding proof shows that the Choquet integral of a simple random variable XD

n X

xi IAi

with x1      xn  xnC1 WD 0

iD1

and disjoint sets A1 ; : : : ; An can be computed as Z X dc D

n n X X .xi  xiC1 /c.Bi / D xi .c.Bi /  c.Bi1 //; iD1

where B0 WD ; and Bi WD

iD1

Si

kD1 Ak

for i D 1; : : : ; n.

}

So far, we have shown that comonotonic monetary risk measures can be identified with Choquet integrals of normalized monotone set functions. Our next goal is to characterize those set functions that induce risk measures with the additional property of convexity. To this end, we will first consider law-invariant risk measures. The following result shows that the risk measures AV@R may be viewed as the extreme points in the convex class of all law-invariant convex risk measures on L1 that are comonotonic. Theorem 4.93. On an atomless probability space, the class of risk measures Z  .X / WD AV@R .X / .d /; 2 M1 .Œ0; 1/; is precisely the class of all law-invariant convex risk measures on L1 that are comonotonic. In particular, any convex risk measure that is law-invariant and comonotonic is also coherent and continuous from above. Proof. Comonotonicity of  follows from Corollary 4.77 and Theorem 4.88. Conversely, let us assume that  is a law-invariant convex risk measure that is also coR monotonic. By Theorem 4.88, .X / D .X / dc for c.A/ WD .IA /. The lawinvariance of  implies that c.A/ is a function of the probability P Œ A , i.e., there

233

Section 4.7 Comonotonic risk measures

exists an increasing function on Œ0; 1 such that .0/ D 0, .1/ D 1, and c.A/ D .P Œ A /. Note that IA[B and IA\B is a pair of comonotone functions for all A; B 2 F . Hence, comonotonicity and subadditivity of  imply c.A \ B/ C c.A [ B/ D .IA\B / C .IA[B / D .IA\B  IA[B / D .IA  IB /

(4.57)

 c.A/ C c.B/: Proposition 4.75 thus implies that is concave. Corollary 4.77 finally shows that the Choquet integral with respect to c can be identified with a risk measure  , where

is obtained from via Lemma 4.69. Now we turn to the characterization of all comonotonic convex risk measures on X. Recall that, for a positively homogeneous monetary risk measure, convexity is equivalent to subadditivity. Also recall that M1;f WD M1;f .; F / denotes the set of all finitely additive normalized set functions Q W F ! Œ0; 1, and that EQ Œ X  denotes the integral of X 2 X with respect to Q 2 M1;f , as constructed in Theorem A.51. Theorem 4.94. For the Choquet integral with respect to a normalized monotone set function c, the following conditions are equivalent: R (a) .X / WD .X / dc is a convex risk measure on X. R (b) .X / WD .X / dc is a coherent risk measure on X. (c) For Qc WD ¹Q 2 M1;f j QŒ A   c.A/ for all A 2 F º, Z X dc D max EQ Œ X  for X 2 X. Q2Qc

(d) The set function c is submodular. In this case, Qc is equal to the maximal representing set Qmax for . Before giving the proof of this theorem, let us state the following corollary, which gives a complete characterization of all comonotonic convex risk measures, and a remark concerning the set Qc in part (c), which is usually called the core of c. Corollary 4.95. A convex risk measure on X is comonotonic if and only if it arises as the Choquet integral of the loss with respect to a submodular, normalized, and monotone set function c. In this case, c is given by c.A/ D .IA /, and  has the representation .X / D max EQ ΠX ; Q2Qc

where Qc D ¹Q 2 M1;f j QŒ A   c.A/ for all A 2 F º is equal to the maximal representing set Qmax .

234

Chapter 4 Monetary measures of risk

R Proof. Theorems 4.88 and 4.94 state that .X / WD .X / dc is a comonotonic coherent risk measure, which can be represented as in the assertion, as soon as c is a submodular, normalized, and monotone set function. Conversely, any comonotonic convex risk measure  is coherent and arises as the Choquet integral of c.A/ WD .IA /, due to Theorem 4.88. Theorem 4.94 then gives the submodularity of c. Remark 4.96. Let c be a normalized, monotone, submodular set function. Theorem 4.94 implies in particular that the core Qc of c is non-empty. Moreover, c can be recovered from Qc : c.A/ D max QΠA  for all A 2 F . Q2Qc

If c has the additional continuity T property that c.An / ! 0 for any decreasing sequence .An / of events such that n An D ;, then this property is shared by any Q 2 Qc , and it follows that Q is -additive. Thus, the corresponding coherent risk R measure .X / D .X / dc admits a representation in terms of -additive probability measures. It follows by Lemma 4.21 that  is continuous from above. } The proof of Theorem 4.94 requires some preparations. The assertion of the following lemma is not entirely obvious, since Fubini’s theorem may fail if Q 2 M1;f is not -additive. Lemma 4.97. For R X 2 X and Q 2 M1;f , the integral EQ Œ X  is equal to the Choquet integral X dQ. P Proof. It is enough to prove the result for X  0. Suppose first that X D niD1 xi IAi is as in Remark 4.92. Then Z n i n h [ i X X .xi  xiC1 /Q Ak D xi QŒ Ai  D EQ Œ X : X dQ D iD1

kD1

iD1

The result for general X 2 X follows by approximating X uniformly with Xn which take Ronly finitely many values, and by using the Lipschitz continuity of both EQ Π  and  dQ with respect to the supremum norm. Lemma 4.98. Let A1 ; : : : ; An be a partition of  into disjoint measurable sets, and suppose that the normalized monotone set function c is submodular. Let Q be the probability measure on F0 WD .A1 ; : : : ; An / with weights QΠAk  WD c.Bk /  c.Bk1 /

for B0 WD ; and Bk WD

k [

Aj ; k  1:

(4.58)

j D1

R P Then X dc  EQ ΠX  for all F0 -measurable X D niD1 xi IAi , and equality holds if the values of X are arranged in decreasing order: x1      xn .

235

Section 4.7 Comonotonic risk measures

RProof. Clearly, it suffices to consider only the case X  0. Then Remark 4.92 implies X dc D EQ Œ X R as soon as the values of X are arranged in decreasing order. Now we prove X dc  EQ Œ X  for arbitrary F0 -measurable X  0. To this end, note that any permutation of ¹1; : : : ; nº induces a probability measure Q on F0 by applying the definition of Q to the re-labeled partitionRA.1/ ; : : : ; A.n/ . If is a permutation such that x.1/      x.n/ , then we have X dc D EQ Œ X , and so the assertion will follow if we can prove that EQ Œ X   EQ Œ X . To this end, it is enough to show that EQ Œ X   EQ Œ X  if  is the transposition of two indices i and i C 1 which are such that xi < xiC1 , because can be represented as a finite product of such transpositions. Note next that EQ Œ X   EQ Œ X  D xi .Q Œ Ai   QŒ Ai / C xiC1 .Q Œ AiC1   QŒ AiC1 /:

(4.59)

To compute the probabilities Q ΠAk , let us introduce B0

WD ; and

Bk

WD

k [

A.j / ;

k D 1; : : : ; n:

j D1

Then Bk D Bk for k ¤ i . Hence,   Q Œ Ai  C Q Œ AiC1  D Q Œ A.i/  C Q Œ A.iC1/  D c.BiC1 /  c.Bi1 /

D c.BiC1 /  c.Bi1 / D QΠAi  C QΠAiC1 :

(4.60)

Moreover, Bi \ Bi D Bi1 , Bi [ Bi D BiC1 , and hence c.Bi1 / C c.BiC1 /  c.Bi / C c.Bi /, due to the submodularity of c. Thus,  QΠAiC1  D c.BiC1 /  c.Bi /  c.Bi /  c.Bi1 / D Q ΠA.i/  D Q ΠAiC1 :

Using (4.59), (4.60), and our assumption xi < xiC1 thus yields EQ ΠX   EQ ΠX . Proof of Theorem 4:94. (a) , (b): According to Proposition 4.84, the property of positive homogeneity is shared by all Choquet integrals, and the implication (b) ) (a) is obvious. (b) ) (c): By Corollary 4.19, .X / D maxQ2Qmax EQ ΠX , where Q 2 M1;f belongs to Qmax if and only if Z EQ ΠX   .X / D X dc for all X 2 X. (4.61) We will now show thatRthis set Qmax coincides with the set Qc . If Q 2 Qmax then, in particular, QΠA   IA dc D c.A/ for all A 2 F . Hence Q 2 Qc . Conversely,

236

Chapter 4 Monetary measures of risk

suppose Q 2 Qc . If X  0 then Z Z 1 Z X dc D c.X > x/ dx  0

1

QΠX > x  dx D EQ ΠX ;

0

where we have used Lemma 4.97. Cash invariance yields (4.61). (c) ) (b) is obvious. (b) ) (d): This follows precisely as in (4.57). (d) ) (b): We have to show that the Choquet integral is subadditive. By Lemma 4.3, it is again enough to prove this for random variables which only take finitely many values. Thus, let A1 ; : : : P ; An be a partition Pof  into finitely many disjoint measurable sets. Let us write X D i xi IAi , Y D i yi IAi , and let us assume that the indices i D 1; : : : ; n are arranged such that x1 C y1      xn C yn . Then the probability measure Q constructed in Lemma 4.98 is such that Z Z Z .X C Y / dc D EQ ΠX C Y  D EQ ΠX  C EQ ΠY   X dc C Y dc: But this is the required subadditivity of the Choquet integral.

4.8

Measures of risk in a financial market

In this section, we will consider risk measures which arise in the financial market model of Section 1.1. In this model, d C 1 assets are priced at times t D 0 and t D 1. Prices at time 1 are modelled as non-negative random variables S 0 ; S 1 ; : : : ; S d on some probability space .; F ; P /, with S 0  1 C r. Prices at time 0 are given by a vector  D .1; /, with  D . 1 ; : : : ;  d /. The discounted net gain of a trading strategy  D . 0 ; / is given by   Y , where the random vector Y D .Y 1 ; : : : ; Y d / is defined by Si   i for i D 1; : : : ; d . Yi D 1Cr As in the previous two sections, risk measures will be defined on the space L1 D ; P /. A financial position X can be viewed as riskless if X  0 or, more generally, if X can be hedged without additional costs, i.e., if there exists a trading strategy  D . 0 ; / such that    D 0 and L1 .; F

XC

 S D X C   Y  0 P -a.s. 1Cr

Thus, we define the following set of acceptable positions in L1 : A0 WD ¹X 2 L1 j 9  2 Rd with X C   Y  0 P -a.s. º:

(4.62)

Section 4.8 Measures of risk in a financial market

237

Proposition 4.99. Suppose that inf¹m 2 R j m 2 A0 º > 1. Then 0 WD A0 is a coherent risk measure. Moreover, 0 is sensitive in the sense of Definition 4:42 if and only if the market model is arbitrage-free. In this case, 0 is continuous from above and can be represented in terms of the set P of equivalent risk-neutral measures 0 .X / D sup E  Œ X :

(4.63)

P  2P

Proof. The fact that 0 is a coherent risk measure follows from Proposition 4.7. If the model is arbitrage-free, then Theorem 1.32 yields the representation (4.63), and it follows that 0 is sensitive and continuous from above. Conversely, suppose that 0 is sensitive, but the market model admits an arbitrage opportunity. Then there are  2 Rd and " > 0 such that 0    Y P -a.s. and A WD ¹  Y  "º satisfies P Œ A  > 0. It follows that   Y  "IA  0, i.e., "IA is acceptable. However, the sensitivity of 0 implies that 0 ."IA / D "0 .IA / > 0 .0/ D 0; where we have used the coherence of 0 , which follows from fact that A0 is a cone. Thus, we arrive at a contradiction. There are several reasons why it may make sense to allow in (4.62) only strategies  that belong to a proper subset S of the class Rd of all strategies. For instance, if the resources available to an investor are limited, only those strategies should be considered for which the initial investment in risky assets is below a certain amount. Such a restriction corresponds to an upper bound on   . There may be other constraints. For instance, short sales constraints are lower bounds on the number of shares in the portfolio. In view of market illiquidity, the investor may also wish to avoid holding too many shares of one single asset, since the market capacity may not suffice to resell the shares. Such constraints will be taken into account by assuming throughout the remainder of this section that S has the following properties: 

0 2 S.



S is convex.



Each  2 S is admissible in the sense that   Y is P -a.s. bounded from below.

Under these conditions, the set AS WD ¹X 2 L1 j 9  2 S with X C   Y  0 P -a.s. º

(4.64)

is non-empty, convex, and contains all X 2 X which dominate some Z 2 AS . Moreover, we will assume from now on that inf¹m 2 R j m 2 AS º > 1:

(4.65)

238

Chapter 4 Monetary measures of risk

Proposition 4.7 then guarantees that the induced risk measure S .X / WD AS .X / D inf¹m 2 R j m C X 2 AS º is a convex risk measure on L1 . Note that (4.65) holds, in particular, if S does not contain arbitrage opportunities in the sense that   Y  0 P -a.s. for  2 S implies P Œ   Y D 0  D 1. Remark 4.100. Admissibility of portfolios is a serious restriction; in particular, it prevents unhedged short sales of any unbounded asset. Note, however, that it is consistent with our notion of acceptability for bounded claims in (4.64), since X C   Y  0 implies   Y  kX k. } Two questions arise: When is S continuous from above, and thus admits a representation (4.32) in terms of probability measures? And, if such a representation exists, how can we identify the minimal penalty function ˛Smin on M1 .P /? In the case S D Rd , both questions were addressed in Proposition 4.99. For general S, only the second question has a straightforward answer, which will be given in Proposition 4.102. As can be seen from the proof of Proposition 4.99, an analysis of the first question requires an extension of the arbitrage theory in Chapter 1 for the case of portfolio constraints. Such a theory will be developed in Chapter 9 in a more general dynamic setting, and we will address both questions for the corresponding risk measures in Corollary 9.32. This result implies the following theorem for the simple one-period model of the present section: Theorem 4.101. In addition to the above assumptions, suppose that the market model is non-redundant in the sense of Definition 1:15 and that S is a closed subset of Rd such that the cone ¹  j  2 S;  0º is closed. Then S is sensitive if and only if S contains no arbitrage opportunities. In this case, S is continuous from above and admits the representation

 EQ ΠX   sup EQ Π  Y  : S .X / D sup (4.66) Q2M1 .P /

2S

In the following proposition, we will explain the specific form of the penalty function in (4.66). This result will not require the additional assumptions of Theorem 4.101. Proposition 4.102. For Q 2 M1 .P /, the minimal penalty function ˛Smin of S is given by ˛Smin .Q/ D sup EQ Œ   Y : 2S

In particular, S can be represented as in (4.66) if S is continuous from above.

239

Section 4.8 Measures of risk in a financial market

Proof. Fix Q 2 M1 .P /. Clearly, the expectation EQ Π  Y  is well defined for each  2 S by admissibility. If X 2 AS , there exists  2 S such that X  Y P -almost surely. Thus, EQ ΠX   EQ Π  Y   sup EQ Π  Y  2S

for any Q 2 M1 .P /. Hence, the definition of the minimal penalty function yields ˛Smin .Q/  sup EQ Œ   Y :

(4.67)

2S

To prove the converse inequality, take  2 S. Note that Xk WD ..  Y / ^ k/ is bounded since  is admissible. Moreover, Xk C   Y D .  Y  k/ I¹Y kº  0; so that Xk 2 AS . Hence, ˛Smin .Q/  EQ Œ Xk  D EQ Œ .  Y / ^ k ; and so ˛Smin .Q/  EQ Œ   Y  by monotone convergence. Exercise 4.8.1. Show that the identity ˛Smin .Q/ D sup EQ Œ   Y  2S

in Proposition 4.102 remains true even for Q 2 M1;f .P / if we assume in addition that Y is P -a.s. bounded. We thus obtain the representation 

S .X / D max EQ ΠX   sup EQ Π  Y  Q2M1;f .P /

2S

}

without assuming continuity from above.

Remark 4.103. Suppose that S is a cone. Then the acceptance set AS is also a cone, and S is a coherent measure of risk. If S is continuous from above, then Corollary 4.37 yields the representation S .X / D

sup EQ ΠX 

max Q2QS

max in terms of the non-empty set QS D ¹Q 2 M1 .P / j ˛Smin .Q/ D 0º. It follows from Proposition 4.102 that for Q 2 M1 .P / max Q 2 QS

if and only if EQ Π  Y   0

for all  2 S.

max If S is sensitive, then the set S cannot contain any arbitrage opportunities, and QS contains the set P of all equivalent martingale measures whenever such measures max exist. More precisely, QS can be described as the set of absolutely continuous supermartingale measures with respect to S; this will be discussed in more detail in the dynamical setting of Chapter 9. }

240

Chapter 4 Monetary measures of risk

Let us now relax the condition of acceptability in (4.64). We no longer insist that the final outcome of an acceptable position, suitably hedged, should always be nonnegative. Instead, we only require that the hedged position is acceptable in terms of a given convex risk measure A with acceptance set A. Thus, we define AN WD ¹X 2 L1 j 9  2 S; A 2 A with X C   Y  A P -a.s. º:

(4.68)

Clearly, A  AN and hence A   WD AN : From now on, we assume that  > 1;

(4.69)

which implies our assumption (4.65) for AS . Proposition 4.104. The minimal penalty function ˛ min for  is given by ˛ min .Q/ D ˛Smin .Q/ C ˛ min .Q/;

(4.70)

where ˛Smin is the minimal penalty function for S and ˛ min is the minimal penalty function for A . Proof. We claim that N ¹X 2 L1 j .X / < 0º ¹X S C A j X S 2 AS ; A 2 Aº A:

(4.71)

If .X / < 0, then there exists A 2 A and  2 S such that X C   Y  A. Therefore X S WD X  A 2 AS . Next, if X S 2 AS then X S C   Y  0 for some  2 S. N Hence, for any A 2 A, we get X S C A C   Y  A 2 A, i.e., X WD X S C A 2 A. In view of (4.71), we have EQ ΠX  

sup

sup

sup EQ ΠX S A 

X S 2AS A2A

XW .X/ 1: Z2L

(a) Show that 1 2 D 2 1 . (b) Show that  WD 1 2 is a convex risk measure on L1 .

242

Chapter 4 Monetary measures of risk

(c) Show that the minimal penalty function ˛ min of  is equal to the sum of the respective minimal penalty functions ˛1min and ˛2min of 1 and 2 : ˛ min .Q/ D ˛1min .Q/ C ˛2min .Q/

for Q 2 M1;f .

(d) Show that  is continuous from below as soon as 1 is continuous from below. } For the rest of this section, we consider the following case study, which is based on [45]. Let us fix a finite class Q0 D ¹Q1 ; : : : ; Qn º of equivalent probability measures Qi  P such that jY j 2 L1 .Qi /; as in [45], we call the measures in Q0 valuation measures. Define the sets B WD ¹X 2 L0 j EQi Œ X  exists and is  0; i D 1; : : : ; n º

(4.72)

and B0 WD ¹X 2 B j EQi Œ X  D 0 for i D 1; : : : ; n º: Note that B0 \ L0C D ¹0º;

(4.73)

since X D 0 P -a.s. as soon as X  0 P -a.s. and EQi ΠX  D 0, due to the equivalence Qi  P . As the initial acceptance set, we take the convex cone A WD B \ L1 :

(4.74)

The corresponding set AN of positions which become acceptable if combined with a suitable hedge is defined as in (4.68) AN WD ¹X 2 L1 j 9  2 Rd with X C   Y 2 B º: Let us now introduce the following stronger version of the no-arbitrage condition K \ L0C D ¹0º, where K WD ¹  Y j  2 Rd º: K \ B D K \ B0 :

(4.75)

In other words, there is no portfolio  2 Rd such that the result satisfies the valuation inequalities in (4.72) and is strictly favorable in the sense that at least one of the inequalities is strict. Note that (4.75) implies the absence of arbitrage opportunities: K \ L0C D K \ B \ L0C D K \ B0 \ L0C D ¹0º;

243

Section 4.8 Measures of risk in a financial market

where we have used (4.73) and B \L0C D L0C . Thus, (4.75) implies, in particular, the existence of an equivalent martingale measure, i.e., P ¤ ;. The following proposition may be viewed as an extension of the “fundamental theorem of asset pricing”. Let us denote by n n °X ± X ˇ ˇ R WD

i Qi i > 0;

i D 1 iD1

iD1

the class of all “representative” models for the class Q0 , i.e., all mixtures such that each Q 2 Q0 appears with a positive weight. Proposition 4.107. The following two properties are equivalent: (a) K \ B D K \ B0 . (b) P \ R ¤ ;. Proof. (b) ) (a): For V 2 K \ B and R 2 R, we have ER Œ V   0. If we can choose R 2 P \ R then we get ER Œ V  D 0, hence V 2 B0 . (a) ) (b): Consider the convex set C WD ¹ER Œ Y  j R 2 Rº  Rd I we have to show that C contains the origin. If this is not the case then there exists  2 Rd such that   x  0 for x 2 C , (4.76) and   x > 0

for some x  2 C ;

see Proposition A.1. Define V WD   Y 2 K. Condition (4.76) implies ER Œ V   0 for all R 2 R, hence V 2 K \ B. Let R 2 R be such that x  D ER Œ Y . Then V satisfies ER Œ V  > 0, hence V … K \ B0 , in contradiction to our assumption (a). We can now state a representation theorem for the coherent risk measure  correN It is a special case of Theorem 4.110 which will be sponding to the convex cone A. proved below. Theorem 4.108. Under assumption (4.75), the coherent risk measure  WD AN corresponding to the acceptance set AN is given by .X / D

sup P  2P \R

E  ΠX :

244

Chapter 4 Monetary measures of risk

Let us now introduce a second finite set Q1  M1 .P / of probability measures Q P with jY j 2 L1 .Q/; as in [45], we call them stress test measures. In addition to the valuation inequalities in (4.72), we require that an admissible position passes a stress test specified by a “floor” .Q/ < 0

for each Q 2 Q1 .

Thus, the convex cone A in (4.74) is reduced to the convex set A1 WD A \ B1 D L1 \ .B \ B1 /; where B1 WD ¹X 2 L0 j EQ Œ X   .Q/ for Q 2 Q1 º: Let AN 1 WD ¹X 2 L1 j 9  2 Rd with X C   Y 2 B \ B1 º denote the resulting acceptance set for positions combined with a suitable hedge. Remark 4.109. The analogue K \ .B \ B1 / D K \ B0

(4.77)

of our condition (4.75) looks weaker, but it is in fact equivalent to (4.75). Indeed, for X 2 K \ B we can find " > 0 such that X1 WD "X satisfies the additional constraints EQ Œ X1   .Q/ for Q 2 Q1 . Since X1 2 K \ B \ B1 , condition (4.77) implies X1 2 K \ B0 , hence X D 1" X1 2 } K \ B0 , since K \ B0 is a cone. Let us now identify the convex risk measure 1 induced by the convex acceptance set AN 1 . Define °X ± X ˇ R1 WD

.Q/  Q ˇ .Q/  0;

.Q/ D 1  R Q2Q

Q2Q

as the convex hull of Q WD Q0 [ Q1 , and define X .R/ WD

.Q/.Q/ Q2Q

for R D

P

Q

.Q/Q 2 R with .Q/ WD 0 for Q 2 Q0 .

Section 4.8 Measures of risk in a financial market

245

Theorem 4.110. Under assumption (4.75), the convex risk measure 1 induced by the acceptance set AN 1 is given by 1 .X / D

.E  ΠX  C .P  //;

sup

(4.78)

P  2P \R1

i.e., 1 is determined by the penalty function ´ C1 for Q … P \ R1 , ˛ 1 .Q/ WD .Q/ for Q 2 P \ R1 . Proof. Let  denote the convex risk measure defined by the right-hand side of (4.78), and let A denote the corresponding acceptance set A WD ¹X 2 L1 j E  Œ X   .P  / for all P  2 P \ R1 º: It is enough to show A D AN 1 . (a): In order to show AN 1  A , take X 2 AN 1 and P  2 P \ R1 . There exists  2 Rd and A1 2 A1 such that X C   Y  A1 . Thus, E  Œ X C   Y   E  Œ A1   .P  /; due to P  2 R1 . Since E  Œ   Y  D 0 due to P  2 P , we obtain E  Œ X   .P  /, hence X 2 A . (b): In order to show A  AN 1 , we take X 2 A and assume that X … AN 1 . This  means that the vector x  D .x1 ; : : : ; xN / with components xi WD EQi Œ X   .Qi / does not belong to the convex cone N C WD ¹.EQi Œ   Y /iD1;:::;N C y j  2 Rd ; y 2 RN CºR ;

where Q D Q0 [ Q1 D ¹Q1 ; : : : ; QN º with N  n. In part (c) of this proof we will show that C is closed. Thus, there exists 2 RN such that

 x  < inf  xI x2C

(4.79)

N see Proposition P A.1. Since C  RC , we obtain i  0 for i D 1; : : : ; N , and we may assume i i D 1 since ¤ 0. Define

R WD

N X iD1

i Qi 2 R1 :

246

Chapter 4 Monetary measures of risk

Since C contains the linear space of vectors .EQi ΠV /iD1;:::;N with V 2 K, (4.79) implies ER ΠV  D 0 for V 2 K, hence R 2 P . Moreover, the right-hand side of (4.79) must be zero, and the condition

 x  < 0 translates into ER ΠX  < .R/; contradicting our assumption X 2 A . (c): It remains to show that C is closed. For  2 Rd we define y./ as the vector in RN with coordinates yi ./ D EQi Π  Y . Any x 2 C admits a representation x D y./ C z ? with z 2 RN C and  2 N , where

N WD ¹ 2 Rd j EQi Œ   Y  D 0 for i D 1; : : : ; N º; and N ? WD ¹ 2 Rd j    D 0 for all  2 N º: Take a sequence xn D y.n / C zn ;

n D 1; 2; : : : ;

N with n 2 N ? and zn 2 RN C , such that xn converges to x 2 R . If lim infn jn j < 1, then we may assume, passing to a subsequence if necessary, that n converges to  2 Rd . In this case, zn must converge to some z 2 RN C , and we have x D y./Cz 2 C . Let us now show that the case limn jn j D 1 is in fact excluded. In that case, ˛n WD .1 C jn j/1 converges to 0, and the vectors n WD ˛n n stay bounded. Thus, we may assume that n converges to  2 N ? . This implies

y./ D lim y.n / D  lim ˛n zn 2 RN C: n"1

n"1

Since  2 N ? and jj D limn jn j D 1, we obtain y./ ¤ 0. Thus, the inequality EQi Œ ./  Y  D yi ./  0 holds for all i and is strict for some i , in contradiction to our assumption (4.75).

4.9

Utility-based shortfall risk and divergence risk measures

In this section, we will establish a connection between convex risk measures and the expected utility theory of Chapter 2.

Section 4.9 Utility-based shortfall risk and divergence risk measures

247

Suppose that a risk-averse investor assesses the downside risk of a financial position X 2 X by taking the expected utility EŒ u.X  /  derived from the shortfall X  , or by considering the expected utility EŒ u.X /  of the position itself. If the focus is on the downside risk, then it is natural to change the sign and to replace u by the function `.x/ WD u.x/. Then ` is a strictly convex and increasing function, and the maximization of expected utility is equivalent to minimizing the expected loss EŒ `.X /  or the shortfall risk EŒ `.X  / . In order to unify the discussion of both cases, we do not insist on strict convexity. In particular, ` may vanish on .1; 0, and in this case the shortfall risk takes the form EŒ `.X  /  D EŒ `.X / : Definition 4.111. A function ` W R ! R is called a loss function if it is increasing and not identically constant. Let us return to the setting where we consider monetary risk measures defined on the class X of all bounded measurable functions on some given measurable space .; F /. Let us fix a probability measure P on .; F /. For a given loss function ` and an interior point x0 in the range of `, we define the following acceptance set: A WD ¹X 2 X j EŒ `.X /   x0 º:

(4.80)

Alternatively, we can write A D ¹X 2 X j EŒ u.X /   y0 º;

(4.81)

where u.x/ D `.x/ and y0 D x0 . The acceptance set A satisfies (4.3) and (4.4). By part (a) of Proposition 4.7 it induces a monetary risk measure  given by .X / D inf¹m 2 R j EŒ `.X  m/  x0 º D inf¹m 2 R j EŒ u.X C m/  y0 º:

(4.82)

This risk measure satisfies (4.31) and hence can be regarded as a monetary risk measure on L1 . When ` is convex or u concave,  is a convex risk measure. It is normalized when x0 D `.0/. Exercise 4.9.1. Let ` be a strictly increasing continuous loss function. Suppose that the risk measure  associated to ` via (4.82) satisfies .X /  .Y /

”

EΠ`.X /   EΠ`.Y / 

for any X; Y 2 L1 . Show that ` is either linear or exponential. Hint: Apply Proposition 2.46. For the rest of this section, we will only consider convex loss functions.

}

248

Chapter 4 Monetary measures of risk

Definition 4.112. The convex risk measure in (4.82) is called utility-based shortfall risk measure. Proposition 4.113. The utility-based shortfall risk measure  is continuous from below. Moreover, the minimal penalty function ˛ min for  is concentrated on M1 .P /, and  can be represented in the form .X / D

max

Q2M1 .P /

.EQ Œ X   ˛ min .Q//:

(4.83)

Proof. We have to show that  is continuous from below. Note first that z D .X / is the unique solution to the equation EΠ`.z  X /  D x0 :

(4.84)

Indeed, that z D .X / solves (4.84) follows by dominated convergence, since the finite convex function ` is continuous. The solution is unique, since ` is strictly increasing on .`1 .x0 /  "; 1/ for some " > 0. Suppose now that .Xn / is a sequence in X which increases pointwise to some X 2 X. Then .Xn / decreases to some finite limit R. Using the continuity of ` and dominated convergence, it follows that EŒ `..Xn /  Xn / ! EŒ `.R  X / : But each of the approximating expectations equals x0 , and so R is a solution to (4.84). Hence R D .X /, and this proves continuity from below. Since  satisfies (4.31), the representation (4.83) follows from Theorem 4.22 and Lemma 4.32. Let us now compute the minimal penalty function ˛ min . Example 4.114. For an exponential loss function `.x/ D e ˇx , the minimal penalty function can be described in terms of relative entropy, and the resulting risk measure coincides, up to an additive constant, with the entropic risk measure introduced in Example 4.34. In fact, .X / D inf¹m 2 R j EŒ e ˇ.mCX/   x0 º D

1 .log EŒ e ˇX   log x0 /: ˇ

In this special case, the general formula (4.18) for ˛ min reduces to the variational formula for the relative entropy H.QjP / of Q with respect to P   1 log x0 ˛ min .Q/ D sup EQ Œ X   log EŒ e ˇX   ˇ ˇ X2X D

1 .H.QjP /  log x0 /I ˇ

Section 4.9 Utility-based shortfall risk and divergence risk measures

249

see Lemma 3.29. Thus, the representation (4.83) of  is equivalent to the following dual variational identity: log EΠe X  D

max

Q2M1 .P /

.EQ ΠX   H.QjP //:

}

In general, the minimal penalty function ˛ min on M1 .P / can be expressed in terms of the Fenchel–Legendre transform or conjugate function ` of the convex function ` defined by ` .z/ WD sup . zx  `.x/ /: x2R

Theorem 4.115. For any convex loss function `, the minimal penalty function in the representation (4.83) is given by      dQ 1 ˛ min .Q/ D inf (4.85) x0 C E ` ; Q 2 M1 .P /: dP

>0 In particular,       dQ 1  x0 C E ` ; EQ ΠX   inf .X / D max dP

>0 Q2M1 .P /

X 2 L1 :

To prepare the proof of Theorem 4.115, we summarize some properties of the functions ` and ` as stated in Appendix A.1. First note that ` is a proper convex function, i.e., it is convex and takes some finite value. We denote by J WD .` /0C its right-continuous derivative. Then, for x; z 2 R, xz  `.x/ C ` .z/

with equality if x D J.z/.

(4.86)

Lemma 4.116. Let .`n / be a sequence of convex loss functions which decreases pointwise to the convex loss function `. Then the corresponding conjugate functions `n increase pointwise to ` . Proof. It follows immediately from the definition of the Fenchel–Legendre transform that each `n is dominated by ` , and that `n .z/ increases to some limit `1 .z/. We have to prove that `1 D ` . The function z 7! `1 .z/ is a lower semicontinuous convex function as the increasing limit of such functions. Moreover, `1 is a proper convex function, since it is dominated by the proper convex function ` . Consider the conjugate function ` 1 of      `1 . Clearly, `1  `, since `1  ` and since ` D ` by Proposition A.6. On the other hand, we have by a similar argument that ` 1  `n for each n. By taking  D ` . n " 1, this shows ` D `, which in turn gives ` 1 1

250

Chapter 4 Monetary measures of risk

Lemma 4.117. The functions ` and ` have the following properties: (a) ` .0/ D  infx2R `.x/ and ` .z/  `.0/ for all z. (b) There exists some z1 2 Œ0; 1/ such that ` .z/ D sup .xz  `.x//

for z  z1 .

x0

In particular, ` is increasing on Œz1 ; 1/. (c)

` .z/ z

! 1 as z " 1.

Proof. Part (a) is obvious. (b): Let N WD ¹z 2 R j ` .z/ D `.0/º. We show in a first step that N ¤ ;. Note that convexity of ` implies that the set S of all z with zx  `.x/  `.0/ for all x 2 R is non-empty. For z 2 S we clearly have ` .z/  `.0/. On the other hand, ` .z/  `.0/ by (a). Now we take z1 WD sup N . It is clear that z1  0. If z > z1 and x < 0, then xz  `.x/  xz1  `.x/  ` .z1 /  `.0/; where the last inequality follows from the lower semicontinuity of ` . But ` .z/ > `.0/, hence sup .xz  `.x// < ` .z/: x x0

for all z,

(4.92)

it follows from (4.86) that lim `.J.z//  x0 < lim.`.J.z// C ` .z// D lim zJ.z/ D 0:

z#0

z#0

z#0

These facts and the continuity of J imply that for large enough n there exists some

n > 0 such that EŒ `.J. n '/I¹'nº /  D x0 : Let us define X n WD J. n '/I¹'nº :

252

Chapter 4 Monetary measures of risk

Then X n is bounded and belongs to A. Hence, it follows from (4.86) and (4.92) that ˛ min .Q/  EQ Œ X n  1 EŒ I¹'nº J. n '/. n '/ 

n 1 D EŒ .`.X n / C ` . n '//  I¹'nº 

n 1 D .x0  `.0/  P Œ ' > n  C EŒ ` . n '/I¹'nº /

n x0  `.0/  :

n

D

Since we assumed that ˛ min .Q/ < 1, the decreasing limit 1 of n must be strictly positive. The fact that ` is bounded from below allows us to apply Fatou’s lemma ˛ min .Q/  lim inf n"1



1 .x0  `.0/  P Œ ' > n  C EŒ ` . n '/I¹'nº /

n

1 .x0 C EΠ` . 1 '/ /:

1

This proves (4.88) under the assumptions (4.89), (4.90), and (4.91). If (4.89) and (4.90) hold, but J is not continuous, then we can approximate the upper semicontinuous function J from above with an increasing continuous function JQ on Œ0; 1/ such that Z z `Q .z/ WD ` .0/ C JQ .y/ dy 0

satisfies ` .z/  `Q .z/  ` ..1 C "/z/

for z  0.

Let `Q WD `Q denote the Fenchel–Legendre transform of `Q . Since ` D ` by Proposition A.6, it follows that  x  Q `  `.x/  `.x/: 1C" Therefore, Q AQ WD ¹X 2 X j EŒ `.X /   x0 º ¹.1 C "/X j X 2 A º DW A" :

253

Section 4.9 Utility-based shortfall risk and divergence risk measures

Q we get that Since we already know that the assertion holds for `,           dQ dQ 1 1 inf x0 C E `  inf x0 C E `Q dP dP

>0

>0 D sup EQ ΠX  Q X2A

 sup EQ ΠX  X2A"

D .1 C "/˛ min .Q/: By letting " # 0, we obtain (4.88). Finally, we remove conditions (4.89) and (4.90). If ` .z/ D C1 for some z, then z must be an upper bound for the slope of `. So we will approximate ` by a sequence .`n / of convex loss functions whose slope is unbounded. Simultaneously, we can handle the case where ` does not take on its infimum. To this end, we choose a sequence n # inf ` such that n  `.0/ < x0 . We can define, for instance, `n .x/ WD `.x/ _ n C

1 x .e  1/C : n

Then `n decreases pointwise to `. Each loss function `n satisfies (4.89) and (4.90). Hence, for any " > 0 there are "n such that 1 > ˛ min .Q/  ˛nmin .Q/ 

1 .x0 C EΠ`n . "n '/ /  "

"n

for each n,

where ˛nmin .Q/ is the penalty function arising from `n . Note that `n % ` by Lemma 4.116. Our assumption ˛ min .Q/ < 1, the fact that inf `n .z/  `n .0/ D `.0/ > x0 ;

z2R

and part (c) of Lemma 4.117 show that the sequence . "n /n2N must be bounded away from zero and from infinity. Therefore, we may assume that "n converges to some

" 2 .0; 1/. Using again the fact that `n .z/  `.0/ uniformly in n and z, Fatou’s lemma yields ˛ min .Q/ C "  lim inf n"1

1 1 .x0 C EΠ`n . "n '/ /  " .x0 C EΠ` . " '/ /: "

n

This completes the proof of the theorem. Example 4.118. Take

´ `.x/ WD

1 p px

if x  0,

0

otherwise,

254

Chapter 4 Monetary measures of risk

where p > 1. Then

´ ` .z/ WD

1 q qz

if z  0,

C1

otherwise,

where q D p=.p  1/ is the usual dual coefficient. We may apply Theorem 4.115 for any x0 > 0. Let Q 2 M1 .P / with density ' WD dQ=dP . Clearly, ˛ min .Q/ D C1 if ' … Lq .; F ; P /. Otherwise, the infimum in (4.85) is attained for   px0 1=q :

Q D EŒ ' q  Hence, we can identify ˛ min .Q/ for any Q P as    dQ q 1=q : ˛pmin .Q/ D .px0 /1=p  E dP Taking the limit p # 1, we obtain the case `.x/ D x C where we measure the risk in terms of the expected shortfall. Here we have dQ min : ˛1 .Q/ D x0  } dP 1 Together with Proposition 4.20, Theorem 4.115 yields the following result for risk measures which are defined in terms of a robust notion of bounded shortfall risk. Here it is convenient to define ` .1/ WD 1. Corollary 4.119. Suppose that Q is a family of probability measures on .; F /, and that `, ` , and x0 are as in Theorem 4:115. We define a set of acceptable positions by A WD ¹X 2 X j EP Œ `.X /   x0 for all P 2 Q º: Then the corresponding convex risk measure can be represented in terms of the penalty function      1 dQ  ˛.Q/ D inf x0 C inf EP ` ; Q 2 M1 .; F /; dP P 2Q

>0 where dQ=dP is the density appearing in the Lebesgue decomposition of Q with respect to P as in Theorem A.13. Example 4.120. In the case of Example 4.114, the corresponding robust problem in Corollary 4.119 leads to the following entropy minimization problem: For a given Q and a set Q of probability measures, find inf H.QjP /:

P 2Q

Note that this problem is different from the standard problem of minimizing H.QjP / with respect to the first variable Q as it appears in Section 3.2. }

Section 4.9 Utility-based shortfall risk and divergence risk measures

255

Example 4.121. Take x0 D 0 in (4.80) and `.x/ WD x. Then ´ 0 if z D 1, ` .z/ WD C1 otherwise. Therefore, ˛.Q/ D 1 if Q ¤ P , and .X / D EŒ X . If Q is a set of probability measures, the “robust” risk measure  of Corollary 4.119 is coherent, and it is given by .X / D sup EP Œ X : } P 2Q

Exercise 4.9.2. Consider a situation in which model uncertainty is described by a parametric family P for  2 ‚. In each model P , the expected utility of a random variable X 2 X is E Œ u.X / , where u W R ! R is a given utility function. In a Bayesian approach, one would choose a prior distribution on ‚. In terms of F. Knight’s distinction between risk and uncertainty, we would now be in a situation of model risk. Risk neutrality with respect to this model risk would be described by the utility functional Z U.X / D

E Πu.X /  .d /I

here we assume that  7! E Œ u.X /  is sufficiently measurable. In order to capture model risk aversion, we could choose another utility function uO W R ! R and consider the utility functional UO .X / defined by Z u. O UO .X // D u.E O Œ u.X / / .d /: Show that UO is quasi-concave, i.e., UO .˛X C .1  ˛/Y /  UO .X / ^ UO .Y / for X; Y 2 L1 and 0  ˛  1. Show next that for u.x/ O D 1  e ˛x , the utility functional UO .X / takes the form O U .X / D .u.X // for a convex risk measure . Then compute the minimal penalty function in the robust representation of  (here you may assume that ‚ is a finite set). Discuss the limit ˛ " 1. } We now explore the relations between shortfall risk and the divergence risk measures introduced in Example 4.36. To this end, let g W Œ0; 1Œ! R [ ¹C1º be a lower semicontinuous convex function satisfying g.1/ < 1 and the superlinear growth condition g.x/=x ! C1 as x " 1. Recall the definition of the g-divergence, h  dQ  i Ig .QjP / WD E g ; dP

Q 2 M1 .P /;

(4.93)

256

Chapter 4 Monetary measures of risk

and of the corresponding divergence risk measure g .X / WD sup .EQ ΠX   Ig .QjP //;

X 2 L1 :

(4.94)

QP

We have seen in Exercise 4.3.3 that  is continuous from below and that Ig .  jP / is its minimal penalty function, so the supremum in (4.94) is actually a maximum. The following representation for g extends the corresponding result for AV@R in Lemma 4.51, where g D 1  I.1=;1/ . Theorem 4.122. Let g  .y/ D supx>0 .xy  g.x// be the Fenchel–Legendre transform of g. Then g .X / D inf .EŒ g  .z  X /   z/; z2R

X 2 L1 :

(4.95)

The proof of Theorem 4.122 is based on Theorem 4.115. In fact, we will see that, in some sense, the two representation (4.85) and (4.95) are dual with respect to each other. We prepare the proof with the following exercises. The first one concerns a nice and sometimes very useful property of convex functions. Exercise 4.9.3. If h is a convex function on Œ0; 1/, then .x; y/ 7! xh

y  x

is a convex function of .x; y/ 2 .0; 1/  Œ0; 1/.

}

Exercise 4.9.4. Let g W Œ0; 1Œ! R [ ¹C1º be a lower semicontinuous convex function satisfying g.1/ < 1 and the superlinear growth condition g.x/=x ! C1 as x " 1. For > 0 let g .x/ WD g.x= /. Then . ; x/ 7! g .x/ is convex by Exercise 4.9.3. Let  .Q/ D Ig .QjP / be the corresponding g -divergence. Show that . ; Q/ 7!  .Q/ is a convex functional and that ´ h. / WD

g .X / D minQ2M1 .P / .EQ ΠX  C  .Q// C1

if > 0, otherwise,

is a lower semicontinuous convex function in if X 2 L1 is fixed.

}

Proof of Theorem 4:122. Let g and h be as in Exercise 4.9.4. Our aim is to compute h.1/. The idea is to use Theorem 4.115 so as to identify the Fenchel–Legendre transform h of h. To this end, we first observe that ` WD g  satisfies the assumptions of Theorem 4.115. Next, ` D g  D g by Proposition A.6. Hence, Theorem 4.115

Section 4.9 Utility-based shortfall risk and divergence risk measures

257

yields that f .x/ WD inf¹m 2 R j EŒ g  .m  X /   xº   h  dQ  i D max EQ Œ X   inf x C E g

dP

>0 Q2M1 .P / D  inf

min

>0 Q2M1 .P /

.EQ ΠX  C x C  .Q//

D  inf . x C h. // D h .x/;

>0

for all x in the interior of g  .R/, which coincides with the interior of dom f . Exercise 4.9.4 hence yields h.1/ D h .1/ D supx .x  f .x//. We have seen in the proof of Proposition 4.113 that x D EΠg  .f .x/  X /  whenever x belongs to the interior of g  .R/. Hence, h.1/ D sup .EΠg  .f .x/  X /   f .x//; x2R

and the assertion follows by noting that the range of f contains all points to the left of kX  k1  x0 , where x0 is the lower bound for all points in which the right-hand derivative of g  is strictly positive.

Part II

Dynamic hedging

Chapter 5

Dynamic arbitrage theory

In this chapter we develop a dynamic version of the arbitrage theory of Chapter 1. Here we will work in a multiperiod setting, where the stochastic price fluctuation of a financial asset is described as a stochastic process in discrete time. Portfolios will be successively readjusted, taking into account the information available at each time. In its weakest form, market efficiency requires that such dynamic trading strategies should not create arbitrage opportunities. In Section 5.2 we show that an arbitragefree model is characterized by the existence of an equivalent martingale measure. Under such a measure, the discounted price processes of the traded assets are martingales, that is, they have the mathematical structure of a fair game. In Section 5.3 we introduce European contingent claims. These are financial instruments whose payoff at the expiration date depends on the behavior of the underlying primary assets, and possibly on other factors. We discuss the problem of pricing such contingent claims in a manner which does not create new arbitrage opportunities. The pricing problem is closely related to the problem of hedging a given claim by using a dynamic trading strategy based on the primary assets. An ideal situation occurs if any contingent claim can be perfectly replicated by the final outcome of such a strategy. In such a complete model, the equivalent martingale measure P  is unique, and derivatives are priced in a canonical manner by taking the expectation of the discounted payoff with respect to the measure P  . Section 5.5 contains a simple case study for completeness, the binomial model introduced by Cox, Ross, and Rubinstein. In this context, it is possible to obtain explicit pricing formulas for a number of exotic options, as explained in Section 5.6. In Section 5.7 we pass to the limiting diffusion model of geometric Brownian motion. Using a suitable version of the central limit theorem, we are led to the general Black–Scholes formula for European contingent claims and to explicit pricing formulas for some exotic options such as lookback options and the up-and-in and up-and-out calls. The general structure of complete models is described in Section 5.4. There it will become clear that completeness is the exception rather than the rule: Typical market models in discrete time are incomplete.

5.1

The multi-period market model

Throughout this chapter, we consider a market model in which d C 1 assets are priced at times t D 0; 1; : : : ; T . The price of the i th asset at time t is modelled as a nonnegative random variable S ti on a given probability space .; F ; P /. The random

262

Chapter 5 Dynamic arbitrage theory

vector S t D .S t0 ; S t / D .S t0 ; S t1 ; : : : ; S td / is assumed to be measurable with respect to a -algebra F t  F . One should think of F t as the class of all events which are observable up to time t . Thus, it is natural to assume that F0  F1      FT :

(5.1)

Definition 5.1. A family .F t / tD0;:::;T of -algebras satisfying (5.1) is called a filtration. In this case, .; F ; .F t / tD0;:::;T ; P / is also called a filtered probability space. To simplify the presentation, we will assume that F0 D ¹;; º and

F D FT :

(5.2)

Let .E; E/ be a measurable space. A stochastic process with state space .E; E/ is given by a family of E-valued random variables on .; F ; P / indexed by time. In our context, the typical parameter sets will be ¹0; : : : ; T º or ¹1; : : : ; T º, and the state space will be some Euclidean space. Definition 5.2. A stochastic process Y D .Y t / tD0;:::;T is called adapted with respect to the filtration .F t / tD0;:::;T if each Y t is F t -measurable. A stochastic process Z D .Z t / tD1;:::;T is called predictable with respect to .F t / tD0;:::;T if each Z t is F t1 measurable. Note that in our definition predictable processes start at t D 1 while adapted processes are also defined at t D 0. In particular, the asset prices S D .S t / tD0;:::;T form an adapted stochastic process with values in Rd C1 . Definition 5.3. A trading strategy is a predictable Rd C1 -valued process  D . 0 ; / D . t0 ;  t1 ; : : : ;  td / tD1;:::;T : The value  ti of a trading strategy  corresponds to the quantity of shares of the i asset held during the t th trading period between t  1 and t . Thus,  ti S t1 is the th i i amount invested into the i asset at time t  1, while  t S t is the resulting value at time t . The total value of the portfolio  t at time t  1 is

i th

 t  S t1 D

d X

i  ti S t1 :

iD0

By time t , the value of the portfolio  t has changed to t  St D

d X iD0

 ti S ti :

263

Section 5.1 The multi-period market model

The predictability of  expresses the fact that investments must be allocated at the beginning of each trading period, without anticipating future price increments. Definition 5.4. A trading strategy  is called self-financing if  t  S t D  tC1  S t

for t D 1; : : : ; T  1.

(5.3)

Intuitively, (5.3) means that the portfolio is always rearranged in such a way that its present value is preserved. It follows that the accumulated gains and losses resulting from the asset price fluctuations are the only source of variations of the portfolio value  tC1  S tC1   t  S t D  tC1  .S tC1  S t /:

(5.4)

In fact,  is self-financing if and only if (5.4) holds for t D 1; : : : ; T  1. It follows through summation over (5.4) that  t  S t D 1  S 0 C

t X

 k  .S k  S k1 /

for t D 1; : : : ; T .

kD1

Here, the constant  1  S 0 can be interpreted as the initial investment for the purchase of the portfolio  1 . Example 5.5. Often it is assumed that the 0th asset plays the role of a locally riskless bond. In this case, one takes S00  1 and one lets S t0 evolve according to a spot rate r t  0: At time t , an investment x made at time t  1 yields the payoff x.1 C r t /. Thus, a unit investment at time 0 produces the value S t0 D

t Y

.1 C rk /

kD1

S0

at time t . An investment in is “locally riskless” if the spot rate r t is known beforehand at time t  1. This idea can be made precise by assuming that the process r is predictable. } Without assuming predictability as in the preceding example, we assume from now on that S t0 > 0 P -a.s. for all t . This assumption allows us to use the 0th asset as a numéraire and to form the discounted price processes X ti WD

S ti ; S t0

t D 0; : : : ; T; i D 0; : : : ; d:

Then X t0  1, and X t D .X t1 ; : : : ; X td / expresses the value of the remaining assets in units of the numéraire. As explained in Remark 1.11, discounting allows comparison of asset prices which are quoted at different times.

264

Chapter 5 Dynamic arbitrage theory

Definition 5.6. The (discounted) value process V D .V t / tD0;:::;T associated with a trading strategy  is given by V0 WD  1  X 0

and

V t WD  t  X t

for t D 1; : : : ; T:

The gains process associated with  is defined as G0 WD 0

and

t X

G t WD

k  .Xk  Xk1 / for t D 1; : : : ; T .

kD1

Clearly, Vt D  t  X t D

t  St ; S t0

so V t can be interpreted as the portfolio value at the end of the t th trading period expressed in units of the numéraire asset. The gains process Gt D

t X

k  .Xk  Xk1 /

kD1

reflects, in terms of the numéraire, the net gains which have accumulated through the trading strategy  up to time t . For a self-financing trading strategy , the identity  t  S t D 1  S 0 C

t X

 k  .S k  S k1 /

(5.5)

kD1

remains true if all relevant quantities are computed in units of the numéraire. This is the content of the following simple proposition. Proposition 5.7. For a trading strategy  the following conditions are equivalent: (a)  is self-financing. (b)  t  X t D  tC1  X t for t D 1; : : : ; T  1. P (c) V t D V0 C G t D  1  X 0 C tkD1 k  .Xk  Xk1 / for all t . Proof. By dividing both sides of (5.3) by S t0 it is seen that condition (b) is a reformulation of Definition 5.4. Moreover, (b) holds if and only if  tC1  X tC1   t  X t D  tC1  .X tC1  X t / D  tC1  .X tC1  X t / for t D 1; : : : ; T  1, and this identity is equivalent to (c).

265

Section 5.1 The multi-period market model

Remark 5.8. The numéraire component of a self-financing trading strategy  satisfies 0  tC1   t0 D . tC1   t /  X t

for t D 1; : : : ; T  1.

(5.6)

Since 10 D V0  1  X0 ;

(5.7)

the entire process  0 is determined by the initial investment V0 and the d -dimensional process . Consequently, if a constant V0 and an arbitrary d -dimensional predictable process  are given, then we can use (5.7) and (5.6) as the definition of a predictable process  0 , and this construction yields a self-financing trading strategy  WD . 0 ; /. In dealing with self-financing strategies , it is thus sufficient to focus on the initial investment V0 and the d -dimensional processes X and . } Remark 5.9. Different economic agents investing into the same market may choose different numéraires. For example, consider the following simple market model in which prices are quoted in euros ( C) as the domestic currency. Let S 0 be a locally riskless C-bond with the predictable spot rate process r 0 , i.e., S t0 D

t Y

.1 C rk0 /;

kD1

and let S 1 describe the price of a locally riskless investment into US dollars ($). Since the price of this $-bond is quoted in C, the asset S 1 is modeled as S t1 D U t 

t Y

.1 C rk1 /;

kD1

where r 1 is the spot rate for a $-investment, and U t denotes the price of 1$ in terms of C, i.e., U t is the exchange rate of the $ versus the C. While it may be natural for European investors to take S 0 as their numéraire, it may be reasonable for an American investor to choose S 1 . This simple example explains why it may be relevant to check which concepts and results of our theory are invariant under a change of numéraire; see, e.g., the discussion at the end of Section 5.2. } Exercise 5.1.1. Consider a market model with two assets which are modeled as usual by the stochastic process S D .S 0 ; S 1 / that is adapted to the filtration .F t / tD0;:::;T . Decide which of the following processes  are predictable and which in general are not. (i)  t D I¹S 1 >S 1 t

t 1 º

;

(ii) 1 D 1 and  t D I¹S 1

1 t 1 >S t 2 º

for t  2;

266

Chapter 5 Dynamic arbitrage theory

(iiii)  t D IA  I¹t >t0 º , where t0 2 ¹0; : : : ; T º and A 2 F t0 ; (iv)  t D I¹S 1 >S 1 º ; t

0

(v) 1 D 1 and  t D 2 t1 I¹S 1

1 t 1 0  > 0:

The existence of such an arbitrage opportunity may be regarded as a market inefficiency in the sense that certain assets are not priced in a reasonable way. In this section, we will characterize those market models which do not allow for arbitrage opportunities. Such models will be called arbitrage-free. The following proposition shows that the market model is arbitrage-free if and only if there are no arbitrage opportunities for each single trading period. Later on, this fact will allow us to apply the results of Section 1.6 to our multi-period model. Proposition 5.11. The market model admits an arbitrage opportunity if and only if there exist t 2 ¹1; : : : ; T º and  2 L0 .; F t1 ; P I Rd / such that   .X t  X t1 /  0

P -a.s.;

and

P Π  .X t  X t1 / > 0  > 0:

(5.8)

Proof. To prove necessity, take an arbitrage opportunity  D . 0 ; / with value process V , and let t WD min¹k j Vk  0 P -a.s., and P Œ Vk > 0  > 0 º: Then t  T by assumption, and either V t1 D 0 P -a.s. or P Œ V t1 < 0  > 0. In the first case, it follows that  t  .X t  X t1 / D V t  V t1 D V t

P -a.s.

Thus,  WD  t satisfies (5.8). In the second case, we let  WD  t I¹V t 1 0. Then  t  .X t  X t1 / is .a/

a martingale increment by condition (b). In particular,  t .a/ and EQ Π t  .X t  X t1 / j F t1  D 0. Hence,

 .X t  X t1 / 2 L1 .Q/

.a/

EQ Œ V t j F t1  I¹j t jaº D EQ Œ V t I¹j t jaº j F t1   EQ Œ  t .a/

D EQ Œ V t I¹j t jaº   t

 .X t X t1 / j F t1 

 .X t  X t1 / j F t1 

D EQ Œ V t1 I¹j t jaº j F t1  D V t1 I¹j t jaº : By sending a " 1, we obtain (5.10). (c) ) (d): By (5.2), every Q-martingale M satisfies M0 D EQ Œ MT j F0  D EQ Œ MT : (d) ) (a): To prove that X ti 2 L1 .Q/ for given i and t , consider the deterministic j process  defined by si WD I¹st º and s WD 0 for j ¤ i . By Remark 5.8,  can be complemented with a predictable process  0 such that  D . 0 ; / is a self-financing strategy with initial investment V0 D X0i . The corresponding value process satisfies VT D V0 C

T X

s  .Xs  Xs1 / D X ti  0:

sD1

From (d) we get EQ ΠX ti  D EQ ΠVT  D V0 D X0i ;

(5.11)

which yields X ti 2 L1 .Q/. i I A  for Condition (a) will follow if we can show that EQ Œ X ti I A  D EQ Œ X t1 given t , i , and A 2 F t1 . To this end, we define a d -dimensional predictable process j  by is WD I¹s 0 such that EQ t Œ X t  X t1 j F t1  D 0: Clearly, PQt is equivalent to P and has a bounded density, since d PQt d PQt d PQtC1 D  dP dP d PQtC1 is the product of two bounded densities. Moreover, if t C1  k  T , Proposition A.12 and the F t -measurability of Z t D d PQt =d PQtC1 imply EQ tC1 Œ .Xk  Xk1 /Z t j Fk1  EQ t Œ Xk  Xk1 j Fk1  D EQ tC1 Œ Z t j Fk1  D EQ tC1 Œ Xk  Xk1 j Fk1  D 0: Hence, (5.14) carries over from PQtC1 to PQt . We can repeat this recursion until finally P  WD PQ1 yields the desired equivalent martingale measure. Clearly, the absence of arbitrage in the market is independent of the choice of the numéraire, while the set P of equivalent martingale measures generally does depend on the numéraire. In order to investigate the structure of this dependence, suppose that the first asset S 1 is P -a.s. strictly positive, so that it can serve as an alternative numéraire. The price process discounted by S 1 is denoted by  0  St S t2 S td S t0 0 1 d Y t D .Y t ; Y t ; : : : ; Y t / WD ; 1; ; : : : ; X t ; t D 0; : : : ; T: D S t1 S t1 S t1 S t1

272

Chapter 5 Dynamic arbitrage theory

Let PQ be the set of equivalent martingale measures for Y . Then PQ ¤ ; if and only if P ¤ ;, according to Theorem 5.16 and the fact that the existence of arbitrage opportunities is independent of the choice of the numéraire. Proposition 5.17. The two sets P and PQ are related via the identity ° ± ˇ d PQ  XT1  PQ D PQ  ˇ D for some P 2 P : dP  X01 Proof. The process X t1 =X01 is a P  -martingale for any P  2 P . In particular, E  Œ XT1 =X01  D 1, and the formula XT1 d PQ  D dP  X01 defines a probability measure PQ  which is equivalent to P . Moreover, by Proposition A.12, 1 EQ  Œ Y t j Fs  D 1  E  Œ Y t  X t1 jFs  Xs 1 D 1  E  Œ X t j Fs  Xs D Y s: Hence, PQ  is an equivalent martingale measure for Y , and it follows that ° ± ˇ d PQ  XT1  PQ  PQ  ˇ D for some P 2 P : dP  X01 Reversing the roles of X and Y yields the identity of the two sets. Remark 5.18. Unless XT1 is P -a.s. constant, the two sets P and PQ satisfy P \ PQ D ;: }

This can be proved as in Remark 1.12.

Exercise 5.2.4. Let X t WD X t1 be the P -a.s. strictly positive discounted price process of a risky asset. The corresponding returns are X t  X t1 ; RQ t WD X t1 so that X t D X0

t Y kD1

We take as filtration F t D .X0 ; : : : ; X t /.

t D 1; : : : ; T;

.1 C RQ k /:

273

Section 5.2 Arbitrage opportunities and martingale measures

(a) Show that X is a P -martingale when the .RQ t / are independent and integrable random variables with EŒ RQ t  D 0. (b) Now give necessary and sufficient conditions on the .RQ t / such that X is a P martingale. (c) Construct an example in which X is a martingale but the .RQ t / are not independent. } Exercise 5.2.5. Let Z1 ; : : : ; ZT be independent standard normal random variables on .; F ; P /, and let F t be the -field generated by Z1 ; : : : ; Z t , where t D 1; : : : ; T . We also let F0 WD ¹;; º. For constants X01 > 0, i > 0, and mi 2 R we now define the discounted price process of a risky asset as the following sequence of log-normally distributed random variables, X t1 WD X01

t Y

e i Zi Cmi ;

t D 0; : : : ; T:

(5.15)

iD1

Construct an equivalent martingale measure for X 1 under which the random variables } X t1 have still a log-normal distribution. Exercise 5.2.6. For a square-integrable random variable X on .; F ; P / and a algebra F0  F , the conditional variance of X given F0 is defined as var.X jF0 / WD EΠ.X  EΠX j F0 /2 j F0 : Show that var.X jF0 / D EΠX 2 j F0   .EΠX j F0 /2 and that var.X / D EΠvar.X jF0 /  C var.EΠX j F0 /:

}

Exercise 5.2.7. Let Y1 and Y2 be jointly normal random variables with mean 0, variance 1, and correlation % 2 .1; 1/. That is, the joint distribution of .Y1 ; Y2 / has the density '.y1 ; y2 / D

1 1  .y 2 Cy 2 2%y1 y2 / e 2.1%2 / 1 2 ; p 2 1  %2

.y1 ; y2 / 2 R2 :

(a) Compute the conditional expectation EΠY2 j Y1 . (b) Compute the conditional variance var.Y2 jY1 /. (c) For constants m; 2 R compute EΠe Y2 Cm j Y1 .

}

274

Chapter 5 Dynamic arbitrage theory

Exercise 5.2.8. Let Y1 and Y2 be as in Exercise 5.2.7. We use the Yi to construct a log-normal price process in analogy to (5.15) X t1

WD

X01

t Y

e i Zi Cmi ;

t D 0; : : : ; 2;

(5.16)

iD1

for constants X01 ; i > 0 and mi 2 R (i D 1; 2). (a) Compute the conditional expectation EΠX21 j X11 . (b) Construct an equivalent martingale measure for the price process in (5.16) when the filtration is the one generated by the process X 1 . } Exercise 5.2.9. Let X0 ; X1 ; : : : describe the discounted prices of a risky asset in a market model with infinite time horizon that is modeled on a filtered probability space .; .F t / tD0;1;::: ; P /. Suppose that every market model X0 ; : : : ; XT with finite time horizon T 2 N is arbitrage-free. (a) Show that there exists a sequence .PT /T D1;2::: of probability measures such that PT is defined on .; FT /, is equivalent to P on FT , and such that the restriction of PT to FT 1 equals PT1 , i.e., PT ΠA  D PT1 ΠA  for all A 2 FT 1 .  (b) Can you give conditions under which PTS arises as the restriction to FT of a  measure P that is defined on F1 WD . t0 F t /? Hint: You may choose a setting in which one can apply the Kolmogorov extension theorem. }

5.3

European contingent claims

A key topic of mathematical finance is the analysis of derivative securities or contingent claims, i.e., of certain assets whose payoff depends on the behavior of the primary assets S 0 ; S 1 ; : : : ; S d and, in some cases, also on other factors. Definition 5.19. A non-negative random variable C on .; FT ; P / is called a European contingent claim. A European contingent claim C is called a derivative of the underlying assets S 0 ; S 1 ; : : : ; S d if C is measurable with respect to the -algebra generated by the price process .S t / tD0;:::;T . A European contingent claim has the interpretation of an asset which yields at time T the amount C.!/, depending on the scenario ! of the market evolution. T is called the expiration date or the maturity of C . Of course, maturities prior to the final trading period T of our model are also possible, but unless it is otherwise mentioned, we will assume that our European contingent claims expire at T . In Chapter 6, we will meet another class of derivative securities, the so-called American contingent claims. As long as there is no risk of confusion between European and American contingent

275

Section 5.3 European contingent claims

claims, we will use the term “contingent claim” to refer to a European contingent claim. Example 5.20. The owner of a European call option has the right, but not the obligation, to buy an asset at time T for a fixed price K, called the strike price. This corresponds to a contingent claim of the form C call D .STi  K/C : Conversely, a European put option gives the right, but not the obligation, to sell the asset at time T for a strike price K. This corresponds to the contingent claim C put D .K  STi /C :

}

Example 5.21. The payoff of an Asian option depends on the average price i Sav WD

1 X i St jT j t2T

of the underlying asset during a predetermined set of periods T  ¹0; : : : ; T º. For instance, an average price call with strike K corresponds to the contingent claim call i Cav WD .Sav  K/C ;

and an average price put has the payoff put i C Cav WD .K  Sav / :

Average price options can be used, for instance, to secure regular cash streams against exchange rate fluctuations. For example, assume that an economic agent receives at each time t 2 T a fixed amount of a foreign currency with exchange rates S ti . In this case, an average price put option may be an efficient instrument for securing the incoming cash stream against the risk of unfavorable exchange rates. An average strike call corresponds to the contingent claim i C .STi  Sav / ;

while an average strike put pays off the amount i .Sav  STi /C :

An average strike put can be used, for example, to secure the risk from selling at time T a quantity of an asset which was bought at successive times over the period T . }

276

Chapter 5 Dynamic arbitrage theory

Example 5.22. The payoff of a barrier option depends on whether the price of the underlying asset reaches a certain level before maturity. Most barrier options are either knock-out or knock-in options. A knock-in option pays off only if the barrier B is reached. The simplest example is a digital option ´ 1 if max0tT S ti  B; C dig WD 0 otherwise, which has a unit payoff if the price processes reaches a given upper barrier B > S0i . Another example is the down-and-in put with strike price K and lower barrier BQ < S0i which pays off ´ Q .K  STi /C if min0tT S ti  B, put Cd&i WD 0 otherwise. A knock-out barrier option has a zero payoff once the price of the underlying asset reaches the predetermined barrier. For instance, an up-and-out call corresponds to the contingent claim ´ .STi  K/C if max0tT S ti < B, call Cu&o WD 0 otherwise; }

see Figure 5.1. Down-and-out and up-and-in options are defined analogously.

B

S01 K

T Figure 5.1. In one scenario, the payoff of the up-and-out call becomes zero because the stock price hits the barrier B before time T . In the other scenario, the payoff is given by .ST  K/C .

277

Section 5.3 European contingent claims

Example 5.23. Using a lookback option, one can trade the underlying asset at the maximal or minimal price that occurred during the life of the option. A lookback call has the payoff STi  min S ti ; 0tT

while a lookback put corresponds to the contingent claim max S ti  STi :

0tT

}

The discounted value of a contingent claim C when using the numéraire S 0 is given by H WD

C : ST0

We will call H the discounted European claim or just the discounted claim associated with C . In the remainder of this text, “H ” will be the generic notation for the discounted payoff of any type of contingent claim. The reader may wonder why we work simultaneously with the notions of a contingent claim and a discounted claim. From a purely mathematical point of view, there would be no loss of generality in assuming that the numéraire asset is identically equal to one. In fact, the entire theory to be developed in Part II can be seen as a discretetime “stochastic analysis” for the d -dimensional process X D .X 1 ; : : : ; X d / and its “stochastic integrals” t X

k  .Xk  Xk1 /

kD1

of predictable d -dimensional processes . However, some of the economic intuition would be lost if we would limit the discussion to this level. For instance, we have already seen the economic relevance of the particular choice of the numéraire, even though this choice may be irrelevant from the mathematician’s point of view. As a compromise between the mathematician’s preference for conciseness and the economist’s concern for keeping track explicitly of economically relevant quantities, we develop the mathematics on the level of discounted prices, but we will continue to discuss definitions and results in terms of undiscounted prices whenever it seems appropriate. From now on, we will assume that our market model is arbitrage-free or, equivalently, that P ¤ ;:

278

Chapter 5 Dynamic arbitrage theory

Definition 5.24. A contingent claim C is called attainable (replicable, redundant ) if there exists a self-financing trading strategy  whose terminal portfolio value coincides with C , i.e., C D  T  S T P -a.s. Such a trading strategy  is called a replicating strategy for C . Clearly, a contingent claim C is attainable if and only if the corresponding discounted claim H D C =ST0 is of the form H D  T  X T D VT D V0 C

T X

 t  .X t  X t1 /;

tD1

for a self-financing trading strategy  D . 0 ; / with value process V . In this case, we will say that the discounted claim H is attainable, and we will call  a replicating strategy for H . The following theorem yields the surprising result that an attainable discounted claim is automatically integrable with respect to every equivalent martingale measure. Note, however, that integrability may not hold for an attainable contingent claim prior to discounting. Theorem 5.25. Any attainable discounted claim H is integrable with respect to each equivalent martingale measure, i.e., E ΠH  < 1

for all P  2 P .

Moreover, for each P  2 P the value process of any replicating strategy satisfies Vt D E ΠH j Ft 

P -a.s. for t D 0; : : : ; T .

In particular, V is a non-negative P  -martingale. Proof. This follows from VT D H  0 and the systems theorem in the form of Theorem 5.14. Remark 5.26. The identity V t D E  ΠH j F t ;

t D 0; : : : ; T;

appearing in Theorem 5.25 has two remarkable implications. Since its right-hand side is independent of the particular replicating strategy, all such strategies must have the same value process. Moreover, the left-hand side does not depend on the choice of P  2 P . Hence, V t is a version of the conditional expectation E  ΠH j F t  for every P  2 P . In particular, E  ΠH  is the same for all P  2 P . }

279

Section 5.3 European contingent claims

Remark 5.27. When applied to an attainable contingent claim C prior to discounting, Theorem 5.25 states that   ˇ 0  C ˇ  t  S t D St E ˇ F t ; t D 0; : : : ; T; ST0 P -a.s. for all P  2 P and for every replicating strategy . In particular, the initial investment which is needed for a replication of C is given by   C : }  1  S 0 D S00 E  ST0 Let us now turn to the problem of pricing a contingent claim. Consider first a discounted claim H which is attainable. Then the (discounted) initial investment  1  X 0 D V0 D E  Œ H 

(5.17)

needed for the replication of H can be interpreted as the unique (discounted) “fair price” of H . In fact, a different price for H would create an arbitrage opportunity. For instance, if H could be sold at time 0 for a price Q which is higher than (5.17), then selling H and buying the replicating portfolio  yields the profit Q   1  X 0 > 0 at time 0, although the terminal portfolio value VT D  T  X T suffices for settling the claim H at maturity T . In order to make this idea precise, let us formalize the idea of an “arbitrage-free price” of a general discounted claim H . Definition 5.28. A real number  H  0 is called an arbitrage-free price of a discounted claim H , if there exists an adapted stochastic process X d C1 such that X0d C1 D  H ; X td C1  0

for t D 1; : : : ; T  1,

and

(5.18)

XTd C1 D H; and such that the enlarged market model with price process .X 0 ; X 1 ; : : : ; X d ; X d C1 / is arbitrage-free. The set of all arbitrage-free prices of H is denoted by ….H /. The lower and upper bounds of ….H / are denoted by inf .H / WD inf ….H /

and

sup .H / WD sup ….H /:

Thus, an arbitrage-free price  H of a discounted claim H is by definition a price at which H can be traded at time 0 without introducing arbitrage opportunities into the market model: If H is sold for  H , then neither buyer nor seller can find an

280

Chapter 5 Dynamic arbitrage theory

investment strategy which both eliminates all the risk and yields an opportunity to make a positive profit. Our aim in this section is to characterize the set of all arbitragefree prices of a discounted claim H . Note that an arbitrage-free price  H is quoted in units of the numéraire asset. The amount that corresponds to  H in terms of currency units prior to discounting is equal to  C WD S00  H ; and  C is an (undiscounted) arbitrage-free price of the contingent claim C WD ST0 H . Theorem 5.29. The set of arbitrage-free prices of a discounted claim H is non-empty and given by ….H / D ¹E  Œ H  j P  2 P and E  Œ H  < 1 º:

(5.19)

Moreover, the lower and upper bounds of ….H / are given by inf .H / D inf E  Œ H  P  2P

and

sup .H / D sup E  ΠH : P  2P

Proof. By Theorem 5.16,  H is an arbitrage-free price for H if and only if we can find an equivalent martingale measure PO for the market model extended via (5.18). PO must satisfy O X i j F t  for t D 0; : : : ; T and i D 1; : : : ; d C 1. X ti D EŒ T O H . Thus, we obtain the incluIn particular, PO belongs to P and satisfies  H D EŒ sion in (5.19). Conversely, if  H D E  Œ H  for some P  2 P , then we can define the stochastic process X td C1 WD E  Œ H j F t ; t D 0; : : : ; T; which satisfies all the requirements of (5.18). Moreover, the same measure P  is clearly an equivalent martingale measure for the extended market model, which hence is arbitrage-free. Thus, we obtain the identity of the two sets in (5.19). To show that ….H / is non-empty, we first fix some measure PQ  P such that Q EŒ H  < 1. For instance, we can take d PQ D c.1 C H /1 dP , where c is the normalizing constant. Under PQ , the market model is arbitrage-free. Hence, Theorem 5.16 yields P  2 P such that dP  =d PQ is bounded. In particular, E  Œ H  < 1 and hence E  Œ H  2 ….H /. The formula for inf .H / follows immediately from (5.19) and the fact that ….H / ¤ ;. The one for sup .H / needs an additional argument. Suppose that P 1 2 P is such that E 1 Œ H  D 1. We must show that for any c > 0 there exists some  2 ….H / with  > c. To this end, let n be such that Q WD E 1 Œ H ^ n  > c, and define X td C1 WD E 1 Œ H ^ n j F t ;

t D 0; : : : ; T:

281

Section 5.3 European contingent claims

Then P 1 is an equivalent martingale measure for the extended market model .X 0; : : : ; X d ; X d C1 /, which hence is arbitrage-free. Applying the already established fact that the set of arbitrage-free prices of any contingent claim is nonempty to the extended market model yields an equivalent martingale measure P  for .X 0 ; : : : ; X d ; X d C1 / such that E  Œ H  < 1. Since P  is also a martingale measure for the original market model, the first part of this proof implies that  WD E  Œ H  2 ….H /. Finally, note that  D E  Œ H   E  Œ H ^ n  D E  Œ XTd C1  D X0d C1 D Q > c: Hence, the formula for sup .H / is proved. Example 5.30. In an arbitrage-free market model, we consider a European call option C call D .ST1  K/C with strike K > 0 and with maturity T . We assume that the numéraire S 0 is the predictable price process of a locally riskless bond as in Example 5.5. Then S t0 is increasing in t and satisfies S00  1. For any P  2 P , Theorem 5.29 yields an arbitrage-free price  call of C call which is given by  call D E 



C call ST0



D E



XT1 

K ST0

C  :

Due to the convexity of the function x 7! x C D x _ 0 and our assumptions on S 0 ,  call can be bounded from below as follows:   C K call  1   E XT  0 ST   C K D S01  E  (5.20) ST0  .S01  K/C : In financial language, this fact is usually expressed by saying that the value of the option is higher than its “intrinsic value” .S01 K/C , i.e., the payoff if the option were exercised immediately. The difference of the price  call of an option and its intrinsic value is often called the “time-value” of the European call option; see Figure 5.2. } Example 5.31. For a European put option C put D .K  ST1 /C , the situation is more complicated. If we consider the same situation as in Example 5.30, then the analogue of (5.20) fails unless the numéraire S 0 is constant. In fact, as a consequence of the put-call parity, the “time value” of a put option whose intrinsic value is large (i.e., the option is “in the money”) usually becomes negative; see Figure 5.3. } Our next aim is to characterize the structure of the set of arbitrage-free prices of a discounted claim H . It follows from Theorem 5.29 that every arbitrage-free price  H

282

Chapter 5 Dynamic arbitrage theory

S01

K

Figure 5.2. The typical price of a call option as a function of S01 is always above the option’s intrinsic value .S01  K/C .

of H must lie between the two numbers inf .H / D inf E  ΠH  and P  2P

sup .H / D sup E  ΠH : P  2P

We also know that inf .H / and sup .H / are equal if H is attainable. The following theorem shows that also the converse implication holds, i.e., H is attainable if and only if inf .H / D sup .H /. Theorem 5.32. Let H be a discounted claim. (a) If H is attainable, then the set ….H / of arbitrage-free prices for H consists of the single element V0 , where V is the value process of any replicating strategy for H . (b) If H is not attainable, then inf .H / < sup .H / and ….H / D .inf .H /; sup .H //: Proof. The first assertion follows from Remark 5.26 and Theorem 5.29. To prove (b), note first that ….H / D ¹E  Œ H  j P  2 P ; E  Œ H  < 1º is an interval because P is a convex set. We will show that ….H / is open by constructing for any  2 ….H / two arbitrage-free prices L and O for H such that L <  < . O To this end, take P  2 P such that  D E  Œ H . We will first construct an equivalent O H  > E  Œ H . Let martingale measure PO 2 P such that EŒ U t WD E  Œ H j F t ;

t D 0; : : : ; T;

283

Section 5.3 European contingent claims

K

S01

Figure 5.3. The typical price of a European put option as a function of S01 compared to the option’s intrinsic value .K  S01 /C .

so that H D U0 C

T X

.U t  U t1 /:

tD1

Since H is not attainable, there must be some t 2 ¹1; : : : ; T º such that U t  U t1 … K t \ L1 .P  /, where K t WD ¹  .X t  X t1 / j  2 L0 .; F t1 ; P I Rd / º: By Lemma 1.69, K t \ L1 .P  / is a closed linear subspace of L1 .; F t ; P  /. Therefore, Theorem A.57 applied with B WD ¹U t  U t1 º and C WD K t \ L1 .P  / yields some Z 2 L1 .; F t ; P  / such that sup¹E  Œ W Z  j W 2 K t \ L1 .P  /º < E  Œ .U t  U t1 / Z  < 1: From the linearity of K t \ L1 .P  / we deduce that E  Œ W Z  D 0 for all W 2 K t \ L1 .P  /,

(5.21)

E  Π.U t  U t1 / Z  > 0:

(5.22)

and hence that

There is no loss of generality in assuming that jZj  1=3, so that ZO WD 1 C Z  E  ΠZ j F t1  can be taken as the density d PO =dP  D ZO of a new probability measure PO  P .

284

Chapter 5 Dynamic arbitrage theory

Since Z is F t -measurable, the expectation of H under PO satisfies O H  D E  Œ H ZO  EŒ D E  Œ H  C E  Œ E  Œ H j F t  Z   E  Œ E  Œ H j F t1  E  Œ Z j F t1   D E  Œ H  C E  Œ U t Z   E  Œ U t1 Z  > E  Œ H ; O H   5 E Œ H  < where we have used (5.22) in the last step. On the other hand, EŒ 3 O H  will yield the desired arbitrage-free price larger than  if we 1. Thus, O WD EŒ have PO 2 P . Let us prove that PO 2 P . For k > t , the F t -measurability of ZO and Proposition A.12 yield that O Xk  Xk1 j Fk1  D E  Œ Xk  Xk1 j Fk1  D 0: EŒ For k D t , (5.21) yields E  Œ .X t  X t1 / Z j F t1  D 0. Thus, it follows from E  Œ ZO j F t1  D 1 that O X t  X t1 j F t1  EŒ D E  Œ .X t  X t1 /.1  E  Œ Z j F t1 / j F t1  C E  Œ .X t  X t1 /Z j F t1  D 0: Finally, if k < t then P  and PO coincide on Fk . Hence O Xk  Xk1 j Fk1  D E  Œ Xk  Xk1 j Fk1  D 0; EŒ and we may conclude that PO 2 P . It remains to construct another equivalent martingale measure PL such that L H  < E  Œ H  D : L WD EŒ

(5.23)

But this is simply achieved by letting d PL d PO WD 2  ;  dP dP  which defines a probability measure PL  P , because the density d PO =dP  is bounded from above by 5=3 and below by 1/3. PL 2 P is then obvious as is (5.23). Remark 5.33. So far, we have assumed that a contingent claim is settled at the terminal time T . A natural way of dealing with an FT0 -measurable payoff C0  0 maturing at some time T0 < T is to apply our results to the corresponding discounted claim H0 WD

C0 ST00

285

Section 5.3 European contingent claims

in the market model with the restricted time horizon T0 . Clearly, this restricted model is arbitrage-free. An alternative approach is to invest the payoff C0 at time T0 into the numéraire asset S 0 . At time T , this yields the contingent claim C WD C0 

ST0 ST00

;

whose discounted claim H D

C C0 D 0 0 ST ST0

is formally identical to H0 . Moreover, our results can be directly applied to H . It is intuitively clear that these two approaches for determining the arbitrage-free prices of C0 should be equivalent. A formal proof must show that the set ….H / is equal to the set ….H0 / WD ¹E0 Œ H0  j P0 2 P0 and E0 Œ H0  < 1 º of arbitrage-free prices of H0 in the market model whose time horizon is T0 . Here, P0 denotes the set of measures P0 on .; FT0 / which are equivalent to P on FT0 and which are martingale measures for the restricted price process .X t / tD0;:::;T0 . Clearly, each P  2 P defines an element of P0 by restricting P  to the -algebra FT0 . In fact, Proposition 5.34 below shows that every element in P0 arises in this way. Thus, the two sets of arbitrage-free prices for H and H0 coincide, i.e., ….H / D ….H0 /: It follows, in particular, that H0 is attainable if and only if H is attainable.

}

Proposition 5.34. Consider the situation described in Remark 5:33 and let P0 2 P0 be given. Then there exists some P  2 P whose restriction to FT0 is equal to P0 . Proof. Let PO 2 P be arbitrary, and denote by ZT0 the density of P0 with respect to the restriction of PO to the -algebra FT0 . Then ZT0 is FT0 -measurable, and dP  WD ZT0 d PO defines a probability measure on F . Clearly, P  is equivalent to PO and to P , and it coincides with P0 on FT0 . To check that P  2 P , it suffices to show that X t  X t1 is a martingale increment under P  for t > T0 . For these t , the density ZT0 is F t1 measurable, so Proposition A.12 implies that O X t  X t1 j F t1  D 0: E  Œ X t  X t1 j F t1  D EŒ

286

Chapter 5 Dynamic arbitrage theory

Example 5.35. Let us consider the situation of Example 5.30, where the numéraire S 0 is a locally riskless bond. Remark 5.33 allows us to compare the arbitrage-free prices of two European call options C0 D .ST10  K/C and C D .ST1  K/C with the same strikes and underlyings but with different maturities T0 < T . As in Example 5.30, we get that for P  2 P E





   0 ˇ C ST0 ˇ C ˇˇ 1 1  ˇ FT0  0 ST0  K E ˇ F T0 ST0 ST0 ST0 

(5.24)

C0 : ST00

Hence, if P  is used to calculate arbitrage-free prices for C0 and C , the resulting price of C0 is lower than the price of C E





C ST0

 E





 C0 : ST00

This argument suggests that the price of a European call option should be an increasing function of the maturity. } Exercise 5.3.1. Let Yk .k D 1; : : : ; T / be independent identically distributed random variables in L1 .; F ; P /, and suppose that the Yk are not P -a.s. constant and satisfy EΠYk  D 0. Let furthermore X t WD X0 C

t X

Yk ;

t D 0; : : : ; T:

kD1

Then X is a P -martingale when we consider the filtration given by F0 D ¹;; º and F t D .Y1 ; : : : ; Y t / for t D 1; : : : ; T . We now enlarge the filtration by adding “insider information” of the terminal value XT . That is, we consider the enlarged filtration FQn WD .Fn [ .XT //: (a) Show that X is no longer a P -martingale with respect to .FQt /. (b) Prove that the process XQ t WD X t 

t1 X kD0

1 .XT  Xk /: T k

is a P -martingale with respect to the enlarged filtration .FQt /.

287

Section 5.4 Complete markets

(c) The insider information of the terminal value XT implies the existence of selffinancing strategies with positive expected profit. Construct a strategy   that maximizes the expected profit T hX

E

 t .X t  X t1 /

i

tD1

within the class of all .FQt /-predictable strategies  with j t j  1 P -a.s. for all t .

5.4

Complete markets

We have seen in Theorem 5.32 that any attainable claim in an arbitrage-free market model has a unique arbitrage-free price. Thus, the situation becomes particularly transparent if all contingent claims are attainable. Definition 5.36. An arbitrage-free market model is called complete if every contingent claim is attainable. Complete market models are precisely those models in which every contingent claim has a unique and unambiguous arbitrage-free price. However, in discrete time, only a very limited class of models enjoys this property. The following characterization of market completeness is sometimes called the “second fundamental theorem of asset pricing”. Theorem 5.37. An arbitrage-free market model is complete if and only if there exists exactly one equivalent martingale measure. In this case, the number of atoms in .; FT ; P / is bounded from above by .d C 1/T . Proof. If the model is complete, then H WD IA for A 2 FT is an attainable discounted claim. It follows from the results of Section 5.3 that the mapping P  7! E  Œ H  D P  Œ A  is constant over the set P . Hence, there can be only one equivalent martingale measure. Conversely, if jP j D 1, then the set ….H / of arbitrage-free prices of every discounted claim H has exactly one element. Hence, Theorem 5.32 implies that H is attainable. To prove the second assertion, note first that the asserted bound on the number of atoms in FT holds for T D 1 by Corollary 1.42. We proceed by induction on T . Suppose that the assertion holds for T  1. By assumption, any bounded FT -measurable random variable H  0 can be written as H D VT 1 C T  .XT  XT 1 /;

288

Chapter 5 Dynamic arbitrage theory

where both VT 1 and T are FT 1 -measurable and hence constant on each atom A of .; FT 1 ; P /. It follows that the dimension of the linear space L1 .; FT ; P Œ  jA/ is less than or equal to d C 1. Thus, Proposition 1.41 implies that .; FT ; P Œ  jA/ has at most d C 1 atoms. Applying the induction hypothesis concludes the proof. Below we state additional characterizations of market completeness. Denote by Q the set of all martingale measures in the sense of Definition 5.13. Then both P and Q are convex sets. Recall that an element of a convex set is called an extreme point of this set if it cannot be written as a non-trivial convex combination of members of this set. Property (d) in the following theorem is usually called the predictable representation property, or the martingale representation property, of the P  -martingale X . Theorem 5.38. For P  2 P the following conditions are equivalent: (a) P D ¹P  º. (b) P  is an extreme point of P . (c) P  is an extreme point of Q. (d) Every P  -martingale M can be represented as a “stochastic integral” of a d dimensional predictable process  M t D M0 C

t X

k  .Xk  Xk1 / for t D 0; : : : ; T .

kD1

Proof. (a) ) (c): If P  can be written as P  D ˛Q1 C .1  ˛/Q2 for ˛ 2 .0; 1/ and Q1 ; Q2 2 Q, then Q1 and Q2 are both absolutely continuous with respect to P  . By defining 1 Pi WD .Qi C P  /; i D 1; 2; 2 we thus obtain two martingale measures P1 and P2 which are equivalent to P  . Hence, P1 D P2 D P  and, in turn, Q1 D Q2 D P  . (c) ) (b): This is obvious since P  Q. (b) ) (a): Suppose that there exists a PO 2 P which is different from P  . We will show below that in this case PO can be chosen such that the density d PO =dP  is bounded by some constant c > 0. Then, if " > 0 is less than 1=c, dP 0 d PO WD 1 C "  "   dP dP defines another measure P 0 2 P different from P  . Moreover, P  can be represented as the convex combination P D

1 " O PC P 0; 1C" 1C"

289

Section 5.4 Complete markets

which contradicts condition (b). Hence, P  must be the unique equivalent martingale measure. It remains to prove the existence of PO 2 P with a bounded density d PO =dP  if there exists some PQ 2 P which is different from P  . Then there exists a set A 2 FT such that P  Œ A  ¤ PQ Œ A . We enlarge our market model by introducing the additional asset X td C1 WD PQ Œ A j F t ; t D 0; : : : ; T; and we take P  instead of P as our reference measure. By definition, PQ is an equivalent martingale measure for .X 0 ; X 1 ; : : : ; X d ; X d C1 /. Hence, the extended market model is arbitrage-free, and Theorem 5.16 guarantees the existence of an equivalent martingale measure PO such that the density d PO =dP  is bounded. Moreover, PO must be different from P  , since P  is not a martingale measure for X d C1 : X0d C1 D PQ Œ A  ¤ P  Œ A  D E  Œ XTd C1 : (a) ) (d): The terminal value MT of a P  -martingale M can be decomposed into the difference of its positive and negative parts MT D MTC  MT : MTC and MT can be regarded as two discounted claims, which are attainable by Theorem 5.37. Hence, there exist two d -dimensional predictable process  C and   such that T X k˙  .Xk  Xk1 / P  -a.s. MT˙ D V0˙ C kD1

for two non-negative constants V0C and V0 . Since the value processes V t˙ WD V0˙ C

t X

k˙  .Xk  Xk1 /

kD1

are P  -martingales by Theorem 5.25, we get that M t D E  ΠMTC  MT j F t  D V tC  V t : This proves that the desired representation of M holds in terms of the d -dimensional predictable process  WD  C    . (d) ) (a): Applying our assumption to the martingale M t WD P  ΠA j F t  shows that H D IA is an attainable contingent claim. Hence, it follows from the results of Section 5.3 that the mapping P  7! P  ΠA  is constant over the set P . Thus, there can be only one equivalent martingale measure.

290

Chapter 5 Dynamic arbitrage theory

Exercise 5.4.1. Consider the sample space  WD ¹1; C1ºT D ¹! D .y1 ; : : : ; yT / j yi 2 ¹1; C1º º and denote by Y t .!/ WD y t , for ! D .y1 ; : : : ; yT /, the projection on the t th coordinate. As filtration we take F0 WD ¹;; º and F t WD .Y1 ; : : : ; Y t / for t D 1; : : : ; T . We consider a financial market model with two assets such that the discounted price process X t WD X t1 D S t1 =S t0 is of the form X t D X0 exp

t X

. k Yk C mk /



kD1

for a constant X0 > 0 and two predictable processes . t / and .m t /. We suppose that 0  jm t j < t for all t . (a) Show that when P is a probability measure on  for which P Œ ¹!º  > 0 for all !, then there exists a unique equivalent martingale measure P  . (b) By using the binary structure of the model, and without using Theorem 5.37, prove the following martingale representation result. If P  is as in (a), every P  -martingale M can be represented as M t D M0 C

t X

k .Xk  Xk1 /

t D 0; : : : ; T ,

kD1

where the predictable process  is given by k D

5.5

Mk  Mk1 : Xk  Xk1

}

The binomial model

A complete financial market model with only one risky asset must have a binary tree structure, as we have seen in Theorem 5.37. Under an additional homogeneity assumption, this reduces to the following particularly simple model, which was introduced by Cox, Ross, and Rubinstein in [58]. It involves the riskless bond S t0 WD .1 C r/t ;

t D 0; : : : ; T;

with r > 1 and one risky asset S 1 D S, whose return R t WD

S t  S t1 S t1

in the t th trading period can only take two possible values a; b 2 R such that 1 < a < b:

291

Section 5.5 The binomial model

Thus, the stock price jumps from S t1 either to the higher value S t D S t1 .1 C b/ or to the lower value S t D S t1 .1 C a/. In this context, we are going to derive explicit formulas for the arbitrage-free prices and replicating strategies of various contingent claims. Let us construct the model on the sample space  WD ¹1; C1ºT D ¹! D .y1 ; : : : ; yT / j yi 2 ¹1; C1º º: Denote by Y t .!/ WD y t the projection on the

t th

for ! D .y1 ; : : : ; yT /

(5.25)

coordinate, and let

´ 1 C Y t .!/ 1  Y t .!/ a Cb D R t .!/ WD a 2 2 b

if Y t .!/ D 1, if Y t .!/ D C1.

The price process of the risky asset is modeled as S t WD S0

t Y

.1 C Rk /;

kD1

where the initial value S0 > 0 is a given constant. The discounted price process takes the form t Y 1 C Rk St : X t D 0 D S0 1Cr St kD1

As filtration we take F t WD .S0 ; : : : ; S t / D .X0 ; : : : ; X t /;

t D 0; : : : ; T:

Note that F0 D ¹;; º, and F t D .Y1 ; : : : ; Y t / D .R1 ; : : : ; R t /

for t D 1; : : : ; T ;

F WD FT coincides with the power set of . Let us fix any probability measure P on .; F / such that P Œ ¹!º  > 0 for all ! 2 . (5.26) Such a model will be called a binomial model or a CRR model. The following theorem characterizes those parameter values a; b; r for which the model is arbitrage-free. Theorem 5.39. The CRR model is arbitrage-free if and only if a < r < b. In this case, the CRR model is complete, and there is a unique martingale measure P  . The martingale measure is characterized by the fact that the random variables R1 ; : : : ; RT are independent under P  with common distribution r a ; t D 1; : : : ; T: P  Œ R t D b  D p  WD ba

292

Chapter 5 Dynamic arbitrage theory

Proof. A measure Q on .; F / is a martingale measure if and only if the discounted price process is a martingale under Q, i.e.,   1 C R tC1 ˇˇ X t D EQ Œ X tC1 j F t  D X t EQ Q-a.s. ˇ Ft 1Cr for all t  T  1. This identity is equivalent to the equation r D EQ Œ R tC1 j F t  D b  QŒ R tC1 D b j F t  C a  .1  QŒ R tC1 D b j F t /; i.e., to the condition QŒ R tC1 D b j F t .!/ D p  D

r a ba

for Q-a.e. ! 2 .

But this holds if and only if the random variables R1 ; : : : ; RT are independent under Q with common distribution QŒ R t D b  D p  . In particular, there can be at most one martingale measure for X . If the market model is arbitrage-free, then there exists an equivalent martingale measure P  . The condition P   P implies p  D P  Œ R1 D b  2 .0; 1/; which holds if and only if a < r < b. Conversely, if a < r < b then we can define a measure P   P on .; F / by P  Œ ¹!º  WD .p  /k  .1  p  /T k > 0 where k denotes the number of occurrences of C1 in ! D .y1 ; : : : ; yT /. Under P  , Y1 ; : : : ; YT , and hence R1 ; : : : ; RT , are independent random variables with common distribution P  Œ Y t D 1  D P  Œ R t D b  D p  , and so P  is an equivalent martingale measure. From now on, we consider only CRR models which are arbitrage-free, and we denote by P  the unique equivalent martingale measure. Remark 5.40. Note that the unique martingale measure P  , and hence the valuation of any contingent claim, is completely independent of the initial choice of the “objective” measure P within the class of measures satisfying (5.26). } Let us now turn to the problem of pricing and hedging a given contingent claim C . The discounted claim H D C =ST0 can be written as H D h.S0 ; : : : ; ST / for a suitable function h.

293

Section 5.5 The binomial model

Proposition 5.41. The value process V t D E  ΠH j F t ;

t D 0; : : : ; T;

of a replicating strategy for H is of the form V t .!/ D v t .S0 ; S1 .!/; : : : ; S t .!//; where the function v t is given by    S1 ST t  ; : : : ; xt : v t .x0 ; : : : ; x t / D E h x0 ; : : : ; x t ; x t S0 S0

(5.27)

Proof. Clearly,     S tC1 ST ˇˇ V t D E  h S0 ; S1 ; : : : ; S t ; S t ; : : : ; St ˇ Ft : St St Each quotient S tCs =S t is independent of F t and has under P  the same distribution as s Y Ss D .1 C Rk /: S0 kD1

Hence (5.27) follows from the standard properties of conditional expectations. Since V is characterized by the recursion VT WD H

and

V t D E  ΠV tC1 j F t ;

t D T  1; : : : ; 0;

we obtain a recursive formula for the functions v t defined in (5.27) vT .x0 ; : : : ; xT / D h.x0 ; : : : ; xT /;

(5.28)

O C .1  p  /  v tC1 .x0 ; : : : ; x t ; x t a/; O v t .x0 ; : : : ; x t / D p   v tC1 .x0 ; : : : ; x t ; x t b/ where aO WD 1 C a

and

bO WD 1 C b:

Example 5.42. If H D h.ST / depends only on the terminal value ST of the stock price, then V t depends only on the value S t of the current stock price V t .!/ D v t .S t .!//: Moreover, the formula (5.27) for v t reduces to an expectation with respect to the binomial distribution with parameter p  ! T t X T t T tk O k b / v t .x t / D h.x t aO .p  /k .1  p  /T tk : k kD0

294

Chapter 5 Dynamic arbitrage theory

In particular, the unique arbitrage-free price of H is given by ! T X T T k O k h.S0 aO .p  /k .1  p  /T k : b / .H / D v0 .S0 / D k kD0

For h.x/ D .x K/C =.1Cr/T or h.x/ D .K x/C =.1Cr/T , we obtain explicit formulas for the arbitrage-free prices of European call or put options with strike price K. For instance, the price of H call WD .ST  K/C =.1 C r/T is given by ! T X 1 call T k O k C T .p  /k .1  p  /T k : b  K/ .S0 aO } .H / D k .1 C r/T kD0

Example 5.43. Denote by M t WD max Ss ; 0st

0  t  T;

the running maximum of S, and consider a discounted claim H D h.ST ; MT /. For instance, H can be an up-and-in or up-and-out barrier option or a lookback put. Then the value process of H is of the form V t D v t .S t ; M t /; where

 M h  S  i T t T t v t .x t ; m t / D E  h x t ; mt _ xt : S0 S0 This follows from (5.27) or directly from the fact that  Su  MT D M t _ S t max ; tuT S t

where max tuT Su =S t is independent of F t and has the same law as MT t =S0 under P  . The same argument works for options that depend on the minimum of the stock price such as lookback calls or down-and-in barrier options. } Exercise 5.5.1. For an Asian option depending on the average price Sav WD

1 X St jT j t2T

during a predetermined set of periods T  ¹0; : : : ; T º, we introduce the process X Ss : A t WD s2T ; st

Show that the value process V t of the Asian option is a function of S t , A t , and t . }

295

Section 5.5 The binomial model

Let us now derive a formula for the hedging strategy  D . 0 ; / of our discounted claim H D h.S0 ; : : : ; ST /. Proposition 5.44. The hedging strategy is given by  t .!/ D  t .S0 ; S1 .!/; : : : ; S t1 .!//; where  t .x0 ; : : : ; x t1 / WD .1 C r/

O  v t .x0 ; : : : ; x t1 ; x t1 a/ O v t .x0 ; : : : ; x t1 ; x t1 b/ : O x t1 b  x t1 aO

Proof. For each ! D .y1 ; : : : ; yT /,  t must satisfy  t .!/.X t .!/  X t1 .!// D V t .!/  V t1 .!/:

(5.29)

In this equation, the random variables  t , X t1 , and V t1 depend only on the first t  1 components of !. For a fixed t , let us define ! C and !  by ! ˙ WD .y1 ; : : : ; y t1 ; ˙1; y tC1 ; : : : ; yT /: Plugging ! C and !  into (5.29) shows O C r/1  X t1 .!// D V t .! C /  V t1 .!/  t .!/  .X t1 .!/ b.1  t .!/  .X t1 .!/ a.1 O C r/1  X t1 .!// D V t .!  /  V t1 .!/: Solving for  t .!/ and using our formula (5.28) for V t , we obtain  t .!/ D .1 C r/

V t .! C /  V t .!  / D  t .S0 ; S1 .!/; : : : ; S t1 .!//: X t1 .!/.bO  a/ O

Remark 5.45. The term  t may be viewed as a discrete “derivative” of the value function v t with respect to the possible stock price changes. In financial language, a hedging strategy based on a derivative of the value process is often called a Delta hedge. } Remark 5.46. Let H D h.ST / be a discounted claim which depends on the terminal value of S by way of an increasing function h. For instance, h can be the discounted payoff function h.x/ D .x  K/C =.1 C r/T of a European call option. Then v t .x/ D E  Œ h.x ST t =S0 /  is also increasing in x, and so the hedging strategy satisfies  t .!/ D .1 C r/t

O  v t .S t1 .!/ a/ v t .S t1 .!/ b/ O  0: O S t1 .!/ b  S t1 .!/ aO

In other words, the hedging strategy for H does not involve short sales of the risky asset. }

296

Chapter 5 Dynamic arbitrage theory

Exercise 5.5.2. Let T0 2 ¹1; : : : ; T  1º and K > 0. The payoff of forward starting call option has the form S C T K : ST0 }

Determine its arbitrage-free price and hedging strategy.

5.6

Exotic derivatives

The recursion formula (5.28) can be used for the numeric computation of the value process of any contingent claim. For the value processes of certain exotic derivatives which depend on the maximum of the stock price, it is even possible to obtain simple closed-form solutions if we make the additional assumption that aO D

1 ; bO

where aO D 1 C a and bO D 1 C b. In this case, the price process of the risky asset is of the form S t .!/ D S0 bO Zt .!/ where, for Yk as in (5.25), Z0 WD 0 and

Z t WD Y1 C    C Y t ;

t D 1; : : : ; T:

Let P denote the uniform distribution P Œ ¹!º  WD

1 D 2T ; jj

! 2 :

Under the measure P , the random variables Y t are independent with common distribution P Œ Y t D C1  D 12 . Thus, the stochastic process Z becomes a standard random walk under P . Therefore, ´ t  2t t Ck if t C k is even, 2 (5.30) P Œ Zt D k  D 0 otherwise. The following lemma is the key to numerous explicit results on the distribution of Z under the measure P ; see, e.g., Chapter III of [110]. For its statement, it will be convenient to assume that the random walk Z is defined up to time T C 1; this can always be achieved by enlarging our probability space .; F /. We denote by M t WD max Zs 0st

the running maximum of Z.

297

Section 5.6 Exotic derivatives

Lemma 5.47 (Reflection principle). For all k 2 N and l 2 N0 , P ΠMT  k and ZT D k  l  D P ΠZT D k C l ; and P ΠMT D k and ZT D k  l  D 2

kCl C1 P ΠZT C1 D 1 C k C l : T C1

Proof. Let  .!/ WD inf¹t  0 j Z t .!/ D kº ^ T: For ! D .y1 ; : : : ; yT / 2  we define .!/ by .!/ D ! if  .!/ D T and by .!/ D .y1 ; : : : ; y.!/ ; y.!/C1 ; : : : ; yT / otherwise, i.e., if the level k is reached before the deadline T . Intuitively, the two trajectories .Z t .!// tD0;:::;T and .Z t ..!/// tD0;:::;T coincide up to  .!/, but from then on the latter path is obtained by reflecting the original one on the horizontal axis at level k; see Figure 5.4.

kCl

k

kl 

T

Figure 5.4. The reflection principle.

Let Ak;l denote the set of all ! 2  such that ZT .!/ D k  l and MT  k. Then  is a bijection from Ak;l to the set ® ¯ MT  k and ZT D k C l ;

298

Chapter 5 Dynamic arbitrage theory

which coincides with ¹ZT D k Clº, due to our assumption l  0. Hence, the uniform distribution P must assign the same probability to Ak;l and ¹ZT D k C lº, and we obtain our first formula. The second formula is trivial in case T C k C l is not even. Otherwise, we let j WD .T C k C l/=2 and apply (5.30) together with part one of this lemma P Œ MT D k; ZT D k  l  D P Œ MT  k; ZT D k  l   P Œ MT  k C 1; ZT D k  l  D P Œ ZT D k C l   P Œ ZT D k C l C 2  ! ! T T T T D2 2 j j C1 ! T C 1 2j C 1  T ; D 2T j C1 T C1 and this expression is equal to the right-hand side of our second formula. Formula (5.30) will change if we replace the uniform distribution P by our martingale measure P  , described in Theorem 5.39 ´ 

P ΠZt D k  D

.p  /

t Ck 2

.1  p  /

t k 2



t

t Ck 2

0



if t C k is even, otherwise.

Let us now show how the reflection principle carries over to P  . Lemma 5.48 (Reflection principle for P  ). For all k 2 N and l 2 N0 , P  ΠMT  k; ZT D k  l  D

 1  p  l

P  ΠZT D k C l  p  p  k P  ΠZT D k  l ; D 1  p

and P  ΠMT D k; ZT D k  l  1  1  p  l k C l C 1  P ΠZT C1 D 1 C k C l  D   p p T C1  p  k k C l C 1 1 P  ΠZT C1 D 1  k  l :   D 1  p 1  p T C1

299

Section 5.6 Exotic derivatives

Proof. We show first that the density of P  with respect to P is given by T CZT T ZT dP  D 2T  .p  / 2 .1  p  / 2 : dP

Indeed, P  puts the weight P  Œ ¹!º  D .p  /k .1  p  /T k to each ! D .y1 ; : : : ; yT / 2  which contains exactly k components with yi D C1. But for such an ! we have ZT .!/ D k  .T  k/ D 2k  T , and our formula follows. From the density formula, we get P  Œ MT  k and ZT D k  l  D 2T .p  /

T Ckl 2

.1  p  /

T Clk 2

P ΠMT  k and ZT D k  l :

Applying the reflection principle and using again the density formula, we see that the probability term on the right is equal to P ΠZT D k C l  D 2T .p  /

T CkCl 2

.1  p  /

T kl 2

P  ΠZT D k C l ;

which gives the first identity. The proof of the remaining ones is analogous. Example 5.49 (Up-and-in call option). Consider an up-and-in call option with payoff ´ .ST  K/C if max0tT S t  B; call Cu&i D 0 otherwise, where B > S0 _ K denotes a given barrier, and where K > 0 is the strike price. Our aim is to compute the arbitrage-free price call .Cu&i /D

1 call E  ΠCu&i : .1 C r/T

Clearly,  call E  ΠCu&i  D E  .ST  K/C I max S t  B 0tT



C

D E Π.ST  K/ I ST  B   C E  .ST  K/C I max S t  B; ST < B : 0tT

The first expectation on the right can be computed explicitly in terms of the binomial distribution. Thus, it remains to compute the second expectation, which we denote by I . To this end, we may assume without loss of generality that B lies within the

300

Chapter 5 Dynamic arbitrage theory

range of possible asset prices, i.e., there exists some k 2 N such that B D S0 bO k . Then, by Lemma 5.48, X I D E  Π.ST  K/C I MT  k; ZT D k  l  l1

X D .S0 bO kl  K/C P  ΠMT  k; ZT D k  l  l1

 p  k X .S0 bO kl  K/C P  ΠZT D k  l  D 1  p l1

 p  k X Q C P  ΠZT D k  l  D bO 2k .S0 bO kl  K/  1p l1

 p  k  B 2 Q C I ST < BQ ; E  Π.ST  K/ D 1  p S0 where

 S 2 S2 0 and BQ WD 0 : KQ D K bO 2k D K B B Hence, we obtain the formula  1 call .Cu&i /D E  Π.ST  K/C I ST  B  .1 C r/T   p  k  B 2  C Q I ST < BQ  : C E Π.ST  K/ 1  p S0

Both expectations on the right now only involve the binomial distribution with parameters p  and T . They can be computed as in Example 5.42, and so we get the explicit formula call / .Cu&i

1 D .1 C r/T

X nk

.S0 bO T 2n  K/C .p  /T n .1  p  /n

nD0

 p  k  B 2 C 1  p S0

T X

T T n

!

Q C .p  /T n .1  p  /n .S0 bO T 2n  K/

nDnk C1

where nk is the largest integer n such that T  2n  k.

T T n

! ; }

Example 5.50 (Up-and-out call option). Consider an up-and-out call option with payoff ´ 0 if max0tT S t  B; call Cu&o D C .ST  K/ otherwise,

301

Section 5.6 Exotic derivatives

where K > 0 is the strike price and B > S0 _ K is an upper barrier for the stock price. As in the preceding example, we assume that B D S0 .1 C b/k for some k 2 N. Let C call WD .ST  K/C denote the corresponding “plain vanilla call”, whose arbitrage-free price is given by .C call / D

1 E  Π.ST  K/C : .1 C r/T

call call C Cu&i , we get from Example 5.49 that Since C call D Cu&o call call .Cu&o / D .C call /  .Cu&i /  1 E  Π.ST  K/C I ST < B  D .1 C r/T   p  k  B 2  C Q Q  E Π.S  K/ I S < B  : T T 1  p S0

where KQ D KS02 =B 2 and BQ WD S02 =B. These expectations can be computed as in Example 5.49. } Exercise 5.6.1. Derive a formula for the arbitrage-free price of a down-and-in put option with payoff ´ 0 if min0tT S t > B; put Cd&i D C otherwise, .K  ST / where K > 0 is the strike price and B < S0 is a lower barrier for the stock price. Then compute the price of the option for the following specific parameter values: T D 3;

S0 D 100;

a D 0:1;

r D 0:05;

B D 70;

K D 90:

}

In the following example, we compute the price of a lookback put option. Example 5.51 (Lookback put option). A lookback put option corresponds to the contingent claim put Cmax WD max S t  ST I 0tT

put

see Example 5.23. In the CRR model, the discounted arbitrage-free price of Cmax is given by  1 put .Cmax /D E  max S t  S0 : T 0tT .1 C r/

302

Chapter 5 Dynamic arbitrage theory

The expectation of the maximum can be computed as E



T X bO k P  ΠMT D k : max S t D S0

0tT

kD0

Lemma 5.48 yields P  ΠMT D k  D

X

P  ΠMT D k; ZT D k  l 

l0

D

X l0

D

1  p  k k C l C 1  P ΠZT C1 D 1  k  l  1  p 1  p T C1

1  p  k 1 E  ΠZT C1 I ZT C1  1  k : 1  p 1  p T C 1

Thus, we arrive at the formula put .Cmax / C S0 T  p  k X S0 Ok b D E  ΠZT C1 I ZT C1  1  k : 1  p .1 C r/T .1  p  /.T C 1/ kD0

As before, one can give explicit formulas for the expectations occurring on the right. } Exercise 5.6.2. Derive a formula for the price of a lookback call option with payoff ST  min S t : 0tT

5.7

}

Convergence to the Black–Scholes price

In practice, a huge number of trading periods may occur between the current time t D 0 and the maturity T of a European contingent claim. Thus, the computation of option prices in terms of some martingale measure may become rather elaborate. On the other hand, one can hope that the pricing formulas in discrete time converge to a transparent limit as the number of intermediate trading periods grows larger and larger. In this section, we will formulate conditions under which such a convergence occurs. Throughout this section, T will not denote the number of trading periods in a fixed discrete-time market model but rather a physical date. The time interval Œ0; T  will T 2T be divided into N equidistant time steps N ; N ; : : : ; NNT , and the date kT N will correspond to the k th trading period of an arbitrage-free market model. For simplicity, we

303

Section 5.7 Convergence to the Black–Scholes price

will assume that each market model contains a riskless bond and just one risky asset. In the N th approximation, the risky asset will be denoted by S .N / , and the riskless bond will be defined by a constant interest rate rN > 1. The question is whether the prices of contingent claims in the approximating market models converge as N tends to infinity. Since the terminal values of the riskless bonds should converge, we assume that lim .1 C rN /N D e rT ;

N "1

where r is a finite constant. This condition is in fact equivalent to the following one: lim N rN D r T:

N "1

.N /

Let us now consider the risky assets. We assume that the initial prices S0 do .N / .N / not depend on N , i.e., S0 D S0 for some constant S0 > 0. The prices Sk are random variables on some probability space .N ; F .N / ; PN /, where PN is a risk-neutral measure for each approximating market model, i.e., the discounted price process .N / Sk .N / Xk WD ; k D 0; : : : ; N; .1 C rN /k .N /

.N /

WD .S1 is a PN -martingale with respect to the filtration Fk remaining conditions will be stated in terms of the returns .N /

.N /

Rk

WD

Sk

.N /

; : : : ; Sk

/. Our

.N /

 Sk1 .N /

;

k D 1; : : : ; N:

Sk1 .N /

.N /

First, we assume that, for each N , the random variables R1 ; : : : ; RN dent under PN and satisfy .N /

1 < ˛N  Rk

 ˇN ;

are indepen-

k D 1; : : : ; N;

for constants ˛N and ˇN such that lim ˛N D lim ˇN D 0:

N "1

N "1

.N /

Second, we assume that the variances varN .Rk / under PN are such that 2 N

N 1 X .N / WD var.Rk / ! 2 2 .0; 1/: T N kD1

The following result can be regarded as a multiplicative version of the central limit theorem.

304

Chapter 5 Dynamic arbitrage theory .N /

Theorem 5.52. Under the above assumptions, the distributions of SN under PN converge weakly to the log-normal distribution with parameters log S0 C rT  12 2 T p and T , i.e., to the distribution of     1 2 ST WD S0 exp WT C r  T ; (5.31) 2 where WT has a centered normal law N.0; T / with variance T . Proof. We may assume without loss of generality that S0 D 1. Consider the Taylor expansion 1 log.1 C x/ D x  x 2 C .x/ x 2 (5.32) 2 where the remainder term  is such that j.x/j  ı.˛; ˇ/

for 1 < ˛  x  ˇ,

and where ı.˛; ˇ/ ! 0 for ˛; ˇ ! 0. Applied to .N /

SN

D

N Y

.N /

.1 C Rk /;

kD1

this yields .N / log SN

D

N  X

.N / Rk

kD1

 1 .N / 2  .Rk / C N ; 2

where jN j  ı.˛N ; ˇN /

N X

.N /

.Rk /2 :

kD1 .N /

 Since PN is a martingale measure, we have EN ΠRk

 EN Œ jN j   ı.˛N ; ˇN /

N X

 D rN , and it follows that

.N /

2 .var.Rk / C rN / ! 0:

kD1

N

In particular, N ! 0 in probability, and the corresponding laws converge weakly to the Dirac measure ı0 . Slutsky’s theorem, as stated in Appendix A.6, asserts that it suffices to show that the distributions of ZN WD

N  X kD1

.N /

Rk

 N X 1 .N / .N /  .Rk /2 DW Yk 2 kD1

305

Section 5.7 Convergence to the Black–Scholes price

converge weakly to the normal law N.rT  12 2 T; 2 T /. To this end, we will check that the conditions of the central limit theorem in the form of Theorem A.37 are satisfied. Note that 1 2 .N / max jYk j  N C N ! 0 2 1kN for N WD j˛N j _ jˇN j, and that 1 2 1  2 Œ ZN  D N rN  . N T C N rN / ! r T  2 T: EN 2 2 Finally, var.ZN / ! 2 T; N

since for p > 2 N X

.N /

p2

 EN ŒjRk jp   N

kD1

N X

.N /

 EN Œ.Rk /2  ! 0:

kD1

Thus, the conditions of Theorem A.37 are satisfied. Remark 5.53. The assumption of independent returns in Theorem 5.52 can be relaxed. Instead of Theorem A.37, we can apply a central limit theorem for martingales under suitable assumptions on the behavior of the conditional variances .n/

var. Rk j Fk1 /I N

}

for details see, e.g., Section 9.3 of [54].

Example 5.54. Suppose the approximating model in the N th stage is a CRR model with interest rate rT rN D ; N .N /

and with returns Rk , which can take the two possible values aN and bN ; see Section 5.5. We assume that aO N D 1 C aN D e 

p

T =N

and

bON D 1 C bN D e 

p T =N

for some given > 0. Since p

N rN ! 0;

p p N aN !  T ;

p

p N bN ! T

as N " 1, (5.33)

306

Chapter 5 Dynamic arbitrage theory

we have aN < rN < bN for large enough N . Theorem 5.39 yields that the N th model is arbitrage-free and admits a unique equivalent martingale measure PN . The measure PN is characterized by .N /

PN ΠRk

 D bN  DW pN D

rN  aN ; b N  aN

and we obtain from (5.33) that 1  lim pN D : 2 N "1 .N /

 Moreover, EN ΠRk N X kD1

 D rN , and we get .N /

 2  2 2 var.Rk / D N.pN bN C .1  pN /aN  rN / ! 2 T N

as N " 1. Hence, the assumptions of Theorem 5.52 are satisfied.

}

Let us consider a derivative which is defined in terms of a function f  0 of the risky asset’s terminal value. In each approximating model, this corresponds to a contingent claim .N / C .N / D f .SN /: Corollary 5.55. If f is bounded and continuous, the arbitrage-free prices of C .N / calculated under PN converge to a discounted expectation with respect to a lognormal distribution, which is often called the Black–Scholes price. More precisely, lim

N "1

 EN



C .N / .1 C rN /N



D e rT E  Πf .ST / e rT Dp 2

Z

1

f .S0 e 

(5.34) p

T yCrT  2 T =2

/e y

2 =2

dy;

1

where ST has the form (5.31) under P  . This convergence result applies in particular to the choice f .x/ D .K  x/C corresponding to a European put option with strike K. Since the put-call parity  EN



.N /

.SN  K/C .1 C rN /N

 D

 EN



.N /

.K  SN /C .1 C rN /N

 C S0 

K .1 C rN /N

holds for each N , the convergence (5.34) is also true for a European call option with the unbounded payoff profile f .x/ D .x  K/C .

307

Section 5.7 Convergence to the Black–Scholes price

Example 5.56 (Black–Scholes formula for the price of a call option). The limit of the .N / arbitrage-free prices of C .N / D .SN  K/C is given by v.S0 ; T /, where e rT v.x; T / D p 2

Z

1

.xe 

p T yCrT  2 T =2

 K/C e y

2 =2

dy:

1

The integrand on the right vanishes for y

x log K C .r  12 2 /T DW d .x; T / DW d : p T

Let us also define x p C .r C 12 2 /T log K ; dC WD dC .x; T / WD d .x; T / C T D p T Rx 2 and let us denote by ˆ.x/ D .2/1=2 1 e y =2 dy the distribution function of the standard normal distribution. Then Z C1 p x 2 v.x; T / D p e .y T / =2 dy  e rT K.1  ˆ.d //; 2 d

and we arrive at the Black–Scholes formula for the price of a European call option with strike K and maturity T v.x; T / D x ˆ.dC .x; T //  e rT Kˆ.d .x; T //: See Figure 5.5 for the plot of the function v.x; t /.

(5.35) }

Remark 5.57. For fixed x and T , the Black–Scholes price of a European call option increases to the upper arbitrage bound x as " 1. In the limit # 0, we obtain the lower arbitrage bound .x  e rT K/C ; see Remark 1.37. } The following proposition gives a criterion for the convergence (5.34) in case f is not necessarily bounded and continuous. It applies in particular to f .x/ D .x  K/C , and so we get an alternative proof for the convergence of call option prices to the Black–Scholes price. Proposition 5.58. Let f W .0; 1/ ! R be measurable, continuous a.e., and such that jf .x/j  c .1 C x/q for some c  0 and 0  q < 2. Then .N /

 Πf .SN /  ! E  Πf .ST /; EN

where ST has the form (5.31) under P  .

308

Chapter 5 Dynamic arbitrage theory

2K 0

K t T

0

Figure 5.5. The Black–Scholes price v.x; t / of a European call option .ST K/C plotted as a function of the initial spot price x D S0 and the time to maturity t .

Proof. Let us note first that by the Taylor expansion (5.32) N Y

.N /

 log EN Π.SN /2  D log

.N /

N

kD1

D

N X

.N /

 .var.1 C Rk / C EN Π1 C Rk 2 /

.N /

log.var.Rk / C .1 C rN /2 / N

kD1

2 2 T C 2N rN C N rN C cQ  N

N X kD1

.N /

2 2 .var.Rk / C 2jrN j C rN / N

for a finite constant c. Q Thus, .N /

 sup EN Π.SN /2  < 1: N

With this property established, the assertion follows immediately from Theorem 5.52 and the Corollaries A.46 and A.47, but we also give the following more elementary proof. To this end, we may assume that q > 0, and we define p WD 2=q > 1. Then .N /

.N /

  sup EN Πjf .SN /jp   c p sup EN Π.1 C SN /2  < 1; N

N

and the assertion follows from Lemma 5.59 below. Lemma 5.59. Suppose . N /N 2N is a sequence of probability measures on R converging weakly to . If f is a measurable and -a.e. continuous function on R such

309

Section 5.7 Convergence to the Black–Scholes price

that

Z c WD sup

jf jp d N < 1

for some p > 1,

N 2N

then

Z

Z f d N !

f d :

Proof. We may assume without loss of generality that f  0. Then fk WD f ^ k is a bounded and -a.e. continuous function for each k > 0. Clearly, Z Z Z f d N D fk d N C .f  k/C d N : Due to part (e) of the portmanteau R theorem in the form of Theorem A.39, the first integral on the right converges to fk d as N " 1. Let us consider the second term on the right Z Z Z 1 c C .f  k/ d N  f d N  p1 f p1 f d N  p1 ; k k ¹f >kº uniformly in N . Hence, Z Z Z fk d D lim fk d N  lim inf f d N N "1

N "1

Z

Z

f d N 

 lim sup

fk d C

N "1

Letting k " 1, we have

R

fk d %

R

c k p1

:

f d , and convergence follows.

Let us now continue the discussion of the Black–Scholes price of a European call option where f .x/ D .x  K/C . We are particularly interested how it depends on the various model parameters. The dependence on the spot price S0 D x can be analyzed via the x-derivatives of the function v.t; x/ appearing in the Black–Scholes formula (5.35). The first derivative .x; t / WD

@ v.x; t / D ˆ.dC .x; t // @x

(5.36)

is called the option’s Delta; see Figure 5.6. In analogy to the formula for the hedging strategy in the binomial model obtained in Proposition 5.44, .x; t / determines the “Delta hedging portfolio” needed for a replication of the call option in continuous time, as explained in (5.45) below. The Gamma of the call option is given by .x; t / WD

@ @2 1 .x; t / D 2 v.x; t / D '.dC .x; t // p I @x @x x t

(5.37)

310

Chapter 5 Dynamic arbitrage theory

1 2K 0

K t T

0

Figure 5.6. The Delta .x; t / of the Black–Scholes price of a European call option.

p 2 see Figure 5.7. Here '.x/ D ˆ0 .x/ D e x =2 = 2 stands as usual for the density of the standard normal distribution. Large Gamma values occur in regions where the Delta changes rapidly, corresponding to the need for frequent readjustments of the Delta hedging portfolio. Note that is always strictly positive. It follows that v.x; t / is a strictly convex function of its first argument.

2K 0

K t T

0

Figure 5.7. The option’s Gamma .x; t /.

Exercise 5.7.1. Prove the formulas (5.36) and (5.37) for Delta and Gamma of a European call option. }

311

Section 5.7 Convergence to the Black–Scholes price

Remark 5.60. On the one hand, 0  .x; t /  1 implies that jv.x; t /  v.y; t /j  jx  yj: Thus, the total change of the option values is always less than a corresponding change in the asset prices. On the other hand, the strict convexity of x 7! v.x; t / together with (A.1) yields that for t > 0 and z > y v.z; t /  v.y; t / v.y; t /  v.0; t / v.y; t / > D zy y0 y and hence

zy v.z; t /  v.y; t / > : v.y; t / y

Similarly, one obtains v.x; t /  v.y; t / xy < v.y; t / y for x < y. Thus, the relative change of option prices is larger in absolute value than the relative change of asset values. This fact can be interpreted as the leverage effect for call options; see also Example 1.43. } Another important parameter is the Theta ‚.x; t / WD

@ x v.x; t / D p '.dC .x; t // C Kr e rt ˆ.d .x; t //I @t 2 t

(5.38)

see Figure 5.8. The fact ‚ > 0 corresponds to our general observation, made in Example 5.35, that arbitrage-free prices of European call options are typically increasing functions of the maturity. Exercise 5.7.2. Prove the formula (5.38) for the Theta of a European call option. Then show that the parameters , , and ‚ are related by the equation 1 ‚.x; t / D rx .x; t / C 2 x 2 .x; t /  r v.x; t / 2 when t > 0.

(5.39) }

Equation (5.39) implies that, for .x; t / 2 .0; 1/  .0; 1/, the function v solves the partial differential equation @v @2 v @v 1 D rx C 2 x 2 2  rv; @t @x 2 @x often called the Black–Scholes equation. Since v.x; t / ! f .x/ D .x  K/C

as t # 0,

(5.40)

(5.41)

v.x; t / is a solution of the Cauchy problem defined via (5.40) and (5.41). This fact is not limited to call options, it remains valid for all reasonable payoff profiles f .

312

Chapter 5 Dynamic arbitrage theory

2K 0

K t T 0 Figure 5.8. The Theta ‚.x; t /.

Proposition 5.61. Let f be a continuous function on .0; 1/ such that jf .x/j  c.1 C x/p for some c; p  0, and define Z 1 p e rt 2 2 f .xe  t yCrt t=2 /e y =2 dy; u.x; t / WD e rt E  Œ f .S t /  D p 2 1 where S t D x exp. W t C rt  2 t =2/ and W t has law N.0; t / under P  . Then u solves the Cauchy problem defined by the Black–Scholes equation (5.40) and the initial condition lim t#0 u.x; t / D f .x/, locally uniformly in x. The proof of Proposition 5.61 is the content of the next exercise. Exercise 5.7.3. In the context Proposition 5.61, use the formula (2.26) for the density of a log-normally distributed random variable to show that Z 1  log y  rt C 2 t =2  log x  1  E Œ f .S t /  D f .y/ dy; p ' p y t t 0 p 2 where '.x/ D e x =2 = 2. Then verify the validity of (5.40) by differentiating under the integral. Use the bound jf .x/j  c.1 C x/p for some c; p  0 to justify the interchange of differentiation and integration and to verify the initial condition lim t#0 u.x; t / D f .x/. } Recall that the Black–Scholes price v.S0 ; T / was obtained as the expectation of the discounted payoff e rT .ST  K/C under the measure P  . Thus, at a first glance, it may come as a surprise that the Rho of the option, %.x; t / WD

@ v.x; t / D Kt e rt ˆ.d .x; t //; @r

(5.42)

313

Section 5.7 Convergence to the Black–Scholes price

is strictly positive, i.e., the price is increasing in r; see Figure 5.9. Note, however, that the measure P  depends itself on the interest rate r, since E  Πe rT ST  D S0 . In a simple one-period model, we have already seen this effect in Example 1.43.

2K 0

K t T

0

Figure 5.9. The Rho %.x; t / of a call option.

The parameter is called the volatility. As we have seen, the Black–Scholes price of a European call option is an increasing function of the volatility, and this is reflected in the strict positivity of V .x; t / WD

p @ v.x; t / D x t '.dC .x; t //I @

(5.43)

see Figure 5.10. The function V is often called the Vega of the call option price, and the functions , , ‚, %, and V are usually called the Greeks (although “vega” does not correspond to a letter of the Greek alphabet). Exercise 5.7.4. Prove the respective formulas (5.42) and (5.43) for Rho and Vega of a European call option. Then derive formulas for the option’s Vanna, @2 v @ @V D D ; @x@ @ @x and the option’s Volga, which is also called Vomma, @V @2 v : D 2 @ @

}

Let us conclude this section with some informal comments on the dynamic picture behind the convergence result in Theorem 5.52 and the pricing formulas in Example 5.56 and Proposition 5.58. The constant r is viewed as the interest rate of a riskfree savings account S t0 D e rt ; 0  t  T:

314

Chapter 5 Dynamic arbitrage theory

2K 0

K t T 0 Figure 5.10. The Vega V .x; t /.

The prices of the risky asset in each discrete-time model are considered as a contin.N / .N / .N / WD Sk at the dates t D kT uous process SQ .N / D .SQ t /0tT , defined as SQ t N , and by linear interpolation in between. Theorem 5.52 shows that the distributions of .N / SQ t converge for each fixed t weakly to the distribution of   1  S t D S0 exp W t C r  2 t ; 2

(5.44)

where W t has a centered normal distribution with variance t . In fact, one can prove convergence in the much stronger sense of a functional central limit theorem: The laws of the processes SQ .N / , considered as C Œ0; T -valued random variables on .N ; F .N / ; PN /, converge weakly to the law of a geometric Brownian motion S D .S t /0tT , where each S t is of the form (5.44), and where the process W D.W t /0tT is a standard Brownian motion or Wiener process. A Wiener process is characterized by the following properties: 

W0 D 0 almost surely,



t 7! W t is continuous,



For each sequence 0 D t0 < t1 <    < tn D T , the increments W t1  W t0 ; : : : ; W tn  W tn1 are independent and have normal distributions N.0; ti  ti1 /;

see, e.g., [171]. This multiplicative version of a functional central limit theorem follows as above if we replace the classical central limit theorem by Donsker’s invariance principle; for details see, e.g., [99]. Sample paths of Brownian motion and geometric Brownian motion can be found in Figures 5.11 and 5.12.

315

Section 5.7 Convergence to the Black–Scholes price

0.5

0.5

1

Figure 5.11. A sample path of Brownian motion.

3

2

1

0.5

1

Figure 5.12. A sample path of geometric Brownian motion.

Geometric Brownian motion is the classical reference model in continuous-time mathematical finance. In order to describe the model more explicitly, we denote by W D .W t /0tT the coordinate process on the canonical path space  D C Œ0; T , defined by W t .!/ D !.t /, and by .F t /0tT the filtration given by F t D .Ws I s  t /. There is exactly one probability measure P on .; FT / such that W is a Wiener process under P , and it is called the Wiener measure. Let us now model the price process of a risky asset as a geometric Brownian motion S defined by (5.44). The discounted price process X t WD

St 2 D S0 e W t  t=2 ; rt e

0  t  T;

316

Chapter 5 Dynamic arbitrage theory

is a martingale under P , since EΠX t j Fs  D Xs EΠe .W t Ws /

2 .ts/=2

 D Xs

for 0  s  t  T . In fact, P is the only probability measure equivalent to P with that property. As in discrete time, uniqueness of the equivalent martingale measure implies completeness of the model. Let us sketch the construction of the replicating strategy for a given European option with reasonable payoff profile f .ST /, for example a call option with strike K. At time t the price of the asset is S t .!/, the remaining time to maturity is T  t , and the discounted price of the option is given by V t .!/ D e rt u.S t .!/; T  t /; where u is the function defined in Proposition 5.61. The process V D .V t /0tT can be viewed as the value process of the trading strategy  D . 0 ; / defined by  t D .S t ; T  t /;

 t0 D e rt u.S t ; T  t /   t X t ;

(5.45)

where  D @u=@x is the option’s Delta. Indeed, if we view  as the number of shares in the risky asset S and  0 as the number of shares in the riskfree savings account S t0 D e rt , then the value of the resulting portfolio in units of the numéraire is given by V t D  t  X t C  t0 D e rt . t  S t C  t0  S t0 /: The strategy replicates the option since VT WD lim e rt u.S t ; T  t / D e rT f .ST / D t"T

f .ST / ; ST0

due to Proposition 5.61. Moreover, its initial cost is given by the Black–Scholes price Z p e rT 1 2 2 V0 D u.S0 ; T / D e rT EŒ f .ST /  D p f .xe  T yCrT  T =2 /e y =2 dy: 2 1 It remains to show that the strategy is self-financing in the sense that changes in the portfolio value are only due to price changes in the underlying assets and do not require any additional capital. To this end, we use Itô’s formula dF .W t ; t / D

 1 @2 F @F @F  .W t ; t / d W t C C .W t ; t / dt @x 2 @x 2 @t

for a smooth function F , see, e.g., [171] or, for a strictly pathwise approach, [117]. Applied to the function F .x; t / D exp. x C rt  2 t =2/, it shows that the price process S satisfies the stochastic differential equation dS t D S t d W t C rS t dt:

(5.46)

317

Section 5.7 Convergence to the Black–Scholes price

Thus, the infinitesimal return dS t =S t is the sum of the safe return r dt and an additional noise term with zero expectation under P  . The strength of the noise is measured by the volatility parameter . Similarly, we obtain dX t D X t d W t D e rt .dS t  rS t dt /:

(5.47)

Applying Itô’s formula to the function F .x; t / D e rt u.exp.x C rt  2 t =2/; T  t / and using (5.46), we obtain d V t D e rt

 1 @u @2 u @u .S t ; t / dS t C e rt 2 S t2 2   ru .S t ; t / dt: @x 2 @x @t

The Black–Scholes partial differential equation (5.40) shows that the term in parenthesis is equal to rS t @u=@x, and we obtain from (5.47) that d Vt D

@u .S t ; t / dX t D  t dX t : @x

More precisely,

Z

t

V t D V0 C

s dXs ; 0

where the integral with respect to X is defined as an Itô integral, i.e., as the limit of non-anticipating Riemann sums X  ti .X tiC1  X ti / ti 2Dn ; ti t

along an increasing sequence .Dn / of partitions of the interval Œ0; T ; see, e.g., [117]. Thus, the Itô integral can be interpreted in financial terms as the cumulative net gain generated by dynamic hedging in the discounted risky asset as described by the hedging strategy . This fact is an analogue of property (c) in Proposition 5.7, and in this sense  D . 0 ; / is a self-financing trading strategy in continuous time. Similarly, we obtain the following continuous-time analogue of (5.5), which describes the undiscounted value of the portfolio as a result of dynamic trading both in the undiscounted risky asset and the riskfree asset Z t Z t rt e V t D V0 C s dSs C s0 dSs0 : 0

0

Perfect replication also works for exotic options C.S / defined by reasonable functionals C on the path space C Œ0; T , due to a general representation theorem for such functionals as Itô integrals of the underlying Brownian motion W or, via (5.47), of the

318

Chapter 5 Dynamic arbitrage theory

process X . Weak convergence on path space implies, in analogy to Proposition 5.61, that the arbitrage-free prices of the options C.S .N / /, computed as discounted expectations under the measure PN , converge to the discounted expectation e rT EŒ C.S /  under the Wiener measure P . On the other hand, the discussion in Section 5.6 suggests that the prices of certain exotic contingent claims, such as barrier options, can be computed in closed form as the Black–Scholes price for some corresponding payoff profile of the form f .ST /. This is illustrated by the following example, where the price of an up-and-in call is computed in terms of the distribution of the terminal stock price under the equivalent martingale measure. Example 5.62 (Black–Scholes price of an up-and-in call option). Consider an upand-in call option ´ .ST  K/C if max0tT S t  B; call Cu&i .S/ D 0 otherwise, where B > S0 _ K denotes a given barrier, and where K > 0 is the strike price. As approximating models we choose the CRR models of Example 5.54. That is, we have interest rates rT rN D N and parameters aN and bN defined by aO N D 1 C aN D e 

p

T =N

and

bON D 1 C bN D e 

p T =N

for some given > 0. Applying the formula obtained in Example 5.49 yields  call EN ΠCu&i .S .N / /  .N /

.N /

 D EN Π.SN  K/C I SN  BN   p  kN  B 2 N .N / .N /  N C EN Π.SN  KQ N /C I SN < BQ N ;  1  pN S0

where

l pN Bm kN D p log S0 T kN k , BN WD S0 bON , BQ N WD S02 =BN D is the smallest integer k such that B  S0 bON kN , and S0 aO N  S 2 0 2kN KQ N D K bON DK : BN

319

Section 5.7 Convergence to the Black–Scholes price

Then we have BN & B; BQ N % BQ D

S02 ; B

and

 S 2 0 KQ N % KQ D K : B

Since f .x/ D .x  K/C I¹xBº is continuous a.e., we obtain .N /

 Π.SN EN

.N /

 K/C I SN

.N /

  BN  D EN Π.SN

.N /

 K/C I SN

B

! EΠ.ST  K/C I ST  B ; due to Proposition 5.58. Combining the preceding argument with the fact that .N / PN ΠKQ N  SN  KQ  ! 0

also gives the convergence of the second expectation .N /

 Π.SN EN

.N / .N / .N /   KQ N /C I SN < BQ N  D EN Π.SN  KQ N /C I SN < BQ 

Q C I ST < BQ : ! EΠ.ST  K/ Next we note that for constants c; d > 0 1 cx 2 C 1  e dx 2c log dx  d; D d x#0 x e  1  cx 2 lim

due to l’Hôpital’s rule. From this fact, one deduces that  B  2r2 1  p  kN

N ! :  1  pN S0 Thus, we may conclude that the arbitrage-free prices 1 E  ΠC call .SQ .N / /  .1 C rN /N N u&i in the N th approximating model converge to    B  2r2 C1

rT C C Q Q e EŒ .ST  K/ I ST < B  : EŒ .ST  K/ I ST  B  C S0 The expectations occurring in this formula are integrals with respect to a log-normal distribution and can be explicitly computed as in Example 5.56. Moreover, our limit is in fact equal to the Black–Scholes price of the up-and-in call option: The functional call Cu&i .  / is continuous in each path in C Œ0; T  whose maximum is different from the value B, and one can show that these paths have full measure for the law of S uncall der P . Hence, Cu&i .  / is continuous P ı S 1 -a.e., and the functional version of Proposition 5.58 yields  call Q .N / call EN Œ Cu&i .S /  ! EŒ Cu&i .S/ ;

so that our limiting price must coincide with the discounted expectation on the right. }

320

Chapter 5 Dynamic arbitrage theory

Remark 5.63. Let us assume, more generally, that the price process S is defined by S t D S0 e W t C˛t ;

0  t  T;

for some ˛ 2 R. Applying Itô’s formula as in (5.46), we see that S is governed by the stochastic differential equation dS t D S t d W t C bS t dt with b D ˛ C 12 2 . The discounted price process is given by 

X t D S0 e W t C.˛r/t D S0 e W t

 2 t=2

with W t D W t C t for D .b  r/= . The process W  is a Wiener process under the measure P   P defined by the density dP  2 D e  WT  T =2 : dP In fact, P  is the unique equivalent martingale measure for X . We can now repeat the arguments above to conclude that the cost of perfect replication for a contingent claim C.S / is given by e rT E  Œ C.S / : } Even in the context of simple diffusion models such as geometric Brownian motion, however, completeness is lost as soon as the future behavior of the volatility parameter is unknown. If, for instance, volatility itself is modeled as a stochastic process, we are facing incompleteness. Thus, the problems of pricing and hedging in discrete-time incomplete markets as discussed in this book reappear in continuous time. Other versions of the invariance principle may lead to other classes of continuous-time models with discontinuous paths, for instance to geometric Poisson or Lévy processes. Discontinuity of paths is another important source of incompleteness. In fact, this has already been illustrated in this book, since discrete-time models can be regarded as stochastic processes in continuous time, where jumps occur at predictable dates.

Chapter 6

American contingent claims

So far, we have studied European contingent claims whose payoff is due at a fixed maturity date. In the case of American options, the buyer can claim the payoff at any time up to the expiration of the contract. First, we take the point of view of the seller, whose aim is to hedge against all possible claims of the buyer. In Section 6.1, this problem is solved under the assumption of market completeness, using the Snell envelope of the contingent claim. The buyer tries to choose the best date for exercising the claim, contingent on the information available up to that time. Since future prices are usually unknown, a formulation of this problem will typically involve subjective preferences. If preferences are expressed in terms of expected utility, the choice of the best exercise date amounts to solving an optimal stopping problem. In the special case of a complete market model, any exercise strategy which maximizes the expected payoff under the unique equivalent martingale measure turns out to be optimal even in an almost sure sense. In Section 6.3, we characterize the set of all arbitrage-free prices of an American contingent claim in an incomplete market model. This involves a lower Snell envelope of the claim, which is analyzed in Section 6.5, using the fact that the class of equivalent martingale measures is stable under pasting. This notion of stability under pasting is discussed in Section 6.4 in a general context, and in Section 6.5 we point out its connection with the time-consistency of dynamic risk measures. This connection will be discussed systematically in Chapter 11. The results on lower Snell envelopes can also be regarded as a solution to the buyer’s optimal stopping problem in the case where preferences are described by robust Savage functionals. Moreover, these results will be used in the theory of superhedging of Chapter 7.

6.1

Hedging strategies for the seller

Throughout this chapter we will continue to use the setting described in Section 5.1. We start by introducing the Doob decomposition of an adapted process and the notion of a supermartingale. Proposition 6.1. Let Q be a probability measure on .; FT /, and suppose that Y is a stochastic process that is adapted to the filtration .F t / tD0;:::;T and satisfies Y t 2 L1 .Q/ for all t . Then there exists a unique decomposition Y D M  A;

(6.1)

322

Chapter 6 American contingent claims

where M is a Q-martingale and A is a process such that A0 D 0 and .A t / tD1;:::;T is predictable. The decomposition (6.1) is called the Doob decomposition of Y with respect to the probability measure Q. Proof. Define A by A t  A t1 WD EQ ΠY t  Y t1 j F t1 

for t D 1; : : : ; T .

(6.2)

Then A is predictable and M t WD Y t C A t is a Q-martingale. Clearly, any process A with the required properties must satisfy (6.2), so the uniqueness of the decomposition follows. Definition 6.2. Let Q be a probability measure on .; FT / and suppose that Y is an adapted process such that Y t 2 L1 .Q/ for all t . Denote by Y D M  A the Doob decomposition of Y . (a) Y is called a Q-supermartingale if A is increasing. (b) Y is called a Q-submartingale if A is decreasing. Clearly, a process is a martingale if and only if it is both a supermartingale and a submartingale, i.e., if and only if A  0. The following exercise gives equivalent characterizations of the supermartingale property of a process Y . Exercise 6.1.1. Let Y be an adapted process with Y t 2 L1 .Q/ for all t . Show that the following conditions are equivalent: (a) Y is a Q-supermartingale. (b) Ys  EQ ΠY t j Fs  for 0  s  t  T . (c) Y t1  EQ ΠY t j F t1  for t D 1; : : : ; T . (d) Y is a Q-submartingale.

}

Exercise 6.1.2. Let Y be a nonnegative Q-supermartingale. Show that for 0  s  } T  t we have Y tCs D 0 Q-a.s. on ¹Y t D 0º. We now return to the market model introduced in Section 5.1. An American option, or American contingent claim, corresponds to a contract which is issued at time 0 and which obliges the seller to pay a certain amount C  0 if the buyer decides at time  to exercise the option. The choice of the exercise time  is entirely up to the buyer, except that the claim is automatically exercised at the “expiration date” of the claim. The American contingent claim can be exercised only once: It becomes invalid as soon as the payoff has been claimed by the buyer. This concept is formalized as follows: Definition 6.3. An American contingent claim is a non-negative adapted process C D .C t / tD0;:::;T on the filtered space .; .F t / tD0;:::;T /.

Section 6.1 Hedging strategies for the seller

323

For each t , the random variable C t is interpreted as the payoff of the American contingent claim if the claim is exercised at time t . The time horizon T plays the role of the expiration date of the claim. The possible exercise times for C are not limited to fixed deterministic times t 2 ¹0; : : : ; T º; the buyer may exercise the claim in a way which depends on the scenario ! 2  of the market evolution. Definition 6.4. An exercise strategy for an American contingent claim C is an FT measurable random variable  taking values in ¹0; : : : ; T º. The payoff obtained by using  is equal to C .!/ WD C.!/ .!/; ! 2 : Example 6.5. An American put option on the i th asset and with strike K > 0 pays the amount put C t WD .K  S ti /C if it is exercised at time t . The payoff at time t of the corresponding American call option is given by C tcall WD .S ti  K/C : Clearly, the American call option is “out of the money” (i.e., has zero payoff) if the corresponding American put is “in the money” (i.e., has non-zero payoff). It is therefore a priori clear that the respective owners of C put and C call will usually exercise their claims at different times. In particular, there will be no put-call parity for American options. } Similarly, one defines American versions of most options mentioned in the examples of Section 5.3. Clearly, the value of an American option is at least as high as the value of the corresponding European option with maturity T . Remark 6.6. It should be emphasized that the concept of American contingent claims can be regarded as a generalization of European contingent claims: If C E is a European contingent claim, then we can define a corresponding American claim C A by ´ 0 if t < T , A (6.3) Ct D E C if t D T . } Example 6.7. A Bermuda option can be exercised by its buyer at each time of a predetermined subset T  ¹0; : : : ; T º. For instance, a Bermuda call option pays the amount .S ti  K/C if it is exercised at some time t 2 T . Thus, a Bermuda option is a financial instrument “between” an American option with T D ¹0; : : : ; T º and a European option with T D ¹T º, just as Bermuda lies between America and Europe; hence the name “Bermuda option”. A Bermuda option can be regarded as a particular American option C that pays the amount C t D 0 for t … T . }

324

Chapter 6 American contingent claims

The process Ht D

Ct ; S t0

t D 0; : : : ; T;

of discounted payoffs of C will be called the discounted American claim associated with C . As far as the mathematical theory is concerned, the discounted American claim H will be the primary object. For certain examples it will be helpful to keep track of the numéraire and, thus, of the payoffs C t prior to discounting. In this section, we will analyze the theory of hedging American claims in a complete market model. We will therefore assume throughout this section that the set P of equivalent martingale measures consists of one single element P  P D ¹P  º: Under this assumption, we will construct a suitable trading strategy that permits the seller of an American claim to hedge against the buyer’s discounted claim H . Let us first try to characterize the minimal amount of capital U t which will be needed at time t 2 ¹0; : : : ; T º. Since the choice of the exercise time  is entirely up to the buyer, the seller must be prepared to pay at any time t the current payoff H t of the option. This amounts to the condition U t  H t . Moreover, the amount U t must suffice to cover the purchase of the hedging portfolio for the possible payoffs Hu for u > t . Since the latter condition is void at maturity, we require UT D HT : At time T  1, our first requirement on UT 1 reads UT 1  HT 1 . The second requirement states that the amount UT 1 must suffice for hedging the claim HT in case the option is not exercised before time T . Due to our assumption of market completeness, the latter amount equals E  Œ HT j FT 1  D E  Œ UT j FT 1 : Thus, UT 1 WD HT 1 _ E  Œ UT j FT 1  is the minimal amount that fulfills both requirements. Iterating this argument leads to the following recursive scheme for U t : UT WD HT ;

U t WD H t _ E  ΠU tC1 j F t  

for t D T  1; : : : ; 0.

(6.4)

Definition 6.8. The process U P WD U defined by the recursion (6.4) is called the Snell envelope of the process H with respect to the measure P  .

325

Section 6.1 Hedging strategies for the seller

Example 6.9. Let H E be a discounted European claim. Then the Snell envelope with respect to P  of the discounted American claim H A associated with H E via (6.3) satisfies  U tP D E  ΠHTA j F t  D E  ΠH E j F t : Thus, U is equal to the value process of a replicating strategy for H E .

}

Clearly, a Snell envelope U Q can be defined for any probability measure Q on .; FT / and for any adapted process H that satisfies the following integrability condition: H t 2 L1 .Q/ for t D 0; : : : ; T . (6.5) In our finite-time setting, this condition is equivalent to  EQ max jH t j < 1: tT

For later applications, the following proposition is stated for a general measure Q. Proposition 6.10. Let H be an adapted process such that (6.5) holds. Then the Snell envelope U Q of H with respect to Q is the smallest Q-supermartingale dominating H : If UQ is another Q-supermartingale such that UQ t  H t Q-a.s. for all t , then Q UQ t  U t Q-a.s. for all t . Q

Q

Proof. It follows from the definition of U Q that U t1  EQ ΠU t j F t1  so that U Q is indeed a supermartingale. If UQ is another supermartingale dominating H , then Q UQ T  HT D UT . We now proceed by backward induction on t . If we already know Q that UQ t  U t , then Q UQ t1  EQ ΠUQ t j F t1   EQ ΠU t j F t1 :

Adding our assumption UQ t1  H t1 yields that Q Q UQ t1  H t1 _ EQ ΠU t j F t1  D U t1 ;

and the result follows. Proposition 6.10 illustrates how the seller can (super-) hedge a discounted American claim H by using the Doob decomposition 

U tP D M t  A t ; 

t D 0; : : : ; T;

of the Snell envelope U P with respect to P  . Then M is a P  -martingale, A is increasing, and .A t / tD1;:::;T is predictable. Since we assume the completeness of

326

Chapter 6 American contingent claims

the market model, Theorem 5.38 yields the representation of the martingale M as the “stochastic integral” of a suitable d -dimensional predictable process  

M t D U0P C

t X

k  .Xk  Xk1 /;

t D 0; : : : ; T:

(6.6)

kD1

It follows that



M t  U tP  H t

for all t .

0

By adding a numéraire component such that  D . 0 ; / becomes a self-financing  trading strategy with initial investment U0P , we obtain a (super-) hedge for H , namely a self-financing trading strategy whose value process V satisfies Vt  Ht

for all t .

(6.7)



Thus, U tP may be viewed as the resulting capital at each time t if we use the selffinancing strategy , combined with a refunding scheme where we withdraw suc cessively the amounts defined by the increments of A. In fact, U tP is the minimal investment at time t for which one can purchase a hedging strategy such that (6.7) holds. This follows from our next result. 

Theorem 6.11. Let H be a discounted American claim with Snell envelope U P . Then there exists a d -dimensional predictable process  such that u X



U tP C

k  .Xk  Xk1 /  Hu

for all u  t P -a.s.

(6.8)

kDtC1

Moreover, any F t -measurable random variable UQ t which, for some predictable ,  satisfies (6.8) in place of U tP is such that  UQ t  U tP

P -a.s.



Thus, U tP is the minimal amount of capital which is necessary to hedge H from time t up to maturity.  Proof. Clearly, U P satisfies (6.8) for  as in (6.6). Now suppose that UQ t is F t measurable, that Q is predictable, and that

Vu WD UQ t C

u X

Qk  .Xk  Xk1 /  Hu

for all u  t P -a.s.

kDtC1 



We show Vu  UuP for all u  t by backward induction. VT  HT D UTP holds P  for some u. Since our market model is by assumption, so assume VuC1  UuC1 complete, Theorem 5.37 implies that Q is bounded. Hence, we get E  ΠVuC1  Vu j Fu  D E  ΠQuC1  .XuC1  Xu / j Fu  D 0

P -a.s.

327

Section 6.2 Stopping strategies for the buyer

It follows that 



P Vu D E  ΠVuC1 j Fu   Hu _ E  ΠUuC1 j Fu  D UuP :

6.2

Stopping strategies for the buyer

In this section, we take the point of view of the buyer of an American contingent claim. Thus, our aim is to optimize the exercise strategy. It is natural to assume that the decision to exercise the claim at a particular time t depends only on the market information which is available at t . This constraint can be formulated as follows: Definition 6.12. A function  W  ! ¹0; 1; : : : ; T º [ ¹C1º is called a stopping time if ¹ D t º 2 F t for t D 0; : : : ; T . In particular, the constant function   t is a stopping time for fixed t 2 ¹0; : : : ; T º. Exercise 6.2.1. Show that a function  W  ! ¹0; 1; : : : ; T º [ ¹C1º is a stopping time if and only if ¹  t º 2 F t for each t . Show next that, if  and are two stopping times, then the following functions are also stopping times:  ^ ;

 _ ;

. C / ^ T:

}

Example 6.13. A typical example of a non-trivial stopping time is the first time at which an adapted process Y exceeds a certain level c  .!/ WD inf¹t  0 j Y t .!/  cº: In fact, ¹  t º D

t [

¹Ys  cº 2 F t

sD0

for t D 0; : : : ; T . This example also illustrates the role of the value C1 in Definition 6.12: We have  .!/ D C1 if, for this particular !, the criterion that triggers  is not met for any t 2 ¹0; : : : ; T º. } Definition 6.14. For any stochastic process Y and each stopping time  we denote by Y  the process stopped in  Y t .!/ WD Y t^.!/ .!/

for ! 2  and for all t 2 ¹0; : : : ; T º.

It follows from the definition of a stopping time that Y  is an adapted process if Y is. Informally, the following basic theorem states that a martingale cannot be turned into a favorable game by using a clever stopping strategy. This result is often called Doob’s stopping theorem or the optional sampling theorem. Recall that we assume F0 D ¹;; º.

328

Chapter 6 American contingent claims

Theorem 6.15. Let M be an adapted process such that M t 2 L1 .Q/ for each t . Then the following conditions are equivalent: (a) M is a Q-martingale. (b) For any stopping time  the stopped process M  is a Q-martingale. (c) EQ Œ M^T  D M0 for any stopping time  . Proof. (a) ) (b): Note that  M tC1  M t D .M tC1  M t / I¹>t º :

Since ¹ > t º 2 F t , we obtain that  EQ Œ M tC1  M t j F t  D EQ Œ M tC1  M t j F t   I¹>t º D 0:

(b) ) (c): This follows simply from the fact that the expectation of M t is constant in t . (c) ) (a): We need to show that if t < T , then EQ ΠMT I A  D EQ ΠM t I A 

(6.9)

for each A 2 F t . Fix such an A and define a stopping time  as ´ t if ! 2 A,  .!/ WD T if ! … A. We obtain that M0 D EQ Œ MT ^  D EQ Œ M t I A  C EQ Œ MT I Ac : Using the constant stopping time T instead of  yields that M0 D EQ Œ MT  D EQ Œ MT I A  C EQ Œ MT I Ac : Subtracting the latter identity from the previous one yields (6.9). Exercise 6.2.2. Let Y D M  A be the Doob decomposition with respect to Q of an adapted process Y with Y t 2 L1 .Q/ (t D 0; : : : ; T ), and let  be a stopping time. } Show that Y  D M   A is the Doob decomposition of Y  . Corollary 6.16. Let U be an adapted process such that U t 2 L1 .Q/ for each t . Then the following conditions are equivalent: (a) U is a Q-supermartingale. (b) For any stopping time  , the stopped process U  is a Q-supermartingale.

329

Section 6.2 Stopping strategies for the buyer

Proof. If U D M  A is the Doob decomposition of U , then Exercise 6.2.2 implies that U  D M   A is the Doob decomposition of U  . This observation and Theorem 6.15 yield the equivalence of (a) and (b). Let us return to the problem of finding an optimal exercise time  for a discounted American claim H . We assume that the buyer chooses the possible exercise times from the set T WD ¹ j  is a stopping time with   T º of all stopping times which do not take the value C1. Assume that the aim of the buyer is to choose a payoff from the class ¹H j  2 T º which is optimal in the sense that it has maximal expectation. Thus, the problem is Maximize EŒ H  among all  2 T .

(6.10)

The analysis of the optimal stopping problem (6.10) does not require any properties of the underlying market model, not even the absence of arbitrage. We may also drop the positivity assumption on H : All we have to assume is that H is an adapted process which satisfies H t 2 L1 .; F t ; P / for all t . (6.11) This relaxed assumption will be useful in Chapter 9, and it allows us to include the interpretation of the optimal stopping problem in terms of the following utility maximization problem: Remark 6.17. Suppose the buyer uses a preference relation on X WD ¹H j  2 T º which can be represented in terms of a Savage representation U.H / D EQ Œ u.H /  where Q is a probability measure on .; F /, and u is a measurable or continuous function; see Section 2.5. Then a natural goal is to maximize the utility U.H / among all  2 T . This is equivalent to the optimal stopping problem (6.10) for the transformed process HQ t WD u.H t /, and with respect to the measure Q instead of P . This utility maximization problem is covered by the discussion in this section as long as HQ t 2 L1 .Q/ for all t . In Remark 6.49 we will discuss the problem of maximizing the more general utility functionals which appear in a robust Savage representation. } Under the assumption (6.11), we can construct the Snell envelope U WD U P of H with respect to P , i.e., U is defined via the recursive formula UT WD HT

and

U t WD H t _ EΠU tC1 j F t ;

Let us define a stopping time min by min WD min¹t  0 j U t D H t º:

t D T  1; : : : ; 0:

330

Chapter 6 American contingent claims

Note that min  T since UT D HT . As we will see in the following theorem, min maximizes the expectation of H among all  2 T . In other words, min is a solution to our optimal stopping problem (6.10). Similarly, we let .t/

min WD min¹u  t j Uu D Hu º; which is a member of the set T t WD ¹ 2 T j   t º: The following theorem uses the essential supremum of a family of random variables as explained in Appendix A.5. Theorem 6.18. The Snell envelope U of H satisfies U t D EŒ H .t / j F t  D ess sup EŒ H j F t : min

2T t

In particular, U0 D EΠHmin  D sup EΠH : 2T

Proof. Since U is a supermartingale under P , Corollary 6.16 shows that for  2 T t U t  EΠU j F t   EΠH j F t : Therefore, U t  ess sup EΠH j F t : 2T t

Hence, the theorem will be proved if we can show that U t D EΠH .t / j F t , which is min in turn implied by the identity U t D EΠU .t / j F t :

(6.12)

min

In order to prove (6.12), let U .t/ denote the stopped process Us.t/ WD Us^ .t / ; min

.t/

and fix some s between t and T . Then Us > Hs on ¹min > sº. Hence, P -a.s. on .t/ ¹min > sº .t/

Us.t/ D Us D Hs _ EΠUsC1 j Fs  D EΠUsC1 j Fs  D EΠUsC1 j Fs : .t/

.t/

.t/

.t/

On the set ¹min  sº one has UsC1 D U .t / D Us , hence Us Thus, U .t/ is a martingale from time t on .t/

min

.t/

D EΠUsC1 j Fs .

Us.t/ D EŒ UsC1 j Fs  for all s 2 ¹t; t C 1; : : : ; T  1º.

331

Section 6.2 Stopping strategies for the buyer

It follows that .t/

.t/

EΠU .t / j F t  D EΠUT j F t  D U t min

D Ut :

This proves the claim (6.12). Definition 6.19. A stopping time   2 T is called optimal .with respect to P / if EΠH   D sup EΠH : 2T

In particular, min is an optimal stopping time in the sense of this definition. The following result implies that min is in fact the minimal optimal stopping time. Proposition 6.20. A stopping time  2 T is optimal if and only if H D U P -a.s., and if the stopped process U  is a martingale. In particular, any optimal stopping time  satisfies   min . Proof. First note that  2 T is optimal if it satisfies the two conditions of the assertion, because then Theorem 6.18 implies that sup EΠH  D U0 D EΠUT  D EΠU  D EΠH :

2T

For the converse implication, we apply the assumption of optimality, the fact that H  U , and the stopping theorem for supermartingales to obtain that U0 D EŒ H   EŒ U   U0 ; so that all inequalities are in fact equalities. It follows in particular that H D U P almost surely. Moreover, the identity EŒ U  D U0 implies that the stopped process U  is a supermartingale with constant expectation U0 , and hence is a martingale. In general, there can be many different optimal stopping times. The largest optimal stopping time admits an explicit description: It is the first time before T for which the Snell envelope U loses the martingale property max WD inf¹t  0 j EŒ U tC1  U t j F t  ¤ 0º ^ T D inf¹t  0 j A tC1 ¤ 0º ^ T: Here, A denotes the increasing process obtained from the Doob decomposition of U under P . Theorem 6.21. The stopping time max is the largest optimal stopping time. Moreover, a stopping time  is optimal if and only if P -a.s.   max and U D H .

332

Chapter 6 American contingent claims

Proof. Let U D M  A be the Doob decomposition of U . Recall from Exercise 6.2.2 that U  D M   A is the Doob decomposition of U  for any stopping time  . Thus, U  is a martingale if and only if A D 0, because A is increasing. Therefore, U  is a martingale if and only if   max , and so the second part of the assertion follows from Proposition 6.20. It remains to prove that max itself is optimal, i.e., that Umax D Hmax . This is clear on the set ¹max D T º. On the set ¹max D t º for t < T one has A t D 0 and A tC1 > 0. Hence, EŒ U tC1  U t j F t  D .A tC1  A t / D A tC1 < 0 on ¹max D t º. Thus, U t > EŒ U tC1 j F t  and the definition of the Snell envelope yields that U t D H t _ EŒ U tC1 j F t  D H t on ¹max D t º. Let us now return to our complete financial market model, where H t is the discounted payoff of an American contingent claim. Thus, an optimal stopping strategy for H maximizes the expected payoff EŒ H . But a stopping time turns out to be the best choice even in a pathwise sense, provided that it is optimal with respect to the unique equivalent martingale measure P  in a complete market model. In order to explain this fact, let us first recall from Section 6.1 the construction of a perfect hedge of H from the seller’s perspective. Let 

UP D M  A 

denote the Doob decomposition of the Snell envelope U P of H with respect to P  . Since P  is the unique equivalent martingale measure in our model, the martingale M has the representation 

M t D U0P C

t X

k  .Xk  Xk1 /;

t D 0; : : : ; T;

kD1

for a d -dimensional predictable process . Clearly, M is equal to the value process of  the self-financing strategy constructed from  and the initial investment U0P . Since M dominates H , this yields a perfect hedge of H from the perspective of the seller: If the buyer exercises the option at some stopping time  , then the seller makes a profit M  H  0. The following corollary states that the buyer can in fact meet the value of the seller’s hedging portfolio, and that this happens if and only if the option  is exercised at an optimal stopping time with respect to P  . In this sense, U0P can be regarded as the unique arbitrage-free price of the discounted American claim H . Corollary 6.22. With the above notation, 

H  M D U0P C

 X

k  .Xk  Xk1 /;

P  -a.s. for all  2 T ,

kD1

and equality holds P  -almost surely if and only if  is optimal with respect to P  .

333

Section 6.2 Stopping strategies for the buyer

Proof. At time  ,



H  UP D M  A  M : 

Moreover, by Theorem 6.21, both H D UP and A D 0 hold P  -a.s. if and only if  is optimal with respect to P  . Let us now compare a discounted American claim H to the corresponding discounted European claim HT , i.e., to the contract which is obtained from H by restricting the exercise time to be T . In particular, we are interested in the relation between American and European put or call options. Let V t WD E  Œ HT j F t  denote the amount needed at time t to hedge HT . Since our market model is complete, V t can also be regarded as the unique arbitrage-free price of the discounted claim HT  at time t . From the seller’s perspective, U tP plays a similar role for the American option. It is intuitively clear that an American claim should be more expensive than the corresponding European one. This is made mathematically precise in the following statement. Proposition 6.23. With the above notation, U tP  dominates H , then U P and V coincide.



 V t for all t . Moreover, if V

Proof. The first statement follows immediately from the supermartingale property  of U P   U tP  E  Œ UTP j F t  D E  Œ HT j F t  D V t : Next, if the P  -martingale V dominates H , then it also dominates the corresponding   Snell envelope U P by Proposition 6.10. Thus V and U P must coincide. Remark 6.24. The situation in which V dominates H occurs, in particular, when the process H is a P  -submartingale. This happens, for instance, if H is obtained by applying a convex function f W Rd ! Œ0; 1/ to the discounted price process X . Indeed, in this case, Jensen’s inequality for conditional expectations implies that E  Œ f .X tC1 / j F t   f .E  Œ X tC1 j F t / D f .X t /:

}

Example 6.25. The discounted payoff of an American call option C tcall D .S t1 K/C is given by   K C call 1 Ht D Xt  0 : St Under the hypothesis that S t0 is increasing in t , (5.24) states that call j F t   H tcall E  ΠH tC1

P  -a.s. for t D 0; : : : ; T  1.

334

Chapter 6 American contingent claims 

In other words, H call is a submartingale, and the Snell envelope U P of H call coincides with the value process    K C ˇˇ  1 Vt D E XT  0 ˇ Ft ST of the corresponding European call option with maturity T . In particular, we have  U0P D V0 , i.e., the unique arbitrage-free price of the American call option is equal to its European counterpart. Moreover, Theorem 6.21 implies that the maximal optimal stopping time with respect to P  is given by max  T . This suggests that, in a complete model, an American call should not be exercised before maturity. } put

Example 6.26. For an American put option C t WD .K  S t1 /C the situation is different, because the argument in (5.24) fails unless S 0 is decreasing. If S 0 is an increasing bond, then the time value   1 C ˇ ˇ 0  .K  ST / W t WD S t E ˇ F t  .K  S t1 /C ST0 of a European put .K  ST1 /C typically becomes negative at a certain time t , corresponding to an early exercise premium W t ; see Figure 5.3. Thus, the early exercise premium is the surplus which an owner of the American put option would have over the value of the European put .K  ST1 /C . The relation between the price of a put option and its intrinsic value can be illustrated in the context of the CRR model. With the notation of Section 5.5, the price process of the risky asset S t D S t1 can be written as S t D S0 ƒ t

for ƒ t WD

t Y

.1 C Rk /

kD1

and with the constant S0  0. Recall that the returns Rk can take only two possible values a and b with 1 < a < b, and that the market model is arbitrage-free if and only if the riskless interest rate r satisfies a < r < b. In this case, the model is complete, and the unique equivalent martingale measure P  is characterized by the fact that it makes R1 ; : : : ; RT independent with common distribution P  ΠRk D b  D p  D Let .x/ WD sup E 2T

C put





r a : ba

.K  xƒ /C .1 C r/

(6.13) 

denote the price of regarded as a function of x WD S0 . Clearly, .x/ is a convex and decreasing function in x. Let us assume that r > 0 and that the parameter a is

335

Section 6.2 Stopping strategies for the buyer

strictly negative. A trivial situation occurs if the option is “far out of the money” in the sense that K x ; .1 C a/T because then S t D xƒ t  K for all t , and the payoff of C put is always zero. In particular, .x/ D 0. If K x (6.14) .1 C b/T then S t D xƒ t  K for all t , and hence     K  .x/ D sup E  x D K  x: .1 C r/ 2T In this case, the price of the American put option is equal to its intrinsic value .Kx/C at time t D 0, and an optimal strategy for the owner would simply consist in exercising the option immediately, i.e., there is no demand for the option in the regime (6.14). Now consider the case K Kx< .1 C a/T of a put option which is “at the money” or “not too far out of the money”. For large put enough t > 0, the probability P  Œ C t > 0  of a non-zero payoff is strictly positive, while the intrinsic value .K  x/C vanishes. It follows that the price .x/ is strictly higher than the intrinsic value, and so it is not optimal for the buyer to exercise the option immediately. Summarizing our observations, we can say that there exists a value x  with K  x < K .1 C b/T such that .x/ D .K  x/C

for x  x  ,

.x/ > .K  x/C

for x  < x < K=.1 C a/T ;

.x/ D 0

for x  K=.1 C a/T ;

and

}

see Figure 6.1.

Remark 6.27. In the context of an arbitrage-free CRR model, we consider a discounted American claim H whose payoff is determined by a function of time and of the current spot price, i.e., H t D h t .S t /

for all t .

336

Chapter 6 American contingent claims

S0

K

Figure 6.1. The price of an American put option as a function of S0 compared to the option’s intrinsic value .K  S0 /C .

Clearly, this setting includes American call and put options as special cases. By using  the same arguments as in the derivation of (5.28), we get that the Snell envelope U P of H is of the form 

U tP D u t .S t /;

t D 0; : : : ; T;

where the functions u t are determined by the recursion uT .x/ D hT .x/

and

u t .x/ D h t .x/ _ .u tC1 .x bO / p  C u tC1 .x a/ O .1  p  //:

Here p  is defined as in (6.13), and the parameters aO and bO are given by aO D 1 C a and bO D 1 C b. Thus, the space Œ0; T   Œ0; 1/ can be decomposed into the two regions Rc WD ¹.t; x/ j u t .x/ > h t .x/º and

Rs WD ¹.t; x/ j u t .x/ D h t .x/º;

and the minimal optimal stopping time min can be described as the first exit time of the space time process .t; S t / from the continuation region Rc or, equivalently, as the first entrance time into the stopping region Rs min D min¹t  0 j .t; S t / … Rc º D min¹t  0 j .t; S t / 2 Rs º:

}

Exercise 6.2.3. Consider a market model with two assets and an American contingent claim. The development of the discounted price process X WD X 1 and the discounted American claim is described by the following diagram.

337

Section 6.3 Arbitrage-free prices

 X2 D 9, H2 D 4 

   X1 D 8, H1 D 1:5  H  H  H



 H

X0 D 5, H0 D 1  H

HH H  X2 D 6, H2 D 1 

H

 HH  H X1 D 4, H1 D 0  H H HH

H H X2 D 3, H2 D 0

The buyer of the American claim uses a probability measure P that assigns equal probability to each of the possible scenarios. Find an optimal stopping strategy that maximizes EΠH  over  2 T . What would be an optimal stopping time p if the buyer p uses the utility function u.x/ D x and thus aims at maximizing EΠH ? Then show that the market model admits a unique risk-neutral measure P  and compute  the corresponding Snell envelope U P . } Exercise 6.2.4. Let H tK WD

.K  S t /C .1 C r/t

be the discounted payoff of an American put option with strike K in a market model with one risky asset S D .S t / tD0;:::T and a riskless asset S t0 D .1 C r/t , where K r > 0. We denote by min the minimal optimal stopping time of the buyer’s problem K to maximize EŒ H  over  2 T . 0

K K (a) Show that min  min P -a.s. when K  K 0 . K (b) Show that ess infK0 min D 0 P -a.s.

(c) Use (b) and the fact that F0 D ¹;; º to conclude that there exists K0  0 such K D 0 P -a.s. for all K  K . that min } 0

6.3

Arbitrage-free prices

In this section, we drop the condition of market completeness, and we develop the notion of an arbitrage-free price  for a discounted American claim H in a general incomplete framework. The basic idea consists in reducing the problem to the determination of the arbitrage-free price for the payoff H which arises from H by fixing the exercise strategy  . The following remark explains that H can be treated like the discounted payoff of a European contingent claim, whose set of arbitrage-free prices is given by ….H / D ¹E  Œ H  j P  2 P ; E  Œ H  < 1 º: (6.15)

338

Chapter 6 American contingent claims

Remark 6.28. As observed in Remark 5.33, a discounted payoff HQ t which is received at time t < T can be regarded as a discounted European claim HQ E maturing at T . HQ E is obtained from HQ t by investing at time t the payoff S t0 HQ t into the numéraire, i.e., by buying HQ t shares of the 0th asset, and by considering the discounted terminal value of this investment: 1 HQ E D 0 . ST0 HQ t / D HQ t : ST In the case of our discounted American claim H which is payed off at the random time  , we can either apply this argument to each payoff HQ t WD H I¹Dt º D H t I¹Dt º ; or directly use a stopping time version of this argument. We conclude that H can be regarded as a discounted European claim, whose arbitrage-free prices are given by (6.15). } Now suppose that H is offered at time t D 0 for a price   0. From the buyer’s point of view there should be at least one exercise strategy  such that the proposed price  is not too high in the sense that    0 for some  0 2 ….H /. From the seller’s point of view the situation looks different: There should be no exercise strategy  0 such that the proposed price  is too low in the sense that  <  0 for all  0 2 ….H 0 /. By adding the assumption that the buyer only uses stopping times in exercising the option, we obtain the following formal definition. Definition 6.29. A real number  is called an arbitrage-free price of a discounted American claim H if the following two conditions are satisfied: 



The price  is not too high in the sense that there exists some  2 T and  0 2 ….H / such that    0 . The price  is not too low in the sense that there exists no  0 2 T such that  <  0 for all  0 2 ….H 0 /.

The set of all arbitrage-free prices of H is denoted ….H /, and we define inf .H / WD inf ….H /

and

sup .H / WD sup ….H /:

Recall from Remark 6.6 that every discounted European claim H E can be regarded as a discounted American claim H A whose payoff is zero if H A is exercised before T , and whose payoff at T equals H E . Clearly, the two sets ….H E / and ….H A / coincide, and so the two Definitions 5.28 and 6.29 are consistent with each other. Remark 6.30. It follows from the definition that any arbitrage-free price  for H must be an arbitrage-free price for some H . Hence, (6.15) implies that  D E  Œ H 

339

Section 6.3 Arbitrage-free prices

for some P  2 P . Similarly, we obtain from the second condition in Definition 6.29 that   infP  2P E  ΠH  for all  2 T . It follows that sup inf E  ΠH     sup sup E  ΠH  2T P  2P

for all  2 ….H /.

(6.16)

2T P  2P

In particular, sup E  ΠH  2T

is the unique arbitrage-free price of H if P  is the unique equivalent martingale measure in a complete market model, and so Definition 6.29 is consistent with the results of the Section 6.1 and 6.2. } Exercise 6.3.1. Show that in every arbitrage-free market model and for any discounted American claim H , inf sup E  ΠH  < 1;

(6.17)

P  2P 2T

and that the set ….H / of arbitrage-free prices is nonempty.

}

Our main goal in this section is to characterize the set ….H /, and to identify the upper and lower bounds in (6.16) with the quantities sup .H / and inf .H /. We will work under the simplifying assumption that H t 2 L1 .P  /

for all t and each P  2 P .

(6.18)



For each P  2 P we denote by U P the corresponding Snell envelope of H , i.e., 

U tP D ess sup E  ΠH j F t : 2T t

With this notation, the right-hand bound in (6.16) can be written as 

sup sup E  ΠH  D sup sup E  ΠH  D sup U0P :

2T P  2P

P  2P 2T

P  2P

In fact, a similar relation also holds for the lower bound in (6.16) 

sup inf E  ΠH  D inf sup E  ΠH  D inf U0P :

2T

P  2P

P  2P

2T

(6.19)

P  2P

The proof that the above interchange of infimum and supremum is indeed justified under assumption (6.18) is postponed to the next section; see Theorem 6.45. Theorem 6.31. Under condition (6.18), the set of arbitrage-free prices for H is a real interval with endpoints inf .H / D inf sup E  ΠH  D sup inf E  ΠH  P  2P 2T

2T P  2P

340

Chapter 6 American contingent claims

and sup .H / D sup sup E  ΠH  D sup sup E  ΠH : P  2P 2T

2T P  2P

Moreover, ….H / either consists of one single point or does not contain its upper endpoint sup .H /. Proof. Let   be a stopping time which is optimal with respect to a given P  2 P .   Then U0P D E  Œ H   D sup 0 2T E  Œ H 0 , and consequently U0P 2 ….H /. Together with the a priori bounds (6.16), we obtain the inclusions 

¹U0P j P  2 P º  ….H /  Œa; b;

(6.20)

where a WD sup inf E  ΠH  and 2T

b WD sup sup E  ΠH :

P  2P

2T P  2P

Moreover, the minimax identity (6.19) shows that a D inf sup E  ΠH  D inf U0P



and

P  2P

P  2P 2T



b D sup U0P : P  2P

Together with (6.20), this yields the identification of inf .H / and sup .H / as a and b.  Now we claim that ¹U0P j P  2 P º is an interval, which, in view of the preceding step, will prove that ….H / is also an interval. Take P0 ; P1 2 P and define P˛ 2 P by P˛ WD ˛P1 C .1  ˛/P0 for 0  ˛  1. By Theorem 6.18, f .˛/ WD U0P˛ is the supremum of the affine functions ˛ 7! E˛ Œ H  D ˛E1 Œ H  C .1  ˛/E0 Œ H ;

 2T:

Thus, f is convex and lower semicontinuous on Œ0; 1, hence continuous; see part (a) of Proposition A.4. Since P is convex, this proves our claim. It remains to exclude the possibility that b belongs to ….H / in case a < b. Suppose by way of contradiction that b 2 ….H /. Then there exist O 2 T and PO 2 P such that O HO  D b D sup sup E  Œ H : EŒ 2T P  2P

In particular, PO attains the supremum of E  ΠHO  for P  2 P . Theorem 5.32 implies that the discounted European claim HO is attainable and that E  ΠHO  is in fact independent of P  2 P . Hence, O HO  D inf E  ΠHO   sup inf E  ΠH ; b D EΠP  2P

 2T P 2P

and we end up with the contradiction b  a. Thus, b cannot belong to ….H /.

341

Section 6.3 Arbitrage-free prices

Comparing the previous result with Theorem 5.32, one might wonder whether ….H / contains its lower bound if inf .H / < sup .H /. At a first glance, it may come as a surprise that both cases inf .H / 2 ….H /

and

inf .H / … ….H /

can occur, as is illustrated by the following simple example. Example 6.32. Consider a complete market model with T D 2, defined on some probability space .0 ; G0 ; P0 /. This model will be enlarged by adding two external states ! C and !  , i.e., we define  WD 0  ¹! C ; !  º and 1 P0 Œ ¹!0 º ; !0 2 0 : 2 We assume that this additional information is revealed at time 2. The enlarged financial market model will then be incomplete, and the corresponding set P of equivalent martingale measures satisfies P Œ ¹.!0 ; ! ˙ /º  WD

P  ¹Pp j 0 < p < 1º; where Pp is determined by Pp Œ 0 ¹! C º  D p. Consider the discounted American claim H defined as ´ 2 if ! D .!0 ; ! C /, H0  0; H1  1; and H2 .!/ WD 0 if ! D .!0 ; !  /. Clearly, ….H / Œ1; 2/. On the other hand, 2  2 is an optimal stopping time for Pp if p > 12 , while 1  1 is optimal for p  12 . Hence, ….H / D Œ1; 2/; and the lower bound inf .H / D 1 is an arbitrage-free price for H . Now consider the discounted American claim HQ defined by HQ t D H t for t D 0; 2 and by HQ 1  0. In this case, we have ….HQ / D .0; 2/: } Theorem 6.31 suggests that an American claim H which admits a unique arbitragefree price should be attainable in an appropriate sense. Corollary 6.22, our hedging result in the case of a complete market, suggests the following definition of attainability. Definition 6.33. A discounted American claim H is called attainable if there exists a stopping time  2 T and a self-financing trading strategy  whose value process V satisfies P -a.s. V t  H t for all t , and V D H . The trading strategy  is called a hedging strategy for H .

342

Chapter 6 American contingent claims

If H is attainable, then a hedging strategy protects the seller not only against those claims H which arise from stopping times  . The seller is on the safe side even if the buyer would have full knowledge of future prices and would exercise H at an arbitrary FT -measurable random time . For instance, the buyer even could choose such that H D max H t : 0tT

In fact, we will see in Remark 7.12 that H is attainable in the sense of Definition 6.33 if and only if V t  H t for all t and V D H for some FT -measurable random time . If the market model is complete, then every American claim H is attainable. Moreover, Theorem 6.11 and Corollary 6.22 imply that the minimal initial investment needed for the purchase of a hedging strategy for H is equal to the unique arbitragefree price of H . In a general market model, every attainable discounted American claim H satisfies our integrability condition (6.18) and has a unique arbitrage-free price which is equal to the initial investment of a hedging strategy for H . This follows from Theorem 5.25. In fact, the converse implication is also true. Theorem 6.34. For a discounted American claim H satisfying (6.18), the following conditions are equivalent: (a) H is attainable. (b) H admits a unique arbitrage-free price .H /, i.e., ….H / D ¹.H /º. (c) sup .H / 2 ….H /. Moreover, if H is attainable, then .H / is equal to the initial investment of any hedging strategy for H . The equivalence of (b) and (c) is an immediate consequence of Theorem 6.31. The remainder of the proof of Theorem 6.34 is postponed to Remark 7.10 because it requires the technique of superhedging, which will be introduced in Section 7.

6.4

Stability under pasting

In this section we define the pasting of two equivalent probability measures at a given stopping time. This operation will play an important role in the analysis of lower and upper Snell envelopes as developed in Section 6.5. In particular, we will prepare for the proof of the minimax identity (6.19), which was used in the characterization of arbitrage-free prices of an American contingent claim. Let us start with a few preparations. Definition 6.35. Let  be a stopping time. The -algebra of events which are observable up to time  is defined as F WD ¹A 2 F j A \ ¹  t º 2 F t for all t º:

343

Section 6.4 Stability under pasting

Exercise 6.4.1. Prove that F is indeed a -algebra. Show next that F D ¹A 2 F j A \ ¹ D t º 2 F t for all tº and conclude that F coincides with F t if   t . Finally show that F  F when is a stopping time with .!/   .!/ for all ! 2 . } The following result is an addendum to Doob’s stopping theorem; see Theorem 6.15: Proposition 6.36. For an adapted process M in L1 .Q/ the following conditions are equivalent: (a) M is a Q-martingale. (b) EQ Œ M j F  D M^ for all  2 T and all stopping times . Proof. (a) ) (b): Take a set A 2 F and let us write EQ Œ M I A  D EQ Œ M I A \ ¹  º  C EQ Œ M I A \ ¹ > º : Condition (b) will follow if we may replace M by M in the rightmost expectation. To this end, note that A \ ¹ D t º \ ¹ > º D A \ ¹ D t º \ ¹ > t º 2 F t : Thus, since the stopped process M  is a martingale by Theorem 6.15, EQ Œ M I A \ ¹ > º  D

T X

EQ Œ MT I A \ ¹ D t º \ ¹ > º 

tD0

D

T X

EQ Œ M t I A \ ¹ D t º \ ¹ > º 

tD0

D EQ Œ M I A \ ¹ > º : (b) ) (a): This follows by taking   t and  s  t . Exercise 6.4.2. Let Z be the density process of a probability measure QQ that is absolutely continuous with respect to Q; see Exercise 5.2.3. Show that for a stopping time , we have QQ Q on F with density given by d QQ ˇˇ ˇ D EQ Œ ZT j F  D ZT ^ : dQ F

}

344

Chapter 6 American contingent claims

Exercise 6.4.3. Show that for a stopping time  , a random variable Y 2 L1 .; F ; Q/, and t 2 ¹0; : : : ; T º, EQ Œ H j F  D EQ Œ H j F t 

Q-a.s. on ¹ D t º.

(6.21) }

We next state the following extension of Theorem 6.18. It provides the solution to the optimal stopping problem posed at any stopping time   T . Proposition 6.37. Let H be an adapted process in L1 .; F ; Q/, and define for  2T T WD ¹ 2 T j   º: Then the Snell envelope U Q of H satisfies Q-a.s. UQ D ess sup EQ Œ H j F ; 2T

and the essential supremum is attained for ./

Q

min WD min¹t   j H t D U t º: Exercise 6.4.4. Prove Proposition 6.37 by using the identity (6.21).

}

Definition 6.38. Let Q1 and Q2 be two equivalent probability measures and take 2 T . The probability measure Q A  WD EQ1 Œ Q2 Œ A j F  ; QŒ

A 2 FT ;

is called the pasting of Q1 and Q2 in . The monotone convergence theorem for conditional expectations guarantees that QQ is indeed a probability measure and that EQQ ΠY  D EQ1 ΠEQ2 ΠY j F   for all FT -measurable Y  0. Note that QQ coincides with Q1 on F , i.e., EQQ ΠY  D EQ1 ΠY 

for all F -measurable Y  0.

Lemma 6.39. For Q1  Q2 , their pasting in 2 T is equivalent to Q1 and satisfies ZT d QQ D ; dQ1 Z where Z is the density process of Q2 with respect to Q1 .

Section 6.4 Stability under pasting

345

Proof. For Y  0, EQQ Œ Y  D EQ1 Œ EQ2 Œ Y j F   i h 1 D EQ1 EQ1 Œ Y ZT j F  Z i hZ T D EQ1 Y ; Z where we have used the martingale property of Z and the fact that Z > 0 Q1 -almost surely. The equivalence of QQ and Q1 follows from ZT > 0 Q1 -almost surely. Lemma 6.40. For Q1  Q2 , let QQ be their pasting in 2 T . Then, for all stopping times  and FT -measurable Y  0, EQQ Œ Y j F  D EQ1 Œ EQ2 Œ Y j F_  j F : Proof. If '  0 is F -measurable, then 'I¹ º is F \ F -measurable. Hence, EQQ Œ Y'I    D EQ1 Œ EQ2 Œ Y j F 'I    D EQ1 Œ EQ1 Œ EQ2 Œ Y j F  j F 'I    D EQQ Œ EQ1 Œ EQ2 Œ Y j F  j F 'I   ; where we have used the fact that QQ coincides with Q1 on F . On the other hand, EQQ Œ Y'I  >  D EQ1 Œ EQ2 Œ EQ2 Œ Y j F ' j F I  >  D EQQ Œ EQ2 Œ Y j F 'I  > : It follows that EQQ Œ Y j F  D EQ1 Œ EQ2 Œ Y j F  j F  I¹ º C EQ2 Œ Y j F  I¹> º ; and this coincides with the right-hand side of the asserted identity. Definition 6.41. A set Q of equivalent probability measures on .; F / is called stable if, for any Q1 ; Q2 2 Q and 2 T , also their pasting in is contained in Q. The condition of stability in the preceding definition is sometimes also called fork convexity, m-stability, or stability under pasting. For the purposes of this book, the most important example of a stable set is the class P of all equivalent martingale measures, but in Section 6.5 we will also discuss the connection between stable sets and dynamic risk measures. Proposition 6.42. P is stable.

346

Chapter 6 American contingent claims

Proof. Take P1 ; P2 2 P and denote by PQ their pasting a given 2 T . Doob’s stopping theorem in the form of Proposition 6.36 and Lemma 6.40 applied with Y WD X ti  0 and   s yield that for s  t Q X t j Fs  D E1 Œ E2 Œ X t j F_s  j Fs  D E1 Œ X_s j Fs  D Xs : EŒ Q X ti  D X i < 1, It follows in particular that each component X ti is in L1 .PQ / since EŒ 0 concluding the proof of PQ 2 P . We conclude this section by an alternative characterization of stable sets. It will be used in Section 11.2. Suppose that 2 T takes at most one value t 2 ¹0; : : : ; T º that is different from T . Then there exists a set B 2 F t such that D t  IB C T  IB c . It follows that the pasting QQ of two equivalent probability measures Q1 and Q2 in is given by Q A  WD EQ1 Œ Q2 Œ A j F t   IB C I QŒ ; A 2 FT : (6.22) A\B c This observation can be used to give the following characterization of stable sets. Proposition 6.43. A set Q of equivalent probability measures is stable if and only if for any t 2 ¹0; : : : ; T º and B 2 F t the probability measure QQ defined in (6.22) belongs again to Q. Proof. We have already seen that QQ 2 Q when Q is stable. For the proof of the converse implication, let 2 T be a stopping time and take Q1 ; Q2 2 Q. We define recursively QQ T WD Q1 and QQ t1 Œ A  WD EQQ t Œ Q2 Œ A j F t1   I¹ Dt 1º C IA\¹ ¤t 1º  for t D T; : : : ; 1. Then QQ 0 2 Q by assumption. We claim that QQ 0 coincides with the pasting of Q1 and Q2 in , and this will prove the assertion. To verify our claim, note that the densities of QQ t with respect to Q1 satisfy the recursion  d QQ t  ZT d QQ t1 D I¹ Dt 1º C I¹ ¤t 1º ; dQ1 dQ1 Z t1 where .Z t / is the density process of Q2 with respect to Q1 . But this implies that ZT d QQ t1 D I C I¹ U 2 º, take QQ 2 Q as in Lemma 6.47. Then

Q

UQ D UQ1  IB c C UQ2  IB D UQ1 ^ UQ2 :

(6.26)

Moreover, if Q1 D Q on F then also QQ D Q on F . Hence, the set O ˆ WD ¹UQ j QO 2 Q and QO D Q on F º #

is such that U D ess inf ˆ. Moreover, (6.26) implies that ˆ is directed downwards, and the second part of Theorem A.33 states the existence of the desired sequence .Qk /  Q. The proof for the essential supremum is analogous. Q

Proof of Theorem 6:46. To prove (6.24), observe first that U t  EQ Œ H t j F t  for each Q 2 Q, so that  holds in (6.24). For the proof of the converse inequality, note that Q  t  min¹u  t j UuQ D Hu º DW  t for Q 2 Q. Q

It was shown in Theorem 6.18 that  t is the minimal optimal stopping time after time t and with respect to Q. It was also shown in the proof of Theorem 6.18 that the Q stopped process .U Q / t is a Q-martingale from time t on. In particular, Q

U t D EQ ΠUQt j F t 

for all Q 2 Q.

(6.27)

Let us now fix some Q 2 Q. Lemma 6.48 yields Qk 2 Q with Qk D Q on F t Q # such that U t k decreases to U t . We obtain  EQ ΠH t j F t  D EQ ΠU#t j F t  D EQ lim UQt k j F t k"1

D lim EQ ΠUQt k j F t  D lim EQk ΠUQt k j F t  k"1

D lim

k"1

k"1

Q Ut k Q



# Ut : Q

Q

Q

Here we have used that H t  U t k  U t 1 and EQ ΠjU t 1 j  D EQ1 ΠjU t 1 j  < 1 together with dominated convergence in the third step, the fact that Qk D Q on F t  F t in the fourth, and (6.27) in the fifth identity.

350

Chapter 6 American contingent claims

Remark 6.49. Suppose the buyer of an American option uses a utility functional of the form inf EQ Πu.Z/ ; Q2Q

where Q is a set of probability measures and u is a measurable function. This may be viewed as a robust Savage representation of a preference relation on discounted asset payoffs; see Section 2.5. Thus, the aim of the buyer is to maximize the utility inf EQ Πu.H / 

Q2Q

of the discounted payoff H among all stopping times  2 T . This generalized utility maximization problem can be solved with the results developed in this section, provided that the set Q is a stable set of equivalent probability measures. Indeed, assume HQ t WD u.H t / 2 L1 .Q/ for all t and each Q 2 Q, and let U Q be the Snell envelope of HQ t with respect to Q 2 Q. Theorem 6.46 states that the generalized optimal stopping problem is solved by the stopping time ® ¯ Q   WD min t  0 j ess inf U t D HQ t ; Q2Q

i.e., #

inf sup EQ Πu.H /  D U0 D inf EQ Πu.H  / :

Q2Q 2T

}

Q2Q

Let us now turn to the analysis of the upper Snell envelope "

Q

U t WD ess sup U t D ess sup ess sup EQ ΠH j F t ; Q2Q

2T t

t D 0; : : : ; T:

Q2Q

In order to simplify the presentation, we will assume from now on that sup EQ ΠjH t j  < 1

for all t .

Q2Q

This condition implies that "

Q

U0 D sup U0  sup sup EQ ΠjH j  < 1: Q2Q

2T Q2Q

Our main result on upper Snell envelopes states that, for stable sets Q, the upper Snell envelope U " satisfies a recursive scheme that is similar to the one for ordinary Snell envelopes. In contrast to (6.4), however, it involves the nonadditive conditional expectation operators ess supQ EQ Π j F t .

351

Section 6.5 Lower and upper Snell envelopes

Theorem 6.50. U " satisfies the following recursive scheme: "

"

"

UT D HT and U t D H t _ ess sup EQ ΠU tC1 j F t ;

t D T  1 : : : ; 0: (6.28)

Q2Q

Proof. The definition of the Snell envelope U Q implies that "

Q

Q

U t D ess sup U t D H t _ ess sup EQ ΠU tC1 j F t : Q2Q

(6.29)

Q2Q

Next, we fix Q 2 Q and denote by Q tC1 .Q/ the set of all QO 2 Q which coincide with Q on F tC1 . According to Lemma 6.48, there are Qk 2 Q tC1 .Q/ such that Qk " Q1 Q1 U tC1 % U tC1 . The fact that EQ ΠjU tC1 j  D EQ1 ΠjU tC1 j  < 1 combined with monotone convergence for conditional expectations shows that "

Q

ess sup EQ ΠU tC1 j F t   ess sup EQ ΠU tC1 j F t  Q2Q

Q2Q

D ess sup

ess sup

O Q2Q Q2Q t C1 .Q/

O Q

EQ ΠU tC1 j F t  Qk

 ess sup lim inf EQ ΠU tC1 j F t  k"1

Q2Q

(6.30)

"

D ess sup EQ ΠU tC1 j F t : Q2Q

In particular, all inequalities are in fact identities. Together with (6.29) we obtain the recursive scheme for U " . The following result shows that the nonadditive conditional expectation operators ess supQ EQ Π j F t  associated with a stable set Q enjoy a consistency property that is similar to the martingale property for ordinary conditional expectations. Theorem 6.51. Let Q be a set of equivalent probability measures and "

V t WD ess sup EQ ΠH j F t ;

t D 0; : : : ; T;

Q2Q

"

for some FT -measurable H  0 such that V0 < 1. If Q is stable then V" D ess sup EQ ΠV" j F  Q2Q

for ;  2 T with   .

352

Chapter 6 American contingent claims

Remark 6.52. Note that, for H as in the theorem and  2 T , V" D

D

T X

ess sup EQ Œ H j F t I¹Dt º

tD0 Q2Q T X

ess sup EQ Œ H j F I¹Dt º

tD0 Q2Q

D ess sup EQ ΠH j F ; Q2Q

}

where we have used (6.21) in the second identity. Proof of Theorem 6:51. By Remark 6.52, V" D ess sup EQ ΠH j F  D ess sup EQ ΠEQ ΠH j F  j F : Q2Q

Q2Q

"

The proof that the right-hand side is equal to ess supQ2Q EQ ΠV j F  is done by first noting that V " is equal to the upper Snell envelope of the process H t given by HT D H and H t D 0 for t < T . Then the same argument as in (6.30) applies. All one has to do is to replace t C 1 by  . Remark 6.53. Let us conclude this section by pointing out the connection between stability under pasting and the time-consistency of dynamic coherent risk measures. Let .Y / WD sup EQ ΠY ; Y 2 L1 .P /; Q2Q

be a coherent risk measure on L1 .P / defined in terms of a set Q of probability measures equivalent to P . In the context of a dynamic financial market model, it is natural to update the initial risk assessment at later times t > 0. If one continues to use Q as a basis to compute the risk but takes into account the available information, one is led to consider the conditional risk measures  t .Y / D ess sup EQ ΠY j F t ;

t D 0; : : : ; T:

(6.31)

Q2Q

The sequence 0 : : : ; T can be regarded as a dynamic coherent risk measure. Such a dynamic risk measure is called time-consistent or dynamically consistent if s . t .Y // D s .Y /

for 0  s  t  T .

(6.32)

When the set Q in (6.31) is a stable set of equivalent probability measures, then Theorem 6.51 implies immediately the time consistency (6.32). The following converse of this statement, and hence a converse of Theorem 6.51, will be given in Theorem 11.22:

Section 6.5 Lower and upper Snell envelopes

353

if . t / is a dynamically consistent sequence of conditional coherent risk measures satisfying certain regularity assumptions, then there exists a stable set Q of equivalent probability measures such that (6.32) holds. An extension of dynamic consistency to dynamic convex risk measures will be given in Section 11.2. Note that Theorem 6.51 shows that in (6.32) the deterministic times s and t can even be replaced by stopping times when Q is stable. }

Chapter 7

Superhedging

The idea of superhedging is to find a self-financing trading strategy with minimal initial investment which covers any possible future obligation resulting from the sale of a contingent claim. If the contingent claim is not attainable, the proof of the existence of such a “superhedging strategy” requires new techniques, and in particular a new uniform version of the Doob decomposition. We will develop this theory for general American contingent claims. In doing so, we will also obtain new results for European contingent claims. In the first three sections of this chapter, we assume that our market model is arbitrage-free or, equivalently, that the set of equivalent martingale measures satisfies P ¤ ;: In the final Section 7.4, we discuss liquid options in a setting where no probabilistic model is fixed a priori. Such options may be used for the construction of specific martingale measures, and also for the purpose of hedging illiquid exotic derivatives.

7.1

P -supermartingales

In this section, H denotes a discounted American claim with sup E  ΠH t  < 1

for all t :

(7.1)

P  2P

Our aim in this chapter is to find the minimal amount of capital U t that will be needed at time t in order to purchase a self-financing trading strategy whose value process satisfies Vu  Hu for all u  t . In analogy to our derivation of the recursive scheme (6.4), we will now heuristically derive a formula for U t . At time T , the minimal amount needed is clearly given by UT D HT : At time T  1, a first requirement is to have UT 1  HT 1 . Moreover, the amount UT 1 must suffice to purchase an FT 1 -measurable portfolio  T such that  T  XT  HT almost surely. An informal application of Theorem 1.32, conditional on FT 1 , shows that UT 1  ess sup E  ΠHT j FT 1 : P  2P

355

Section 7.1 P -supermartingales

Hence, the minimal amount UT 1 is equal to the maximum of HT 1 and this essential supremum. An iteration of this argument yields the recursive scheme UT D HT

and

U t D H t _ ess sup E  ΠU tC1 j F t  P  2P

for t D T  1; : : : ; 0. By combining Proposition 6.42 and Theorem 6.50, we can identify U as the upper Snell envelope 

"

U t D ess sup U tP D ess sup ess sup E  ΠH j F t  P  2P

2T t

P  2P 

of H with respect to the stable set P , where U P denotes the Snell envelope of H with respect to P  . In the first three sections of this chapter, we will in particular give a rigorous version of the heuristic argument above. Note first that condition (7.1) implies that 

sup .H / D sup U0P D sup sup E  ΠH  < 1; P  2P

P  2P 2T

where we have used the identification of the upper bound sup .H / of the arbitragefree prices of H given in Theorem 6.31. It will turn out that the following definition applies to the upper Snell envelope if we choose Q D P . Definition 7.1. Suppose that Q is a non-empty set of probability measures on .; FT /. An adapted process is called a Q-supermartingale if it is a supermartingale with respect to each Q 2 Q. Analogously, we define the notions of a Q-submartingale and of a Q-martingale. In Theorem 5.25, we have already encountered an example of a P -martingale, namely the value process of the replicating strategy of an attainable discounted European claim. Theorem 7.2. The upper Snell envelope U " of H is the smallest P -supermartingale that dominates H . Proof. For each P  2 P the recursive scheme (6.28) implies that P  -a.s. "

"

"

U t  H t _ E  ΠU tC1 j F t   E  ΠU tC1 j F t : "

Since U0 is a finite constant due to our integrability assumption (7.1), induction on " t shows that U t is integrable with respect to each P  2 P and hence is a P -supermartingale dominating H .

356

Chapter 7 Superhedging

" If UQ is another P -supermartingale which dominates H , then UQ T  HT D UT . " Moreover, if UQ tC1  U tC1 for some t , then " UQ t  H t _ E  ΠUQ tC1 j F t   H t _ E  ΠU tC1 j F t :

Thus,

" " UQ t  H t _ ess sup E  ΠU tC1 j F t  D U t ; P  2P

and backward induction shows that UQ dominates U " . For European claims, Theorem 7.2 takes the following form. Corollary 7.3. Let H E be a discounted European claim such that sup E  ΠH E  < 1: P  2P

Then

"

V t WD ess sup E  ΠH E j F t ;

t D 0; : : : ; T;

P  2P

is the smallest P -supermartingale whose terminal value dominates H E . Remark 7.4. Note that the proof of Theorem 7.2 did not use any special properties of the set P . Thus, if Q is an arbitrary set of equivalent probability measures, the process U defined by the recursion UT D HT

and

U t D H t _ ess sup EQ ΠU tC1 j F t  Q2Q

is the smallest Q-supermartingale dominating the adapted process H .

7.2

}

Uniform Doob decomposition

The aim of this section is to give a complete characterization of all non-negative P supermartingales. It will turn out that an integrable and non-negative process U is a P -supermartingale if and only if it can be written as the difference of a P -martingale N and an increasing adapted process B satisfying B0 D 0. This decomposition may be viewed as a uniform version of the Doob decomposition since it involves simultaneously the whole class P . It will turn out that the P -martingale N has a special structure: It can be written as a “stochastic integral” of the underlying process X , which defines the class P . On the other hand, the increasing process B is only adapted, not predictable as in the Doob decomposition with respect to a single measure.

357

Section 7.2 Uniform Doob decomposition

Theorem 7.5. For an adapted, non-negative process U , the following two statements are equivalent: (a) U is a P -supermartingale. (b) There exists an adapted increasing process B with B0 D 0 and a d -dimensional predictable process  such that U t D U0 C

t X

k  .Xk  Xk1 /  B t

P -a.s. for all t .

kD1

Proof. First, we prove the easier implication (b) ) (a). Fix P  2 P and note that VT WD U0 C

T X

k  .Xk  Xk1 /  UT  0:

kD1

Hence, V is a P -martingale by Theorem 5.14. It follows that U t 2 L1 .P  / for all t . Moreover, for P  2 P E  Œ U tC1 j F t  D E  Œ V tC1  B tC1 j F t   V t  B t D U t ; and so U is a P -supermartingale. The proof of the implication (a) ) (b) is similar to the proof of Theorem 5.32. We must show that for any given t 2 ¹1; : : : ; T º, there exist  t 2 L0 .; F t1 ; P I Rd / and R t 2 L0C .; F t ; P / such that U t  U t1 D  t  .X t  X t1 /  R t : This condition can be written as U t  U t1 2 K t  L0C .; F t ; P /; where K t is as in (5.12). There is no loss of generality in assuming that P is itself a martingale measure. In this case, U t  U t1 is contained in L1 .; F t ; P / by the definition of a P -supermartingale. Assume that U t  U t1 … C WD .K t  L0C .; F t ; P // \ L1 .P /: Absence of arbitrage and Lemma 1.68 imply that C is closed in L1 .; F t ; P /. Hence, Theorem A.57 implies the existence of some Z 2 L1 .; F t ; P / such that ˛ WD sup EŒ Z W  < EŒ Z .U t  U t1 /  DW ı < 1:

(7.2)

W 2C

In fact, we have ˛ D 0 since C is a cone containing the constant function 0. Lemma 1.58 implies that such a random variable Z must be non-negative and must satisfy EŒ .X t  X t1 / Z j F t1  D 0:

(7.3)

358

Chapter 7 Superhedging

In fact, we can always modify Z such that it is bounded from below by some " > 0 and still satisfies (7.2). To see this, note first that every W 2 C is dominated by a term of the form  t  .X t  X t1 /. Hence, our assumption P 2 P , the integrability of W , and an application of Fatou’s lemma yield that EŒ W   EŒ  t  .X t  X t1 /   lim inf EŒ I¹j t jcº  t  .X t  X t1 /   0: c"1

Thus, if we let Z " WD " C Z, then Z " also satisfies EŒ Z " W   0 for all W 2 C . If we chose " small enough, then EŒ Z " .U t  U t1 /  is still larger than 0; i.e., Z " also satisfies (7.2) and in turn (7.3). Therefore, we may assume from now on that our Z with (7.2) is bounded from below by some constant " > 0. Let Z t1 WD EŒ Z j F t1 ; and define a new measure PQ  P by Z d PQ WD : dP Z t1 We claim that PQ 2 P . To prove this, note first that Xk 2 L1 .PQ / for all k, because the density d PQ =dP is bounded. Next, let   Z ˇˇ 'k WD E F ˇ k ; k D 0; : : : ; T: Z t1 If k ¤ t , then 'k1 D 'k ; this is clear for k > t , and for k < t it follows from   EŒ Z j F t1  ˇˇ 'k D E ˇ Fk D 1: Z t1 Thus, for k ¤ t Q Xk  Xk1 j Fk1  D EŒ

1 'k1

EΠ.Xk  Xk1 / 'k j Fk1 

D EΠXk  Xk1 j Fk1  D 0: If k D t , then (7.3) yields that Q Xk  Xk1 j Fk1  D EΠHence PQ 2 P .

1 Z t1

EΠ.X t  X t1 / Z j F t1  D 0:

Section 7.3 Superhedging of American and European claims

359

Q U t  U t1 j F t1   0, and we get Since PQ 2 P , we have EŒ Q EŒ Q U t  U t1 j F t1  Z t1  0  EŒ Q .U t  U t1 / Z t1  D EŒ D EŒ .U t  U t1 / Z  D ı: This, however, contradicts the fact that ı > 0. Remark 7.6. The decomposition in part (b) of Theorem 7.5 is sometimes called the optional decomposition of the P -supermartingale U . The existence of such a decomposition was first proved by El Karoui and Quenez [105] and D. Kramkov [182] in a continuous-time framework where B is an “optional” process; this explains the terminology. }

7.3

Superhedging of American and European claims

Let H be a discounted American claim such that sup E  ΠH t  < 1

for all t ,

P  2P

which is equivalent to the condition that the upper bound of the arbitrage-free prices of H is finite sup .H / D sup sup E  ΠH  < 1: P  2P 2T

Our aim in this section is to construct self-financing trading strategies such that the seller of H stays on the safe side in the sense that the corresponding portfolio value is always above H . Definition 7.7. Any self-financing trading strategy  whose value process V satisfies Vt  Ht

P -a.s. for all t

is called a superhedging strategy for H . Sometimes, a superhedging strategy is also called a superreplication strategy. According to Definition 6.33, H is attainable if and only if there exist  2 T and a superhedging strategy whose value process satisfies V D H P -almost surely. Lemma 7.8. If H is not attainable, then the value process V of any superhedging strategy satisfies P ΠV t > H t for all t  > 0:

360

Chapter 7 Superhedging

Proof. We introduce the stopping time  WD inf¹t  0 j H t D V t º: Then P Œ  D 1  D P Œ V t > H t for all t . Suppose that P Œ  D 1  D 0. In this case, V D H P -a.s so that we arrive at the contradiction that H must be an attainable American claim. Let us now turn to the question whether superhedging strategies exist. In Section 6.1, we have already seen how one can use the Doob decomposition of the Snell  envelope U P of H together with the martingale representation of Theorem 5.38 in  order to obtain a superhedging strategy for the price U0P , where P  denotes the unique equivalent martingale measure in a complete market model. We have also  seen that U0P is the minimal amount for which such a superhedging strategy is avail able, and that U0P is the unique arbitrage-free price of H . The same is true of any attainable American claim in an incomplete market model. In the context of a non-attainable American claim H in an incomplete financial market model, the P  -Snell envelope will be replaced with the upper Snell envelope U " of H . The uniform Doob decomposition will take over the roles played by the usual Doob decomposition and the martingale representation theorem. Since U " is a P -supermartingale by Theorem 7.2, the uniform Doob decomposition states that U " takes the form " Ut

D

" U0

C

t X

s  .Xs  Xs1 /  B t

(7.4)

sD1

 Ht for some predictable process  and some increasing process B. Thus, the self-financing trading strategy  D . 0 ; / defined by  and the initial capital "

 1  X D D U0 D sup .H / is a superhedging strategy for H . Moreover, if VQ is the value process of any superhedging strategy, then Lemma 7.8 implies that VQ0 > E  ΠH  for all  2 T and each P  2 P . In particular, VQ0 is larger than any arbitrage-free price for H , and it follows that VQ0  sup .H /. Thus, we have proved: Corollary 7.9. There exists a superhedging strategy with initial investment sup .H /, and this is the minimal amount needed to implement a superhedging strategy. We will call sup .H / the cost of superhedging of H . Sometimes, a superhedging strategy is also called a superreplication strategy, and one says that sup .H / is the cost of superreplication or the upper hedging price of H . Recall, however, that sup .H /

Section 7.3 Superhedging of American and European claims

361

is typically not an arbitrage-free price for H . In particular, the seller cannot expect to receive the amount sup .H / for selling H . On the other hand, the process B in the decomposition (7.4) can be interpreted as a refunding scheme: Using the superhedging strategy , the seller may withdraw successively the amounts defined by the increments of B. With this capital flow, the " hedging portfolio at time t has the value U t  H t . Thus, the seller is on the safe side at no matter when the buyer decides to exercise the option. As we are going to show in Theorem 7.13 below, this procedure is optimal in the sense that, if started at any time t , it requires a minimal amount of capital. Remark 7.10. Suppose sup .H / belongs to the set ….H / of arbitrage-free prices for H . By Theorem 6.31, this holds if and only if sup .H / is the only element of ….H /. In this case, the definition of ….H / yields a stopping time  2 T and some P  2 P such that sup .H / D E  Œ H : Now let V be the value process of a superhedging strategy bought at V0 D sup .H /. It follows that E  Œ V  D sup .H /. Hence, V D H P -a.s., so that H is attainable in the sense of Definition 6.33. This observation completes the proof of Theorem 6.34. } Remark 7.11. If the American claim H is not attainable, then sup .H / is not an arbitrage-free price of H . Thus, one may expect the existence of arbitrage opportunities if H would be traded at the price sup .H /. Indeed, selling H for sup .H / and buying a superhedging strategy  creates such an arbitrage opportunity: The balance at t D 0 is zero, but Lemma 7.8 implies that the value process V of  cannot be reached by any exercise strategy , i.e., we always have V  H

and

P ΠV > H  > 0:

(7.5)

Note that (7.5) is not limited to exercise strategies which are stopping times but holds for arbitrary FT -measurable random times W  ! ¹0; : : : ; T º. In other words, sup .H / is too expensive even if the buyer of H would have full information about the future price evolution. } Remark 7.12. The argument of Remark 7.11 implies that an American claim H is attainable if and only if there exists an FT -measurable random time W  ! ¹0; : : : ; T º such that H D V , where V the value process of a superhedging strategy. In other words, the notion of attainability of American claims does not need the restriction to stopping times. } We already know that sup .H / is the smallest amount for which one can buy a superhedging strategy at time 0. The following “superhedging duality theorem” extends

362

Chapter 7 Superhedging "

this result to times t > 0. To this end, denote by U t .H / the set of all F t -measurable random variables UQ t  0 for which there exists a d -dimensional predictable process Q such that UQ t C

u X

Qk  .Xk  Xk1 /  Hu

for all u  t P -a.s.

(7.6)

kDtC1 "

"

Theorem 7.13. The upper Snell envelope U t of H is the minimal element of U t .H /. More precisely "

"

(a) U t 2 U t .H /, "

"

(b) U t D ess inf U t .H /. Proof. Assertion (a) follows immediately from the uniform Doob decomposition of " " the P -supermartingale U " . As to part (b), we clearly get U t  ess inf U t .H / " from (a). For the proof of the converse inequality, take UQ t 2 U t .H / and choose a " predictable process Q for which (7.6) holds. We must show that the set B WD ¹U t  UQ t º satisfies P Œ B  D 1. Let " " UO t WD U t ^ UQ t D U t  IB C UQ t  IB c : " " Then UO t  U t , and our claim will follow if we can show that U t  UO t . Let  denote the predictable process obtained from the uniform Doob decomposition of the P -supermartingale U " , and define ´ if s  t , Os WD s s  IB C Qs  IB c if s > t . " With this choice, UO t satisfies (7.6), i.e., UO t 2 U t .H /. Let " VOs WD U0 C

s X

Ok  .Xk  Xk1 /:

kD1 " Us

Then VOs  for all s  t . In particular VOt  UO t , and hence VOT  HT , which O implies that V is a P -martingale; see Theorem 5.25. Hence, Doob’s stopping theorem implies "

U t D ess sup ess sup E  ΠH j F t  P  2P

2T t

 h i X ˇ Ok  .Xk  Xk1 / ˇ F t  ess sup ess sup E  UO t C P  2P

2T

D UO t ; which concludes the proof.

kDtC1

363

Section 7.3 Superhedging of American and European claims

We now take the point of view of the buyer of the American claim H . The buyer allocates an initial investment  to purchase H , and then receives the amount H  0. The objective is to find an exercise strategy and a self-financing trading strategy  with initial investment , such that the portfolio value is covered by the payoff of the claim. In other words, find  2 T and a self-financing trading strategy with value process V such that V0 D  and V C H  0. As shown below, the maximal  for which this is possible is equal to #

inf .H / D sup inf E  ΠH  D inf sup E  ΠH  D U0 ; 2T

P  2P

P  2P

2T

where #

U t D ess inf U tP



P  2P

D ess inf ess sup E  ΠH j F t  P  2P

2T t

D ess sup ess inf E  ΠH j F t  2T t

P  2P

is the lower Snell envelope of H with respect to the stable set P . More generally, we will consider the buyer’s problem for arbitrary t  0. To this end, denote by # U t .H / the set of all F t -measurable random variables UQ t  0 for which there exists a d -dimensional predictable process Q and a stopping time 2 T t such that UQ t 

 X

Q k  .Xk  Xk1 /  H

P -a.s.

kDtC1 #

#

Theorem 7.14. U t is the maximal element of U t .H /. More precisely #

#

(a) U t 2 U t .H /, #

#

(b) U t D ess sup U t .H /. Proof. (a): Let  be a superhedging strategy for H with initial investment sup .H /, and denote by V the value process of . The main idea of the proof is to use that V t  H t  0 can be regarded as a new discounted American claim, to which we can apply Theorem 7.13. However, we must take care of the basic asymmetry of the hedging problem for American options: The seller of H must hedge against all possible exercise strategies, while the buyer must find only one suitable stopping time. # It will turn out that a suitable stopping time is given by  t WD inf¹u  t j Uu D Hu º. With this choice, let us define a modified discounted American claim HQ by HQ u D .Vu  Hu /  I¹uDt º ;

u D 0; : : : ; T:

364

Chapter 7 Superhedging

Clearly HQ   HQ  t for all 2 T t . It follows that ess sup ess sup E  ΠHQ  j F t  D ess sup E  ΠHQ  t j F t  P  2P

2T t

P  2P

D V t  ess inf E  ΠH t j F t  P  2P #

D Vt  Ut ; where we have used that V is a P -martingale in the second and Theorem 6.46 in # the third step. Thus, V t  U t is equal to the upper Snell envelope UQ " of HQ at time Q t . Let  be the d -dimensional predictable process obtained from the uniform Doob decomposition of UQ " . Then, due to part (a) of Theorem 7.13, #

Vt  Ut C

u X

Qk  .Xk  Xk1 /  HQ u D .Vu  Hu /  I¹uDt º

for all u  t .

kDtC1

Thus,  WD Q   is as desired. # (b): Part (a) implies the inequality  in (b). To prove its converse, take UQ t 2 U t , a d -dimensional predictable process , Q and 2 T t such that  X

UQ t 

Q k  .Xk  Xk1 /  H

P -a.s.

kDtC1

We will show below that E

 h X

ˇ i Q k  .Xk  Xk1 / ˇ F t D 0

for all P  2 P .

(7.7)

kDtC1

Given this fact, we obtain that UQ t  E  ΠH j F t   ess sup E  ΠH j F t  2T t # for all P  2 P . Taking the essential infimum over P  2 P thus yields UQ t  U t and in turn (b). To prove (7.7), let

GQ s WD I¹st C1º

s X

I¹k º Q k  .Xk  Xk1 /;

s D 0; : : : ; T:

kDtC1

Then GQ T  UQ t  H  H 2 L1 .P  / for all P  , and Theorem 5.14 implies that GQ is a P -martingale. Hence (7.7) follows.

365

Section 7.3 Superhedging of American and European claims

We conclude this section by stating explicitly the corresponding results for European claims. Recall from Remark 6.6 that every discounted European claim H E can be regarded as the discounted American claim. Therefore, the results we have obtained so far include the corresponding “European” counterparts as special cases. Corollary 7.15. For any discounted European claim H E such that sup E  Œ H E  < 1; P  2P

there exist two d -dimensional predictable processes  and  such that P -a.s. 

ess sup E ΠH

E

j Ft  C

P  2P

k  .Xk  Xk1 /  H E ;

(7.8)

k  .Xk  Xk1 /  H E :

(7.9)

kDtC1 

ess inf E ΠH P  2P

T X

E

T X

j Ft  

kDtC1

Remark 7.16. For t D 0, (7.8) takes the form sup E  ΠH E  C P  2P

T X

k  .Xk  Xk1 /  H E

P -a.s.

kD1

Thus, the self-financing trading strategy  arising from  and the initial investment  1  X 0 D supP  2P E  ΠH E  allows the seller to cover all possible obligations without any downside risk. Similarly, (7.9) yields an interpretation of the self-financing trading strategy  which arises from  and the initial investment 1  X 0 D  inf E  ΠH E : P  2P

The latter quantity corresponds to the largest loan the buyer can take out and still be sure that, by using the trading strategy , this debt will be covered by the payoff H E . } Remark 7.17. Let H be a discounted European claim such that sup E  ΠH E  < 1: P  2P

O H  D sup .H /. If  D . 0 ; / is a superhedging Suppose that PO 2 P is such that EŒ strategy for H , then O H C HO WD EŒ

T X kD1

k  .Xk  Xk1 /

366

Chapter 7 Superhedging

satisfies HO  H  0. Hence, HO is an attainable discounted claim, and it follows from Theorem 5.25 that O HO  D EŒ O H : EŒ This shows that HO and H are identical and that H is attainable. We have thus obtained another proof of Theorem 5.32. } As the last result in this section, we formulate the following “superhedging duality theorem”, which states that the bounds in (7.8) and (7.9) are optimal. Corollary 7.18. Suppose that H E is a discounted European claim with sup E  Œ H E  < 1: P  2P "

Denote by U t .H E / the set of all F t measurable random variables UQ t for which there exists a d -dimensional predictable process Q such that UQ t C

T X

Qk  .Xk  Xk1 /  H E

P -a.s.

kDtC1

Then

"

ess sup E  ΠH E j F t  D ess inf U t .H E /: P  2P

# By U t .H E / we denote the set of all F t measurable random variables UQ t for which there exists a d -dimensional predictable process Q such that

UQ t 

T X

Q k  .Xk  Xk1 /  H E

P -a.s.

kDtC1

Then

#

ess inf E  ΠH E j F t  D ess sup U t .H E /: P  2P

Remark 7.19. Define A as the set of financial positions Z 2 L1 .; FT ; P / which are acceptable in the sense that there exists a d -dimensional predictable process  such that ZC

T X

k  .Xk  Xk1 /  0

P -a.s.

kD1

As in Section 4.8, this set A induces a coherent risk measure  on L1 .; FT ; P / .Z/ D inf¹m 2 R j m C Z 2 Aº;

Z 2 L1 .; FT ; P /:

Section 7.3 Superhedging of American and European claims

367

Corollary 7.18 implies that  can be represented as .Z/ D sup E  ΠZ : P  2P

We therefore obtain a multiperiod version of Proposition 4.99.

}

Remark 7.20. Often, the superhedging strategy in a given incomplete model can be identified as the perfect hedge in an associated “extremal” model. As an example, consider a one-period model with d discounted risky assets given by bounded random variables X 1 ; : : : ; X d . Denote by the distribution of X D .X 1 ; : : : ; X d / and by . / the convex hull of the support of . The closure K WD . / of . / is convex and compact. We know from Section 1.5 that the model is arbitrage-free if and only if the price system  D . 1 ; : : : ;  d / is contained in the relative interior of . /, and the equivalent martingale measures can be identified with the measures   with barycenter . Consider a derivative H D h.X / given by a convex function h on K. The cost of superhedging is given by Z sup h d  D inf¹˛./ j ˛ affine on K, ˛  h -a.s. º; 

which is a special case of the duality result of Theorem 1.32. Since ¹˛  hº is convex and closed, the condition .˛  h/ D 1 implies ˛  h on K. Denote by M./ the class of all probability measures on K with barycenter . For any affine function ˛ with ˛  h on K, and for any Q 2 M./ we have Z Z h d Q  ˛ d Q D ˛./: Thus, O h./ D

Z sup

h d Q

(7.10)

2M./ Q

where we define for f 2 C.K/ fO WD inf¹˛ j ˛ affine on K, ˛  f -a.s. º: The supremum in (7.10) is attained since M./ is weakly compact. More precisely, it is attained by any measure O 2 M./ on K which is maximal with respect to the balayage order 0. We have seen in Example 1.38 that inf .H / and sup .H / coincide with the universal arbitrage bounds of Remark 1.37 .S01  K/C D inf .H /

and

sup .H / D S01 :

Thus, the superhedging strategy for the seller consists in the trivial hedge of buying the asset at time 0, while the corresponding strategy for the buyer is a short-sale of the asset in case the option is in the money, i.e., if S01 > K. }

7.4

Superhedging with liquid options

In practice, some derivatives such as put or call options are traded so frequently that their prices are quoted just like those of the primary assets. The prices of such liquid options can be regarded as an additional source of information on the expectations of the market as to the future evolution of asset prices. This information can be exploited in various ways. First, it serves to single out those martingale measures P  which are

369

Section 7.4 Superhedging with liquid options

compatible with the observed options prices, in the sense that the observed prices coincide with the expectations of the discounted payoff under P  . Second, liquid options may be used as instruments for hedging more exotic options. Our aim in this section is to illustrate these ideas in a simple setting. Assume that there is only one risky asset S 1 such that S01 is a positive constant, and that S 0 is a riskless bond with interest rate r D 0. Thus, the discounted price process of the risky asset is given by X t D S t1  0 for t D 0; : : : ; T . As the underlying space of scenarios, we use the product space  WD Œ0; 1/T : We define X t .!/ D x t for ! D .x1 ; : : : ; xT / 2 , and denote by F t the -algebra generated by X0 ; : : : ; X t ; note that F0 D ¹;; º. No probability measure P is given a priori. Let us now introduce a linear space X of FT -measurable functions as the smallest linear space such that the following conditions are satisfied: (a) 1 2 X. (b) .X t  Xs / IA 2 X for 0  s < t  T and A 2 Fs . (c) .X t  K/C 2 X for K  0 and t D 1; : : : ; T . The functions in the space X will be interpreted as (discounted) payoffs of liquid derivatives. The constant 1 in (a) corresponds to a unit investment into the riskless bond. The function X t  Xs in (b) corresponds to the payoff of a forward contract on the risky asset, issued at time s for the price Xs and expiring at time t . The decision to buy such a forward contract at time s may depend on the market situation at time s; this is taken into account by allowing for payoffs .X t  Xs / IA with A 2 Fs . Linearity of X together with conditions (a) and (b) implies that Xt 2 X

for all t .

Finally, condition (c) states that call options with any possible strike and any maturity up to time T can be used as liquid securities. Suppose that a linear pricing rule ˆ is given on X. The value ˆ.Y / will be interpreted as the market price of the liquid security Y 2 X. The price of a liquid call option with strike K and maturity t will be denoted by C t .K/ WD ˆ..X t  K/C /: Assumption 7.22. We assume that ˆ W X ! R is a linear functional which satisfies the following conditions: (a) ˆ.1/ D 1. (b) ˆ.Y /  0 if Y  0.

370

Chapter 7 Superhedging

(c) ˆ..X t  Xs / IA / D 0 for all 0  s < t  T and A 2 Fs . (d) C t .K/ D ˆ..X t  K/C / ! 0 as K " 1 for all t . The first two conditions must clearly be satisfied if the pricing rule ˆ shall not create arbitrage opportunities. Condition (c) states that Xs is the fair price for a forward contract issued at time s. This condition is quite natural in view of Theorem 5.29. In our present setting, it can also be justified by the following simple replication argument. At time s, take out a loan Xs .!/ and use it for buying the asset. At time t , the asset is worth X t .!/ and the loan must be paid back, which results in a balance X t .!/  Xs .!/. Since this investment strategy requires zero initial capital, the price of the corresponding payoff should also be zero. The continuity condition (d) is also quite natural. Our first goal is to show that any such pricing rule ˆ is compatible with the paradigm that arbitrage-free prices can be identified as expectations with respect to some martingale measure for X . More precisely, we are going to construct a martingale measure P  such that ˆ.Y / D E  Œ Y  for all Y 2 X. On the one hand, this will imply regularity properties of ˆ. On the other hand, this will yield an extension of our pricing rule ˆ to a larger space of payoffs including path-dependent exotic options. As a first step in this direction, we have the following result. Lemma 7.23. For each t , there exists a unique probability measure t on Œ0; 1/ such that for all K  0 Z C t .K/ D ˆ..X t  K/C / D .x  K/C t .dx/: In particular, t has the mean Z x t .dx/ D X0 : Proof. Since K 7! .X t  K/C is convex and decreasing, linearity and positivity of ˆ imply that the function  t .K/ WD ˆ..X t  K/C / is convex and decreasing as well. Hence, there exists a decreasing right-continuous function f W Œ0; 1/ ! Œ0; 1/ such that Z K C t .K/ D C t .0/  f .x/ dx Z D X0 

0 K

f .x/ dx; 0

i.e., f .K/ is equal to the right-hand derivative of C t .K/ at K. Our fourth condition on ˆ yields Z 1

f .x/ dx D X0 ; 0

371

Section 7.4 Superhedging with liquid options

so that f .x/ & 0 as x " 1. Hence, there exists a positive measure t on .0; 1/ such that f .x/ D t ..x; 1// for x > 0. Fubini’s theorem implies Z

Z

1

x t .dx/ D

f .y/ dy D X0

.0;1/

0

and Z

K

Z

C t .K/ D X0  Z D

0

.0;1/

I¹y 0 such that " WD "C.1"/  2 R0 , and the expectation EŒ "  must be strictly larger than EŒ  . This, however, contradicts the maximality of EŒ  . Now let  be any admissible strategy whose value process V satisfies V0  v. If V denotes the corresponding success ratio, then so



H

D H ^ VT  VT :

V

The P -martingale property of V yields that for all P  2 P , E ΠH  Therefore,

V

V

  E  ΠVT  D V0  v:

(8.13)

is contained in R0 and it follows that EŒ

V

  EŒ



:

(8.14)

Consider the superhedging strategy   of H  D H   and denote by V  its value process. Clearly,   is an admissible strategy. Moreover, V0 D sup .H  / D sup E  ΠH 



 D v:

P  2P

Thus, (8.14) yields that

V

satisfies EŒ

V

  EŒ



:

(8.15)

On the other hand, VT dominates H  , so H

V

D H ^ VT  H ^ H  D H 



:

Therefore, V  dominates  on the set ¹H > 0º. Moreover, any success ratio is equal to one on ¹H D 0º, and we obtain that V    P -almost surely. According to (8.15), this can only happen if the two randomized tests V  and  coincide P almost everywhere. This proves that   solves the hedging problem (8.9) and (8.10).

387

Section 8.2 Hedging with minimal shortfall risk

8.2

Hedging with minimal shortfall risk

Our starting point in this section is the same as in the previous one: At time T , an investor must pay the discounted random amount H  0. A complete elimination of the corresponding risk would involve the cost sup .H / D sup E  ΠH  P  2P

of superhedging H , but the investor is only willing to put up a smaller amount v 2 .0; sup .H //: This means that the investor is ready to take some risk: Any “partial” hedging strategy whose value process V satisfies the capital constraint V0  v will generate a nontrivial shortfall .H  VT /C : In the previous section, we constructed trading strategies which minimize the shortfall probability P Œ VT < H  among the class of trading strategies whose initial investment is bounded by v, and which are admissible in the sense of Definition 8.1, i.e., their terminal value VT is non-negative. In this section, we assess the shortfall in terms of a loss function, i.e., an increasing function ` W R ! R which is not identically constant. We assume furthermore that `.x/ D 0 for x  0

and

EΠ`.H /  < 1:

A particular role will be played by convex loss functions, which correspond to risk aversion in view of the shortfall; compare the discussion in Section 4.9. Definition 8.8. Given a loss function ` satisfying the above assumptions, the shortfall risk of an admissible strategy with value process V is defined as the expectation EΠ`.H  VT /  D EΠ`. .H  VT /C /  of the shortfall weighted by the loss function `. Our aim is to minimize the shortfall risk among all admissible strategies satisfying the capital constraint V0  v. Alternatively, we could minimize the cost under a given bound on the shortfall risk. In other words, the problem consists in constructing strategies which are efficient with respect to the trade-off between cost and shortfall risk. This generalizes our discussion of quantile hedging in the previous Section 8.1, which corresponds to a minimization of the shortfall risk with respect to the nonconvex loss function `.x/ D I.0;1/ .x/:

388

Chapter 8 Efficient hedging

Remark 8.9. Recall our discussion of risk measures in Chapter 4. From this point of view, it is natural to quantify the downside risk in terms of an acceptance set A for hedged positions. As in Section 4.8, we denote by AN the class of all positions X such that there exists an admissible strategy  with value process V such that V0 D 0

and

X C VT  A P -a.s.

for some A 2 A. Thus, the downside risk of the position H takes the form N .H / D inf¹m 2 R j m  H 2 Aº: Suppose that the acceptance set A is defined in terms of shortfall risk, i.e., A WD ¹X 2 L1 j EŒ `.X  /   x0 º; where ` is a convex loss function and x0 is a given threshold. Then .H / is the smallest amount m such that there exists an admissible strategy  whose value process V satisfies V0 D m and EŒ `..H  VT /C /   x0 : For a given m, we are thus led to the problem of finding a strategy  which minimizes the shortfall risk under the cost constraint V0  m. In this way, the problem of quantifying the downside risk of a contingent claim is reduced to the construction of efficient hedging strategies as discussed in this section. } As in the preceding section, the construction of the optimal hedging strategy is carried out in two steps. The first one is to solve the “static” problem of minimizing EŒ `.H  Y /  among all FT -measurable random variables Y  0 which satisfy the constraints sup E  Œ Y   v: P  2P

If Y  solves this problem, then so does YQ WD H ^ Y  . Hence, we may assume that 0  Y   H or, equivalently, that Y  D H  for some randomized test  , which belongs to the set R of all FT -measurable random variables with values in Œ0; 1. Thus, the static problem can be reformulated as follows: Find a randomized test  2 R which minimizes the “shortfall risk” EŒ `. H.1  among all

/ /

(8.16)

2 R subject to the constraints E ΠH

v

for all P  2 P .

(8.17)

The next step is to fit the terminal value VT of an admissible strategy to the optimal profile H  . It turns out that this step can be carried out without any further assumptions on our loss function `. Thus, we assume at this point that the optimal  of step one is granted, and we construct the corresponding optimal strategy.

389

Section 8.2 Hedging with minimal shortfall risk

Theorem 8.10. Given a randomized test  which minimizes (8.16) subject to (8.17), a superhedging strategy   for the modified discounted claim H  WD H



with initial investment sup .H  / has minimal shortfall risk among all admissible strategies  which satisfy the capital constraint  1  X 0  v. Proof. The proof extends the last argument in the proof of Theorem 8.7. As a first step, we take any admissible strategy  such that the corresponding value process V satisfies the capital constraint V0  v. Denote by VT I H ¹VT 0º.

(8.19)

390

Chapter 8 Efficient hedging

Proof. The proof is similar to the one of Proposition 3.36. Let R0 denote the set of all randomized tests which satisfy the constraints (8.19). Take n 2 R0 such that EŒ`. H.1  n / / converges to the infimum of the shortfall risk, and use Lemma 1.70 to select convex combinations Q n 2 conv¹ n ; nC1 ; : : : º which converge P -a.s. to some Q 2 R. Since ` is continuous and increasing, Fatou’s lemma implies that EŒ `. H.1  Q / /  lim inf EŒ `. H.1  Q n / / D inf EŒ `. H.1  2R0

n"1

/ /;

where we have used the convexity of ` to conclude that EŒ`. H.1  Q n / / tends to the same limit as EŒ`. H.1  n / /. Fatou’s lemma also yields that for all P  2 P E  Œ H Q   lim inf E  Œ H Q n   v: n"1

Hence Q 2 R0 , and we conclude that uniqueness part is obvious.



WD Q is the desired minimizer. The

Remark 8.12. The proof shows that the analogous existence result holds if we use a robust version of the shortfall risk defined as sup EQ Π`. H.1 

/ /;

Q2Q

where Q is a class of equivalent probability measures; see also Remark 3.37 and Sections 8.2 and 8.3. } Combining Proposition 8:11 and Theorem 8.10 yields existence and uniqueness of an optimal hedging strategy under risk aversion in a general arbitrage-free market model. Corollary 8.13. Assume that the loss function ` is strictly convex on Œ0; 1/. Then there exists an admissible strategy which is optimal in the sense that it minimizes the shortfall risk among all admissible strategies  subject to the capital constraint  1  X 0  v. Moreover, any optimal strategy requires the exact initial investment v, and its success ratio is P -a.s. equal to 

where



 I¹H >0º C I¹H D0º ;

denotes the solution of the static problem constructed in Proposition 8:11.

Proof. The existence of an optimal strategy follows by combining Proposition 8:11 and Theorem 8.10. Strict convexity of ` implies that  is P -a.s. unique on ¹H > 0º. Since ` is strictly increasing on Œ0; 1/,  and the success ratio V  of any optimal

391

Section 8.2 Hedging with minimal shortfall risk

strategy   must coincide P -a.s. on ¹H > 0º. On ¹H D 0º, the success ratio equal to 1 by definition. Since ` is strictly increasing on Œ0; 1/, we must have that sup E  Œ H



V

is

 D v;

P  2P

for otherwise we could find some " > 0 such that " WD " C .1  "/  would also satisfy the constraints (8.17). Since we have assumed that v < sup .H /, the constraints (8.17) imply that  ¥ 1 and hence that EŒ `. H.1 

" / /

< EΠ`. H.1 



/ /:

This, however, contradicts the optimality of  . Since the value process V  of an optimal strategy is a P -martingale, and since VT  H

V

DH



;

we conclude from the above that v  V0 D sup E  ΠVT   sup E  ΠH P  2P



 D v:

P  2P

Thus, V0 is equal to v. Beyond the general existence statement of Proposition 8.11, it is possible to obtain an explicit formula for the optimal solution of the static problem if the market model is complete. Recall that we assume that the loss function `.x/ vanishes for x  0. In addition, we will also assume that ` is strictly convex and continuously differentiable on .0; 1/. Then the derivative `0 of ` is strictly increasing on .0; 1/. Let J denote the inverse function of `0 defined on the range of `0 , i.e., on the interval .a; b/ where a WD limx#0 `0 .x/ and b WD limx"1 `0 .x/. We extend J to a function J C W Œ0; 1 ! Œ0; 1 by setting ´ C1 for y  b, C J .y/ WD 0 for y  a. From now on, we assume also that P D ¹P  º; i.e., P  is the unique equivalent martingale measure in a complete market model. Its density will be denoted by dP  '  WD : dP

392

Chapter 8 Efficient hedging

Theorem 8.14. Under the above assumptions, the solution of the static optimization problem of Proposition 8:11 is given by J C .c '  / ^ 1 P -a.s. on ¹H > 0º; H where the constant c is determined by the condition E  Œ H   D v. 

D1

Proof. The problem is of the same type as those considered in Section 3.3. It can in fact be reduced to Corollary 3.43 by considering the random utility function u.x; !/ WD `.H.!/  x/;

0  x  H.!/:

Just note that the shortfall risk EŒ `.H  Y /  coincides with the negative expected utility EŒu.Y; /  for any profile Y such that 0  Y  H . Moreover, since our market model is complete, it has a finite structure by Theorem 5.37, and so all integrability conditions are automatically satisfied. Thus, Corollary 3.43 states that the optimal profile H  WD Y  which maximizes the expected utility EŒ u.Y; /  under the constraints 0  Y  H and E  Œ Y   v is given by H  .!/ D I C .c '  .!/; !/ ^ H.!/ D .H.!/  J C .c '  .!///C : Dividing by H yields the formula for the optimal randomized test

.

Corollary 8.15. In the situation of Theorem 8:14, suppose that the objective probability measure P is equal to the martingale measure P  . Then the modified discounted claim takes the simple form H D H



D .H  J C .c  //C :

Example 8.16. Consider the discounted payoff H of a European call option .STi  K/C with strike K under the assumption that the numéraire S 0 is a riskless bond, i.e., that S t0 D .1 C r/t for a certain constant r  0. If the assumptions of Corollary 8.15 hold, then the modified profile H  is the discounted value of the European call option struck at KQ WD K C J C .c  /  .1 C r/T , i.e., H D

Q C .STi  K/ : .1 C r/T

}

Example 8.17. Consider an exponential loss function `.x/ D .e ˛x  1/C for some ˛ > 0. In this case,   1 y C C J .y/ D log ; y  0; ˛ ˛ and the optimal profile is given by   1 c'  C  H DH log ^ H: } ˛ ˛

393

Section 8.2 Hedging with minimal shortfall risk

Example 8.18. If ` is the particular loss function `.x/ D

xp ; p

x  0;

for some p > 1, then the problem is to minimize a lower partial moment of the difference VT  H . Theorem 8:14 implies that it is optimal to hedge the modified claim H p D H  .cp  '  /1=.p1/ ^ H (8.20) } where the constant cp is determined by E  ΠH p D v. Let us now consider the limit p " 1 in (8.20), corresponding to ever increasing risk aversion with respect to large losses. Proposition 8.19. Let us consider the loss functions `p .x/ D

xp ; p

x  0;

for p > 1. As p " 1, the modified claims H L1 .P  / to the discounted claim

 p

of (8.20) converge P -a.s. and in

.H  c1 /C where the constant c1 is determined by E  Π.H  c1 /C  D v:

(8.21)

Proof. Let .p/ be shorthand for 1=.p  1/ and note that .'  / .p/ ! 1

P -a.s. as p " 1. .pn /

Hence, if .pn / is a sequence for which cpn lim H

n"1

 pn

converges to some cQ 2 Œ0; 1, then

D H  cQ ^ H D .H  cQ /C :

Hence, E ΠH

 pn

 ! E  Π.H  cQ /C :

Since each term on the left-hand side equals v, we must have E  Π.H  cQ /C  D v; which determines cQ uniquely as the constant c1 of (8.21).

394

Chapter 8 Efficient hedging

Example 8.20. If the discounted claim H in Proposition 8.19 is the discounted payoff of a call option with strike K, and the numéraire is a riskless bond as in Example 8.16, then the limiting profile limp"1 H p is equal to the discounted call with the higher strike price K C c1  ST0 . } In the remainder of this section, we consider loss functions which are not convex but which correspond to risk neutrality and to risk-seeking preferences. Let us first consider the risk-neutral case. Example 8.21. In the case of risk neutrality, the loss function is given by `.x/ D x

for x  0.

Thus, the task is to minimize the expected shortfall EΠ.H  VT /C  under the capital constraint V0  v. Let P  be the unique equivalent martingale measure in a complete market model. Then the static problem corresponding to Proposition 8:11 is to maximize the expectation EΠH under the constraint that



2 R satisfies E ΠH

  v:

We can define two equivalent measures Q and Q by H dQ D dP EΠH 

and

dQ H : D   dP E ŒH 

The problem then becomes the hypothesis testing problem of maximizing EQ Œ under the side condition v EQ Œ   ˛ WD  : E ŒH 



Since the density dQ=dQ is proportional to the inverse of the density '  D dP =dP , Theorem A.31 implies that the optimal test takes the form  1

D I¹'  0  D 1. As in Example 8.21, we then conclude that the optimal test must be of the form I¹1>c '  H 1q º C   I¹1Dc '  H 1q º q

q

(8.23)

for certain constants cq and . Under the simplifying assumption that P Π1 D cq '  H 1q  D 0;

(8.24)

the formula (8.23) reduces to  q

By taking D for EΠ`.H.1 

 q

´ 1 D 0

on ¹1 > cq '  H 1q º; otherwise.

(8.25)

we obtain an identity in (8.22), and so q must be a minimizer //  under the constraint that E  ΠH   v. }

396

Chapter 8 Efficient hedging

In our last result of this section, we recover the knock-out option H  I¹1>c H '  º ; 0

which was obtained as the solution to the problem of quantile hedging by taking the limit q # 0 in (8.25). Intuitively, decreasing q corresponds to an increasing appetite for risk in view of the shortfall. Proposition 8.23. Let us assume for simplicity that (8.24) holds for all q 2 .0; 1/, that P Œ H > 0  D 1, and that there exists a unique constant c0 such that E  Œ H  I¹1>c H '  º  D v:

(8.26)

0

Then the solutions

 q

of (8.25) converge P -a.s. to the solution  0

D I¹1>c H '  º 0

of the corresponding problem of quantile hedging as constructed in Proposition 8:3. Proof. Take any sequence qn # 0 such that .cqn /1=.1qn / converges to some cQ 2 Œ0; 1. Then lim qn D I¹1>cH : Q ' º n"1

Hence, E ΠH

 qn

 ! E  Œ H  I¹1>cH : Q ' º

Since we assumed (8.24) for all q 2 .0; 1/, the left-hand terms are all equal to v, and it follows from (8.26) that cQ D c0 . This establishes the desired convergence.

8.3

Efficient hedging with convex risk measures

As in the previous sections of this chapter, we consider the shortfall .H  VT /C arising from hedging the discounted claim H with a self-financing trading strategy with initial capital V0 D v 2 .0; sup .H //: In this section, our aim is to minimize the shortfall risk ..H  VT /C /; where  is a given convex risk measure as discussed in Chapter 4. Here we assume that  is defined on a suitable function space, such as Lp .; F ; P /, so that the shortfall

397

Section 8.3 Efficient hedging with convex risk measures

risk is well-defined and finite; cf. Remark 4.44. In particular, we assume that .Y / D .YQ / whenever Y D YQ P -a.s. As in the preceding two sections, the construction of the optimal hedging strategy can be carried out in two steps. The first step is to solve the static problem of minimizing ..H  Y /C / over all FT -measurable random variables Y  0 that satisfy the constraint sup E  ΠY   v: P  2P

If Y  solves this problem, then so does H ^ Y  . Hence, we may assume that 0  Y   H , and we can reformulate the problem as minimize .Y  H / subject to 0  Y  H and sup E  ΠY   v:

(8.27)

P  2P

The next step is to fit the terminal value VT of an admissible strategy to the optimal profile Y  . It turns out that this step can be carried out without any further assumptions on our risk measure . Thus, we assume at this point that the optimal Y  of step one is granted, and we construct the corresponding optimal strategy. Proposition 8.24. A superhedging strategy for a solution Y  of (8.27) with initial investment sup .Y  / has minimal shortfall risk among all admissible strategies whose value process satisfies the capital constraint V0  v. Proof. Let V be the value process of any admissible strategy such that V0  v. Due to Doob’s systems theorem in the form of Theorem 5.14, V is a martingale under any P  2 P , and so sup E  Œ VT  D V0  v: P  2P

Thus, Y WD H ^ VT satisfies the constraints in (8.27), and we get ..H  VT /C / D .Y  H /  .Y   H /: Next let V  be the value process of a superhedging strategy for Y  with initial investment V0 D sup .Y  / D sup E  ΠY  : P  2P

Then we have

V0

 v and

VT

 0. Moreover, VT  Y  P -a.s., and thus

.Y   H / D ..H  Y  /C /  ..H  VT /C /: This concludes the proof.

398

Chapter 8 Efficient hedging

Let us now return to the static problem defined by (8.27). Proposition 8.25. If  is lower semicontinuous with respect to P -a.s. convergence of random variables in the class ¹Y j 0  Y  H º and .Y / < 1 for one such Y , then there exists a solution of the static optimization problem (8.27). In particular, there exists a solution if H is bounded and  is continuous from above. Proof. Take a sequence Yn with 0  Yn  H and supP  2P E  Œ Yn   v such that .Yn  H / converges to the infimum A of the shortfall risk. We can use Lemma 1.70 to select convex combinations Zn 2 conv¹Yn ; YnC1 ; : : : º which converge P -a.s. to some random variable Z. Then 0  Z  H and Fatou’s lemma yields that E  Œ Z   lim inf E  Œ Zn   v n"1

for all P  2 P . Lower semicontinuity of  implies that .Z  H /  lim inf .Zn  H /: n"1

Moreover, the right-hand side is equal to A, due to the convexity of . Hence, Z is the desired minimizer. Combining Proposition 8:25 and Proposition 8.24 yields the existence of a riskminimizing hedging strategy in a general arbitrage-free market model. So far, all arguments were practically the same as in the preceding two sections. Beyond the general existence statement of Proposition 8.25, it is sometimes possible to obtain an explicit formula for the optimal solution of the static problem if the market model is complete and so P D ¹P  º. In this case, the static optimization problem (8.27) simplifies to minimize .Y  H / subject to 0  Y  H and E  Œ Y   v: By substituting Z for H  Y , this is equivalent to the problem Q minimize .Z/ subject to 0  Z  H and E  Œ Z   v;

(8.28)

where vQ WD E  ΠH   v. We will now discuss this problem in the case  D AV@R . Our approach relies on the general idea that a minimax problem can be transformed into a standard minimization problem by using a duality result for the expression involving the maximum. In the case of AV@R , we can use the following representation of AV@R from Lemma 4.51,

AV@R .Z/ D

1 1 min.EΠ.Z  r/C  C r/ D min.EΠ.Z  r/C  C r/ (8.29)

r2R

r0

Section 8.3 Efficient hedging with convex risk measures

399

for Z  0. Our discussion of problem (8.28) will be valid also beyond the setting of a complete discrete-time market model, whose underlying probability space has necessarily a discrete structure by Theorem 5.37. In fact it applies whenever P and P  are two equivalent probability measures on a given measurable space .; F /. This is important in view of the application of the next theorem in Example 3.50. Theorem 8.26. Suppose that H 2 L1 .P / and denote by ' WD dP  =dP the price density of P  with respect to P . Then the problem (8.28) admits a solution for  D AV@R which is of the form Z  D H I¹'>cº C .H ^ r  /I¹'cº C H I¹'Dcº for constants c > 0 and  2 Œ0; 1. This is indeed a special case of (8.30). Now we consider the case r  > 0. Note first that we must have Z   H ^ r  . Indeed, let us assume P Œ Z  < H ^ r   > 0. Then we could obtain a strictly lower risk AV@R .Z  / either by decreasing the level r  in case P Œ Z   H ^ r   D 1 or, in case P Œ Z  > H ^ r   > 0, by shifting mass of Z  from ¹Z  > H ^ r  º to the set ¹Z  < H ^ r  º.

400

Chapter 8 Efficient hedging

Thus, we can solve our problem by minimizing EΠ.ZO C H ^ r   r  /C  subject to 0  ZO  H  H ^ r 

and

E  ΠZO   vO WD vQ  E  ΠH ^ r  :

Any ZO satisfying these constraints must be concentrated on ¹H > r  º, so that the problem is equivalent to minimize EŒ ZO  subject to 0  ZO  H  H ^ r  and E  Œ ZO   v. O

(8.33)

But this problem is equivalent to the one for r  D 0 if we replace H by H  H ^ r  . Hence, it is solved by ZO  D .H  H ^ r  /I¹'>cº C .H  H ^ r  /I¹'Dcº for some constants c > 0 and  2 Œ0; 1. It follows that Z  D ZO  CH ^r  D H I¹'>cº C.H ^r  /I¹' 1 . Then the solution Y  of problem (8.28) is P -a.s. unique. Moreover, there exists a critical capital level v  such that Z  D I¹'>q' .t0 /º

P -a.s. for vQ  v  ;

Q and where t0 is determined by the condition E  Œ Z   D v, Z  D r  C .1  r  /I¹'>q' .t /º where r  D 1  equation

1vQ ˆ.t /

> 0, ˆ.t / WD

Rt 0

P -a.s. for vQ > v  ,

q' .s/ ds, and t is the unique solution of the

q' .t /.t  1 C / D ˆ.t /: Finally, the critical capital level v  is equal to 1  ˆ.t /. Proof. We fix vQ 2 .0; 1/. For a constant c we then let Zr D r C .1  r/I¹'>cº ;

(8.34)

401

Section 8.3 Efficient hedging with convex risk measures

where r D r.c/  0 is such that E  ΠZr  D v, Q i.e., r.c/ D

vQ  EΠ'I ' > c  : EΠ'I '  c 

This makes sense as long as c  c0 , where c0 is defined via vQ D EŒ 'I ' > c0 . Theorem 8.26 states that a solution of our problem can be found within the class ¹Zr.c/ j c  c0 º. Thus, we have to minimize Z 1 Z 1

AV@R .Zr.c/ / D qZr.c/ .s/ ds D r.c/ C .1  r.c// I¹q' .s/>cº ds 1

1

over c  c0 . Here we have used Lemma A.23 in the second identity. This minimization problem can be simplified further be using the reparameterization c D q' .t /, which is one-to-one according to our assumptions. Indeed, by letting %.t / WD r.q' .t // D 1 

1  vQ ; ˆ.t /

we simply have to minimize the function Z R.t / WD AV@R .Z%.t/ / D %.t / C .1  %.t //

1

1

I.t;1 .s/ ds

D %.t / C .1  %.t //.  .t  1 C /C / D  .1  v/ Q

.t  1 C /C ˆ.t /

over t  t0 WD F' .c0 /. For t  1  , we get R.t / D , which cannot be optimal. We show next that the function ‰.t / WD

t 1C ˆ.t /

has a unique maximizer t 2 .1  ; 1, which will define the solution as soon as t  t0 and as long as t D t0 does not give a better result. To this end, we note first that ˆ.t /  .t  1 C /q' .t / ‰ 0 .t / D : (8.35) ˆ.t /2 The numerator of this expression is strictly larger than zero for t  1  and equal to ˆ.1  / > 0 at t D 1  . Moreover, for t > 1  , Z 1

Z t ˆ.t /  .t  1 C /q' .t / D q' .s/ ds C q' .s/  q' .t / ds; 0

1

which is easily seen to be strictly decreasing in t . For t " 1 this expression converges to 1  k'k1 , which is strictly negative due to our assumption k'k1 > 1 . Hence,

402

Chapter 8 Efficient hedging

the numerator in (8.35) has a unique zero t 2 .1  ; 1/, which is the unique solution of the equation q' .t /.t  1 C / D ˆ.t /; and this solution t is consequently the unique maximizer of ‰. If t  t0 , then R has no minimizer on .t0 ; 1, and it follows that t  D t0 is its minimizer. Let us compare R.t / with R.t0 / in case t > t0 . We have R.t / D  .1  v/ Q

.t  1 C /C .t  1 C /C D  ˆ.t0 / ˆ.t / ˆ.t /

and R.t0 / D  .t0 C  1/C D  ˆ.t0 /

.t0  1 C /C : ˆ.t0 /

Since t is the unique maximizer of the function t 7! .t  1 C /C =ˆ.t /, we thus see that R.t / is strictly smaller than R.t0 /. Hence the solution is defined by t  WD t0 _ t : Clearly, t is independent of v, Q while t0 decreases from 1 to 0 as vQ increases from 0 to 1. Thus, by taking v  as the capital level for which t D t0 , we see that the optimal solution has the form ´ I for vQ  v  ,  ' .t0 /º Z D ¹'>q r  C .1  r  /I¹'>q' .t /º for vQ > v  , 1vQ > 0. where r  D 1  ˆ.t / Finally, when vQ is equal to the critical capital level v  , we must have that t D t0 . But t0 was defined to be the solution of 1  ˆ.t / D v, Q and so v  D 1  ˆ.t /.

Let us now point out the connections of the preceding theorem with robust statistical test theory as explained in Section 3.5 and in particular in Remark 3.54. To this end, let R denote the set of all measurable functions W  ! Œ0; 1, which will be interpreted as randomized statistical tests; see Remark A.32. Problem (8.28) for H D 1 can then be rewritten as minimize sup EQ Œ

 subject to

2 R and E  Œ

  v, Q

Q2Q

where Q is the set of all probability measures Q P whose density dQ=dP is P -a.s. bounded by 1= . The solution  of this problem is described in Theorem 8.27. It is easy to see that  must also solve the following dual problem: maximize E  Œ

 subject to

2 R and sup EQ ΠQ2Q

  ˛;

Section 8.3 Efficient hedging with convex risk measures

403

where ˛ D supQ2Q EQ Œ  . Thus,  is an optimal randomized test for testing the hypothesis P  against the composite null hypothesis Q ; see Remark 3.54. When Q admits a least-favorable measure Q0 with respect to P  in the sense of Definition 3.48, then  must also be a standard Neyman–Pearson test for testing the hypothesis P  against the null hypothesis Q0 . By Theorem A.31, it must hence be of the form  D I¹ Dcº C I¹ >cº ; where c > 0 and  2 Œ0; 1 are constants and D

dP  : dQ0

Under the assumptions of Theorem 8.27, this is the case when  D c.' _ q' .t //, where c is a suitable constant. It follows that dQ0 1 ' D  : dP c ' _ q' .t / We have by (8.34), E

h

i ' 1 EΠ'I '  q' .t /  D P Π' > q' .t /  C ' _ q' .t / q' .t / D 1  t C

1 ˆ.t / D : q' .t /

This yields that c D , and we see that dQ0 1 ' D  dP

' _ q' .t /

(8.36)

defines a probability measure Q0 2 Q with D

dP  D .' _ q' .t //: dQ0

The following result now follows immediately from Theorem 8.27: Corollary 8.28. For a measure P   P satisfying the assumptions of Theorem 8:27, the measure Q0 in (8.36) is a least-favorable measure for ˇ dQ ± ° 1 ˇ Q D Q 2 M1 .P / ˇ  P -a.s. dP

in the sense of Definition 3:48.

Chapter 9

Hedging under constraints

So far, we have focused on frictionless market models, where asset transactions can be carried out with no limitation. In this chapter, we study the impact of market imperfections generated by convex trading constraints. Thus, we develop the theory of dynamic hedging under the condition that only trading strategies from a given class S may be used. In Section 9.1 we characterize those market models for which S does not contain arbitrage opportunities. Then we take a direct approach to the superhedging duality for American options. To this end, we first derive a uniform Doob decomposition under constraints in Section 9.2. The appropriate upper Snell envelopes are analyzed in Section 9.3. In Section 9.4 we derive a superhedging duality under constraints, and we explain its role in the analysis of convex risk measures in a financial market model.

9.1

Absence of arbitrage opportunities

In practice, it may be reasonable to restrict the class of trading strategies which are admissible for hedging purposes. As discussed in Section 4.8, there may be upper bounds on the capital invested into risky assets, or upper and lower bounds on the number of shares of an asset. Here we model such portfolio constraints by a set S of d -dimensional predictable processes, viewed as admissible investment strategies into risky assets. Throughout this chapter, we will assume that S satisfies the following conditions: (a) 0 2 S. (b) S is predictably convex: If ;  2 S and h is a predictable process with 0  h  1, then the process h t  t C .1  h t /  t ;

t D 1; : : : ; T;

belongs to S. (c) For each t 2 ¹1; : : : ; T º, the set S t WD ¹ t j  2 Sº is closed in L0 .; F t1 ; P I Rd /. (d) For all t ,  t 2 S t implies  t? 2 S t .

405

Section 9.1 Absence of arbitrage opportunities

In order to explain condition (d), let us recall from Lemma 1.66 that each  t 2 L0 .; F t1 ; P I Rd / can be uniquely decomposed as  t D  t C  t? ;

where  t 2 N t and  t? 2 N t? ,

and where N t D ¹ t 2 L0 .; F t1 ; P I Rd / j  t  .X t  X t1 / D 0 P -a.s. º; N t? D ¹ t 2 L0 .; F t1 ; P I Rd / j  t   t D 0 P -a.s. for all  t 2 N t º: Remark 9.1. Under condition (d), we may replace  t .X t X t1 / by  t? .X t X t1 /, and  t?  .X t  X t1 / D 0 P -a.s. implies  t? D 0. Note that condition (d) holds if the price increments satisfy the following non-redundance condition: For all t 2 ¹1; : : : ; T º and  t 2 L0 .; F t1 ; P I Rd /,  t  .X t  X t1 / D 0

P -a.s.

H)

t D 0

P -a.s.

(9.1) }

Example 9.2. For each t let C t be a closed convex subset of Rd such that 0 2 C t . Take S as the class of all d -dimensional predictable processes  such that  t 2 C t P -a.s. for all t . If the non-redundance condition (9.1) holds, then S satisfies conditions (a) through (d). This case includes short sales constraints and restrictions on the size of a long position. } Example 9.3. Let a; b be two constants such that 1  a < 0 < b  1, and take S as the set of all d -dimensional predictable processes such that a   t  X t1  b

P -a.s. for t D 1; : : : ; T .

This class S corresponds to constraints on the capital invested into risky assets. If we assume that the non-redundance condition (9.1) holds, then S satisfies conditions (a) through (d). More generally, instead of the two constants a and b, one can take dynamic margins defined via two predictable processes .a t / and .b t /. } Let S denote the set of all self-financing trading strategies  D . 0 ; / which arise from an investment strategy  2 S, i.e., S D ¹ D . 0 ; / j  is self-financing and  2 S º: In this section, our goal is to characterize the absence of arbitrage opportunities in S. The existence of an equivalent martingale measure P  2 P is clearly sufficient. Under an additional technical assumption, a condition which is both necessary and sufficient will involve a larger class PS  P . In order to introduce these conditions, we need some preparation.

406

Chapter 9 Hedging under constraints

Definition 9.4. An adapted stochastic process Z on .; F ; .F t /; Q/ is called a local Q-martingale if there exists a sequence of stopping times .n /n2N  T such that n % T Q-a.s., and such that the stopped processes Z n are Q-martingales. The sequence .n /n2N is called a localizing sequence for Z. In the same way, we define local supermartingales and local submartingales. Remark 9.5. If Q is a martingale measure for the discounted price process X , then the value process V of each self-financing trading strategy  D . 0 ; / is a local Q-martingale. To prove this, one can take the sequence n WD inf¹t  0 j j tC1 j > n º ^ T as a localizing sequence. With this choice, j t j  n on ¹n  t º, and the increments n V tn  V t1 D I¹n t º  t  .X t  X t1 /;

t D 1; : : : ; T;

of the stopped process V n are Q-integrable and satisfy n j F t1  D I¹n t º  t  EQ Œ X t  X t1 j F t1  D 0: EQ Œ V tn  V t1

}

The following proposition is a generalization of an argument which we have already used in the proof of Theorem 5.25. Throughout this chapter, we will assume that F0 D ¹;; º and FT D F . Proposition 9.6. A local Q-supermartingale Z whose negative part Z t is integrable for each t 2 ¹1; : : : ; T º is a Q-supermartingale. Proof. Let .n / be a localizing sequence. Then Z tn  

T X

Zs 2 L1 .Q/:

sD0

In view of limn Z tn D Z t , Fatou’s lemma for conditional expectations implies that Q-a.s. n EQ Œ Z t j F t1   lim inf EQ Œ Z tn j F t1   lim inf Z t1 D Z t1 : n"1

n"1

We get in particular that EQ Œ Z t   Z0 < 1. Thus Z t 2 L1 .Q/, and the assertion follows. Exercise 9.1.1. Let Z be a local Q-martingale with ZT  0. Show that in our situation, where F0 D ¹;; º, Z is a Q-martingale.

Section 9.1 Absence of arbitrage opportunities

407

Definition 9.7. By PS we denote the class of all probability measures PQ  P such that X t 2 L1 .PQ / for all t , (9.2) and such that the value process of any trading strategy in S is a local PQ -supermartingale. Remark 9.8. If S contains all self-financing trading strategies  D . 0 ; / with bounded , then PS coincides with the class P of all equivalent martingale measures. To prove this, let PQ 2 PS , and note that the value process V of any such  is a PQ -supermartingale by (9.2) and by Proposition 9.6. The same applies to the strategy , so V is in fact a PQ -martingale, and Theorem 5.14 shows that PQ is a martingale measure for X . } Our first goal is to extend the “fundamental theorem of asset pricing” to our present setting; see Theorem 5.16. Let us introduce the positive cone R WD ¹  j  2 S;  0º generated by S. Accordingly, we define the cones R and R t . Clearly, R contains no arbitrage opportunities if and only if S is arbitrage-free. We will need the following condition on the L0 -closure RO t of R t : for each t , RO t \ L1 .; F t ; P I Rd /  R t .

(9.3)

This condition clearly holds if R t itself is closed in L0 and in particular if S t D R t for all t . Theorem 9.9. Under condition (9.3), there are no arbitrage opportunities in S if and only if PS is non-empty. In this case, there exists a measure PQ 2 PS which has a bounded density d PQ =dP . Example 9.10. In the situation of Example 9.2, condition (9.3) will be satisfied as soon as the cones generated by the convex sets C t are closed in Rd . This case includes short sales constraints and constraints on the size of a long position, which are modeled by taking C t D Œa1t ; b t1       Œadt ; b td  for certain numbers akt ; b tk such that 1  akt  0  b tk  1. Example 9.11. Consider now the situation of Example 9.3. We claim that S does not contain arbitrage opportunities if and only if the unconstrained market is arbitragefree, so that we have PS D P . To prove this, note that the existence of an arbitrage opportunity in the unconstrained market is equivalent to the existence of some t and some F t1 -measurable  t such that  t  .X t  X t1 /  0 P -a.s. and P Œ  t  .X t  X t1 / > 0  > 0 (see Proposition 5.11). Next, there exists a constant c > 0 such that these properties are shared by Qt WD  t I¹j t X t 1 jcº and in turn by "Qt , where " > 0. But "Qt 2 S t if " is small enough.

408

Chapter 9 Hedging under constraints

As to the proof of Theorem 9.9, we will first show that the condition PS ¤ ; implies the absence of arbitrage opportunities in S. Proof of sufficiency in Theorem 9:9. Suppose PQ is a measure in PS , and V is the value process of a trading strategy in S such that VT  0 P -almost surely. Combining Lemma 9.12 below with Proposition 9.6 shows that V is a PQ -supermartingale. Q VT , so V cannot be the value process of an arbitrage opportunity. Hence V0  EŒ Lemma 9.12. Suppose that PS ¤ ; and that V is the value process of a trading strategy in S such that VT  0 P -almost surely. Then V t  0 P -a.s. for all t . Proof. The assertion will be proved by backward induction on t . We have VT  0 by assumption, so let us assume that V t  0 P -a.s. for some t . For  D . 0 ; / 2 S .c/ with value process V , we let s WD s I¹js jcº for c > 0 and for all s. Then the value process V .c/ of  .c/ is a PQ -supermartingale for any fixed PQ 2 PS . Furthermore, .c/

V t1 I¹j t jcº D V t I¹j t jcº   t .c/

  t

.c/

 .X t  X t1 /

 .X t  X t1 / .c/

D V t1  V t : The last term on the right belongs to L1 .PQ /, so we may take the conditional expectaQ  j F t1  on both sides of the inequality. We get tion EŒ Q V .c/  V t.c/ j F t1   0 V t1 I¹j t jcº  EŒ t1

PQ -a.s.

By letting c " 1, we obtain V t1  0. Let us now prepare for the proof that the condition PS ¤ ; is necessary. First we argue that the absence of arbitrage opportunities in S is equivalent to the absence of arbitrage opportunities in each of the embedded one-period models, i.e., to the nonexistence of  t 2 S t such that  t  .X t  X t1 / amounts to a non-trivial positive gain. This observation will allow us to apply the techniques of Section 1.6. Let us denote S 1 WD ¹ 2 S j  is boundedº: Similarly, we define S t1 WD ¹ t j  2 S 1 º D S t \ L1 .; F t1 ; P I Rd /:

409

Section 9.1 Absence of arbitrage opportunities

Lemma 9.13. The following conditions are equivalent: (a) There exists an arbitrage opportunity in S. (b) There exist t 2 ¹1; : : : ; T º and  t 2 S t such that  t  .X t  X t1 /  0 P -a.s.,

and

P Π t  .X t  X t1 / > 0  > 0:

(9.4)

(c) There exist t 2 ¹1; : : : ; T º and  t 2 S t1 which satisfies (9.4). Proof. The proof is essentially the same as the one of Proposition 5.11. In order to apply the results of Section 1.6, we introduce the convex sets K tS WD ¹ t  .X t  X t1 / j  t 2 S t º; for t 2 ¹1; : : : ; T º. Lemma 9.13 shows that S contains no arbitrage opportunities if and only if the condition K tS \ L0C D ¹0º (9.5) holds for all t 2 ¹1; : : : ; T º. Lemma 9.14. Condition (9.5) implies that K tS  L0C .; F t ; P / is a closed convex subset of L0 .; F t ; P /. Proof. The proof is essentially the same as the one of Lemma 1.68. Only the following additional observation is required: If . n / is sequence in S t , and if ˛ and are two F t1 -measurable random variables such that 0  ˛  1 and is integer-valued, then  WD ˛  2 S t . Indeed, predictable convexity of S and our assumption that 0 2 S imply that n X ˛ I¹ Dkº  k 2 S t kD1

for each n, and the closedness of S t in L0 .; F t1 ; P I Rd / yields D˛

1 X

I¹ Dkº  k 2 S t :

kD1

From now on, we will assume that EΠjXs j  < 1

for all s.

(9.6)

For the purpose of proving Theorem 9.9, this can be assumed without loss of generality: If (9.6) does not hold, then we replace P by an equivalent measure P 0 which

410

Chapter 9 Hedging under constraints

has a bounded density dP 0 =dP and for which the price process X is integrable. For instance, we can take T h X i dP 0 D c exp  jXs j dP; sD1

where c denotes the normalizing constant. If there exist a measure PQ  P 0 such that each value process for a strategy in S is a local PQ -supermartingale and such that the density d PQ =dP 0 is bounded, then PQ 2 PS , and the density d PQ =dP is bounded as well. Lemma 9.15. If S contains no arbitrage opportunities and condition (9.3) holds, then for each t 2 ¹1; : : : ; T º there exists some Z t 2 L1 .; F t ; P / such that Z t > 0 P -a.s. and such that EŒ Z t  t  .X t  X t1 /   0 for all  2 S 1 .

(9.7)

Proof. Recall that R does not contain arbitrage opportunities if and only if S is arbitrage-free. Hence, for each t , K tR \ L0C .; F t ; P / D ¹0º by Lemma 9.13. By the equivalence of conditions (a) and (c) of same lemma and condition (9.3), we even get O K tR \ L0C .; F t ; P / D ¹0º (9.8) where RO t denotes again the L0 -closure of R t . The cone RO t satisfies all conditions required from S t , and hence Lemma 9.14 implies that each O

O

C tR WD .K tR  L0C .; F t ; P // \ L1 is a closed convex cone in L1 which contains L1 C .; F t ; P /. Furthermore, it follows from (9.8) and the argument in the proof of “(a) , (b)” of Theorem 1.55 O O that C tR \ L0C D ¹0º, so C tR satisfies the assumptions of the Kreps–Yan theorem, which is stated in Theorem 1.62. We conclude that there exist Z t 2 L1 .; F t ; P / O such that P Œ Z t > 0  D 1, and such that EŒ Z t W   0 for each W 2 C tR . As O  t  .X t  X t1 / 2 C tR for each  2 S 1 , Z t has property (9.7). Now we can complete the proof of Theorem 9.9 by showing that the absence of arbitrage opportunities in S implies the existence of a measure PQ that belongs to the class PS and has a bounded density d PQ =dP . Proof of necessity in Theorem 9:9. Suppose that S does not contain arbitrage opportunities. We are going to construct the desired measure PQ via backward recursion. First we consider the case t D T . Take a bounded random variable ZT > 0 as constructed in Lemma 9.15, and define a probability measure PQT by d PQT ZT D : dP EŒ ZT 

411

Section 9.1 Absence of arbitrage opportunities

Clearly, PQT is equivalent to P , and X t 2 L1 .PQT / for all t . We claim that EQ T ΠT  .XT  XT 1 / j FT 1   0

for all  2 S 1 .

(9.9)

To prove this claim, consider the family ˆ WD ¹EQ T Œ T  .XT  XT 1 / j FT 1  j  2 S 1 º: For ; Q 2 S 1 , let A WD ¹EQ T Œ T  .XT  XT 1 / j FT 1  > EQ T Œ QT  .XT  XT 1 / j FT 1 º; and define  0 by  t0 D 0 for t < T and T0 WD T IA C QT IAc : The predictable convexity of S implies that  0 2 S 1 . Furthermore, we have EQ T Œ T0  .XT  XT 1 / j FT 1  D EQ T Œ T  .XT  XT 1 / j FT 1  _ EQ T Œ QT  .XT  XT 1 / j FT 1 : Hence, the family ˆ is directed upwards in the sense of Theorem A.33. By virtue of that theorem, ess sup ˆ is the increasing limit of a sequence in ˆ. By monotone convergence, we get  EQ T ess sup EQ T Œ T  .XT  XT 1 / j FT 1  2S 1

D sup EQ T ΠEQ T ΠT  .XT  XT 1 / j FT 1   2S 1

1 sup EΠT  .XT  XT 1 / ZT  D EΠZT  2S 1

(9.10)

 0; where we have used (9.7) in the last step. Since S contains 0, it follows that ess sup EQ T ΠT  .XT  XT 1 / j FT 1  D 0

PQT -a.s.,

2S 1

which yields our claim (9.9). Now we apply the previous argument inductively: Suppose we already have a probability measure PQtC1  P with a bounded density d PQtC1 =dP such that EQ tC1 ΠjXs j  < 1 for all s,

412

Chapter 9 Hedging under constraints

and such that EQ tC1 Πk  .Xk  Xk1 / j Fk1   0 P -a.s. for k  t C 1 and  2 S 1 . (9.11) Then we may apply Lemma 9.15 with P replaced by PQtC1 , and we get some strictly positive ZQ t 2 L1 .; F t ; PQtC1 / satisfying (9.7) with PQtC1 in place of P . We now proceed as in the first step by defining a probability measure PQt  PQtC1  P as ZQ t d PQt : D EQ tC1 ΠZQ t  d PQtC1 Then PQt has bounded densities with respect to both PQtC1 and P . In particular, EQ t ΠjXs j  < 1 for all s. Moreover, the F t -measurability of d PQt =d PQtC1 implies that (9.11) is satisfied for PQt replacing PQtC1 . Repeating the arguments that led to (9.9) yields EQ t Π t  .X t  X t1 / j F t1   0 for all  2 S 1 . After T steps, we arrive at the desired measure PQ WD PQ1 2 PS .

9.2

Uniform Doob decomposition

The goal of this section is to characterize those non-negative adapted processes U which can be decomposed as U t D U0 C

t X

k  .Xk  Xk1 /  B t ;

(9.12)

kD1

where the predictable d -dimensional process  belongs to S, and where B is an adapted and increasing process such that B0 D 0. In the unconstrained case where S consists of all strategies, we have seen in Section 7.2 that such a decomposition exists if and only if U is a supermartingale under each equivalent martingale measure P  2 P . In our present context, a first guess might be that the role of P is now played by PS . Since each value process of a strategy in S is a local PQ -supermartingale for each PQ 2 PS , any process U which has a decomposition (9.12) is also a local PQ supermartingale for PQ 2 PS . Thus, one might suspect that the latter property would also be sufficient for the existence of a decomposition (9.12). This, however, is not the case, as is illustrated by the following simple example. Example 9.16. Consider a one-period market model with the riskless bond S00  S10  1 and with one risky asset S 1 . We assume that S01  1 and that S11 takes the values S11 .!  / D 12 and S11 .! C / D 32 on  WD ¹!  ; ! C º. We choose any measure P on  which assigns positive mass to both ! C and !  . If we let S D Œ0; 1, then a measure PQ belongs to PS if and only if PQ Œ ¹! C º  2 .0; 12 . Thus, for any positive

413

Section 9.2 Uniform Doob decomposition

initial value U0 , the process defined by U1 .!  / WD 0 and U1 .! C / WD 2U0 is a PS supermartingale. If U can be decomposed according to (9.12), then we must be able to write 2U0 D U1 .! C / D U0 C   .S11 .! C /  S01 .! C //  B1 .! C / for some B1 .! C /  0. This requirement is equivalent to U0  =2. Hence the decomposition (9.12) fails for U0 > 1=2. } The reason for the failure of the decomposition (9.12) for certain PS -supermartingales is that PS does not reflect the full structure of S; the definition of PS depends only on the cone ¹  j > 0;  2 Sº generated by S. In the approach we are going to present here, the structure of S will be reflected by a stochastic process which we associate to any measure Q P . Definition 9.17. For a measure Q P , the upper variation process for S is the increasing process AQ defined by Q

Q

A0 WD 0 and

Q

A tC1  A t WD ess supΠtC1  .EQ ΠX tC1 j F t   X t / 2S

for t D 0; : : : ; T  1. By QS we denote the set of all Q  P such that Q

EQ ΠAT  < 1 and such that EQ ΠjX tC1  X t j j F t  < 1

P -a.s. for all t .

Clearly, the upper variation process of any measure Q  P satisfies Q

Q

A tC1  A t D ess supΠtC1  .EQ ΠX tC1 j F t   X t /; 2S 1

where S 1 are the bounded processes in S. Hence for Q 2 QS and  2 S 1 , the condition EQ ΠjX tC1  X t j j F t  < 1 guarantees that  tC1  .EQ ΠX tC1 j F t   X t / D EQ Π tC1  .X tC1  X t / j F t ; and it follows that Q

Q

A tC1  A t D ess sup EQ Π tC1  .X tC1  X t / j F t  for Q 2 QS .

(9.13)

2S 1

In particular, we have Q

Q AP T D 0 for P 2 PS ,

(9.14)

P  PS  QS :

(9.15)

which implies the inclusion

414

Chapter 9 Hedging under constraints

Proposition 9.18. If Q 2 QS , and V is the value process of a trading strategy in S, then V  AQ is a local Q-supermartingale. Proof. Let V be the value process of  D . 0 ; / 2 S. Denote by n .!/ the first time t at which j tC1 .!/j > n or EQ Œ jX tC1  X t j j F t .!/ > n: If such a t does not exist, let n .!/ WD T . Then n is a stopping time. Since n jV tC1  V tn j  I¹n t C1º j tC1 j  jX tC1  X t j 2 L1 .Q/;

V tn belongs to L1 .Q/, and n EQ Œ V tC1  V tn j F t  D I¹n t C1º  tC1  .EQ Œ X tC1 j F t   X t / n  .AQ /tC1  .AQ /t n :

This proves that V n  .AQ /n is a Q-supermartingale. Let us identify the class QS in some special cases. Remark 9.19. If S 1 consists of all bounded predictable processes  with non-negative components, then QS D PS . To prove this, take Q 2 QS , and note first that AQ  0, due to (9.13) and the fact that S is a cone. Thus, value processes of strategies in S are local Q-supermartingales by Proposition 9.18. By taking  2 S such that j  ti  1 and  t  0 for j ¤ i , we get that X i is a local Q-supermartingale, and Proposition 9.6 implies that X i is a Q-supermartingale. In particular, X ti is Q-integrable, and we conclude Q 2 PS . } Remark 9.20. If S 1 consists of all bounded predictable processes , then QS D P . This follows by combining Remarks 9.8 and 9.19. } Example 9.21. Suppose our market model contains just one risky asset, and S consists of all predictable processes  such that at  t  bt

P -a.s. for all t ,

where a and b are two given predictable processes with 1 < a t  0  b t < 1 P -a.s. If we assume in addition that EΠjX tC1  X t j j F t  > 0 P -a.s., then the nonredundance condition (9.1) holds, and S satisfies the assumptions (a) through (d) stated at the beginning of this chapter. If Q  P is any probability measure such that EQ ΠjX tC1  X t j j F t  < 1

P -a.s. for all t ,

(9.16)

415

Section 9.2 Uniform Doob decomposition

then ess sup EQ Π tC1  .X tC1  X t / j F t  2S 1

D b tC1 .EQ Œ X tC1  X t j F t /C  a tC1 .EQ Œ X tC1  X t j F t / ˛

  EQ Z t1 ess sup EQ Π t  .X t  X t1 / j F t1  2S Q

Q

D EQ ŒZ t1 .A t  A t1 / : Thus, we cannot have Q

Q

EQ ΠU t  U t1 j F t1   A t  A t1

P -a.s.,

so U  AQ cannot be a Q-supermartingale, in contradiction to our hypothesis (a).

9.3

Upper Snell envelopes

From now on, we assume the condition PS ¤ ;. Let H be a discounted American claim. Our goal is to construct a superhedging strategy for H that belongs to our class S of admissible strategies. The uniform Doob decomposition suggests that we should

418

Chapter 9 Hedging under constraints

find an adapted process U  H such that U  AQ is a Q-supermartingale for each Q 2 QS . If we consider only one such Q, then the minimal process U which satisfies these requirements is given by UQ Q C AQ , where Q UQ t WD ess sup EQ ΠH  AQ  j F t  t D 0; : : : ; T;

(9.22)

2T t

is the Snell envelope of H  AQ with respect to Q. Thus, one may guess that Q

Q

ess sup.UQ t C A t /;

t D 0; : : : ; T;

Q2QS

is the minimal process U which dominates H and for which U  AQ is a Q-supermartingale for each Q 2 QS . Let us assume that Q sup UQ 0 D sup sup EQ ΠH  AQ   < 1:

Q2QS

Q2QS 2T

Note that this condition holds if H is bounded. Definition 9.23. The process " Q Q UQ t WD ess sup.A t C UQ t / Q2QS

Q  D ess sup A t C ess sup EQ ΠH  AQ  j Ft  ; Q2QS

t D 0; : : : ; T;

2T t

will be called the upper QS -Snell envelope of H . The main result of this section confirms our guess that UQ " is the process we are looking for. Theorem 9.24. The upper QS -Snell envelope of H is the smallest process U  H such that U  AQ is a Q-supermartingale for each Q 2 QS . For a European claim, we have the following additional result. Proposition 9.25. For a discounted European claim H E with Q

sup EQ ΠH E  AT  < 1;

Q2QS

the upper QS -Snell envelope takes the form " Q Q UQ t D ess sup.EQ ΠH E  AT j F t  C A t /; Q2QS

t D 0; : : : ; T:

419

Section 9.3 Upper Snell envelopes

Proposition 9.25 will follow from Lemma 9.30 below. The next result provides a scheme for the recursive calculation of UQ " . It will be used in the proof of Theorem 9.24. Proposition 9.26. For fixed Q0 2 QS let Q t .Q0 / denote the set of all Q 2 QS which coincide with Q0 on F t . Then UQ " satisfies the following recursion formula: " Q Q " Q UQ t  A t 0 D .H t  A t 0 / _ ess sup EQ ΠUQ tC1  A tC1 j F t ; t D 0; : : : ; T  1: Q2Q t .Q0 /

The proofs of this proposition and of Theorem 9.24 will be given at the end of this section. Let us recall the following concepts from Section 6.4. The pasting of two probability measures Q1  Q2 in a stopping time  2 T D ¹ j is a stopping time  T º is the probability measure Q A  D EQ1 Œ Q2 Œ A j F  ; QŒ

A2F:

It was shown in Lemma 6.40 that, for all stopping times and FT -measurable Y  0, EQQ ΠY j F  D EQ1 ΠEQ2 ΠY j F_  j F :

(9.23)

Recall also that a set Q of equivalent probability measures on .; F / is called stable if for any pair Q1 ; Q2 2 Q and all  2 T the corresponding pasting also belongs to Q. A technical inconvenience arises from the fact that our set QS may not be stable. We must introduce a further condition on  which guarantees that the pasting of Q1 ; Q2 2 QS in  also belongs to QS . Lemma 9.27. For  2 T , the pasting QQ of Q1 ; Q2 2 QS in  satisfies EQQ ΠjX tC1  X t j j F t  < 1 P -a.s., and its upper variation process is given by Q Q

Q

Q2

1 A t D A t^ C .A t

2  AQ  /  I¹t º C EQ2 Œ jX tC1  X t j j F t I¹t º ; and each of the two conditional expectations is finite almost surely.

(9.24)

420

Chapter 9 Hedging under constraints

Q Q As above, (9.23) yields Now we will compute the upper variation process AQ of Q.

EQQ Œ  t  .X tC1  X t / j F t  D EQ1 Œ  t  .X tC1  X t / j F t I¹>t º C EQ2 Œ  t  .X tC1  X t / j F t I¹t º : Taking the essential supremum over  t 2 S t1 gives Q Q

Q Q

Q

Q

Q

Q

1 2 A tC1  A t D .A tC1  A t 1 /I¹>t º C .A tC1  A t 2 /I¹t º ;

Q Q

and from this our formula for A t follows. In a final step, we show that QQ belongs to QS under condition (9.24). We must Q Q

show that EQQ ΠAT  < 1. Let Z t denote the density of Q2 with respect to Q1 on F t . Q Q

Then, by our formula for AT , Q Q

Q

Q2 2 1 EQQ ΠAT  D EQQ ΠAQ  C AT  A  Q

Q2 2 1 D EQ1 ΠAQ  C EQ2 ΠAT  A j F   i h 1 Q Q 2 EQ1 Π.AT 2  AQ / Z j F   EQ1 ΠAT 1  C EQ1  T  Z 1 Q Q  EQ1 ΠAT 1  C EQ2 ΠAT 2 ; "

which is finite for Q1 ; Q2 2 QS . Lemma 9.28. Suppose we are given Q1 ; Q2 2 QS , a stopping time  2 T , and a set B 2 F such that dQ2 =dQ1 jF  " a.s. on B. Let QQ be the pasting of Q1 and Q2 in the stopping time WD  IB C T IB c : Then QQ 2 QS , and the Snell envelopes associated with these three measures by (9.22) are related as follows: Q Q 1 2 Q Q1 C AQ Q Q2 C AQ UQ Q C AQ  D .U  /  IB c C .U  /  IB

P -a.s.

(9.25)

Proof. We have dQ2 =dQ1 jF  ", hence QQ 2 QS follows from Lemma 9.27. Let Q  be a stopping time in the set T of all stopping times   . The formula for AQ in Lemma 9.27 yields Q

Q1 Q1 Q1 Q2 Q2 AQ D A C .A  A /IB c C .A  A /IB :

Moreover, (9.23) implies that EQQ ΠY j F  D EQ1 ΠY j F IB c C EQ2 ΠY j F IB

421

Section 9.3 Upper Snell envelopes

for all random variables Y such that all conditional expectations make sense. Hence, Q

Q

Q EQQ ΠH  AQ j F  C A Q1 Q2 Q2 1 D .EQ1 ΠH  AQ j F  C A / IB c C .EQ2 ΠH  A j F  C A / IB :

Whenever 1 ; 2 are stopping times in T , then  WD 1 IB c C 2 IB is also a stopping time in T . Conversely, every  2 T can be written in that way for stopping times 1 and 2 . Thus, taking the essential supremum over all 1 and 2 and applying Proposition 6.36 yields (9.25). Q Q

Q

Qk

In fact, we have A D A 1 in (9.25), as we will have A lemma.

Q

D A 0 in the following

Lemma 9.29. For any Q0 2 QS ,  2 T , and ı > 0, there exist a set ƒı 2 F such that Q0 Œ ƒı   1  ı and measures Qk 2 QS such that Qk D Q0 on F and k Q" % ess sup.UQ Q C AQ UQ Qk C AQ  / D U 

P -a.s. on ƒı :

Q2QS

Proof. By Theorem A.33 and its proof, there exists a sequence .Qn0 /  QS such that 0

0

Q Q lim max.UQ  n C A n / D UQ "

k"1 nk

P -a.s.

We will recursively define measures Qk 2 QS and sets ƒkı 2 F such that ƒkı  ƒık1 , Q0 Œ ƒkı   1  .1  2k /ı, and 0

0

Q Q k UQ Qk C AQ D max.UQ  n C A n /  nk

P -a.s. on ƒkı .

T By letting ƒı WD k ƒkı , this will imply the first part of the assertion. We start this recursion in k D 0 by taking Q0 and ƒ0ı WD . 0 implies that there exists some For Qk given, the equivalence of Qk and QkC1 " > 0 such that the set ˇ ² ³ 0 ˇ dQkC1 ˇ D WD  " 2 F dQk ˇF satisfies Q0 Œ D   1  2.kC1/ ı. Thus, ƒkC1 WD ƒkı \ D satisfies Q0 Œ ƒkC1  ı ı .kC1/ 1  .1  2 /ı. We now define a set 0 QkC1

B WD ¹UQ 

0 QkC1

C A

k > UQ Qk C AQ  º \ D;

422

Chapter 9 Hedging under constraints

0 and consider the pasting QkC1 of Qk and QkC1 in the stopping time WD  IB C T IB c . By Lemma 9.28, QkC1 2 QS and 0

0

Q Q kC1 k Q QkC1 CAQ UQ  kC1 CA kC1 D .UQ Qk CAQ /  IB   /  IB c C.U 0 QkC1

k Q D .UQ Qk CAQ  / _ .U 0 Qn

D max .UQ  nkC1

0 Qn

CA

0 QkC1

CA

/

P -a.s. P -a.s. on D P -a.s. on ƒkC1 . ı

/

Now we can proceed to proving the main results in this section. Proof of Proposition 9:26. For Q0 2 QS and t 2 ¹t; : : : ; T º, Q t .Q0 / denotes the set of all Q 2 QS which coincide with Q0 on F t . By Lemma 9.29 and by the definition of UQ Q as the Snell envelope of H  AQ , " Q Q UQ t  A t 0 D ess sup UQ t Q2Q t .Q0 /

Q Q D ess sup ..H t  A t / _ EQ ΠUQ tC1 j F t / Q2Q t .Q0 /

Q

Q

D .H t  A t 0 / _ ess sup EQ ΠUQ tC1 j F t : Q2Q t .Q0 /

Q

"

Q

Since UQ tC1  UQ tC1  A tC1 , we get " Q Q " Q UQ t  A t 0  .H t  A t 0 / _ ess sup EQ ΠUQ tC1  A tC1 j F t :

(9.26)

Q2Q t .Q0 /

For the proof of the converse inequality, let us fix an arbitrary Q 2 Q t .Q0 /. For any ı > 0, Lemma 9.29 yields a set ƒı 2 F tC1 with measure QŒ ƒı   1  ı and Qk " Q Qk 2 Q tC1 .Q/ such that UQ tC1 % UQ tC1  A tC1 P -a.s. on ƒı . Since Qk coincides with Q on F tC1 , we have P -a.s. on ƒı "

Q

Q

Q

k k EQ ΠUQ tC1  A tC1 j F t  D lim EQ ΠUQ tC1 j F t  D lim EQk ΠUQ tC1 j Ft 

k"1

k"1

Qk

 lim sup UQ t k"1



ess sup Q Q2Q t C1 .Q/

Q Q UQ t

Q Q " Q  ess sup UQ t D UQ t  A t Q Q2Q t .Q/

" Q D UQ t  A t 0 : "

By taking ı # 0 and by recalling UQ t  H t , we arrive at the converse of the inequality (9.26).

423

Section 9.3 Upper Snell envelopes

Proof of Theorem 9:24. Since Q0 2 QS is obviously contained in Q t .Q0 /, the recursion formula of Proposition 9.26 yields " Q Q " Q0 " Q0 UQ t  A t 0  .H t  A t 0 / _ EQ0 ΠUQ tC1  A tC1 j F t   EQ0 ΠUQ tC1  A tC1 j F t ; "

i.e., UQ t  AQ0 is indeed a Q0 -supermartingale for each Q0 2 QS . We also know that UQ " dominates H . Let U be any process which dominates H and for which U  AQ is a Q-supermartingale for each Q 2 QS . For fixed Q, the Q-supermartingale U  AQ dominates H AQ and hence also UQ Q , since UQ Q is the smallest Q-supermartingale dominating H  AQ by Proposition 6.10. It follows that Q

Q

"

U t  ess sup.UQ t C A t / D UQ t

P -a.s. for all t .

Q2QS

Proposition 9.25 we will be implied by taking    T in the following lemma. Lemma 9.30. Let H be a discounted American claim whose payoff is zero if it is not exercised at a given stopping time   2 T , i.e., H t .!/ D 0 if t ¤   .!/. Then its upper QS -Snell envelope is given by " Q Q UQ t D I¹  t º ess sup.EQ Œ H   A  j F t  C A t /;

t D 0; : : : ; T:

Q2QS

Proof. By definition, " Q UQ t D ess sup ess sup.EQ ΠH  AQ  j F t  C A t /: 2T t

Q2QS

Since each process AQ is increasing, it is clearly optimal to take  D t on ¹   t º. Hence, ´ 0 on ¹  < t º, " UQ t D H t on ¹  D t º. So we have to show that choosing     is optimal on ¹  > t º. If 2 T t is a stopping time with P Œ >    > 0, then  WD ^   is as least as good as , since each process AQ is increasing. So it remains to exclude the case that there exists a stopping time 2 T t with    on ¹  > t º and P Œ <    > 0, such that yields a strictly better result than   . In this case, there exists some Q1 2 QS such that Q1

1 EQ1 ΠH  AQ  j Ft  C At

Q

Q

> ess sup.EQ ΠH   A  j F t  C A t / Q2QS

(9.27)

424

Chapter 9 Hedging under constraints

with strictly positive probability on ¹  > t º. Take any PQ 2 PS and " > 0, and define ² Q ˇ ³ d P ˇˇ B" WD " : dQ1 ˇF Now let Q" be the pasting Q1 and PQ in the stopping time IB" C T IB c . According "

Q"

to Lemmas 9.27 and 9.28, Q" 2 QS , and its upper variation process satisfies A t Q Q" Q A t 1 and A D A 1 as well as Q"

Q

1 A  D A 1 IB c C AQ  IB" "

D

P -a.s. on ¹  > t º.

By using our assumption that H  H  , we get Q1

1 EQ1 ΠH  AQ  j Ft  C At

Q"

Q"

 EQ" ΠH   A  j F t  C A t

P -a.s. on B" .

By letting " # 0, the P -measure of B" becomes arbitrarily close to 1, and we arrive at a contradiction to (9.27).

9.4

Superhedging and risk measures

Let H be a discounted American claim such that " Q UQ 0 D sup UQ 0 D sup sup EQ ΠH  AQ   < 1: Q2QS

Q2QS 2T

Our aim in this section is to construct superhedging strategies for H which belong to our set S of admissible trading strategies. Recall that a superhedging strategy for H is any self-financing trading strategy whose value process dominates H . If applied " with t D 0, the following theorem shows that UQ 0 is the minimal amount for which a superhedging strategy is available. Q " .H / the set of all F t -measurable random variables U t  0 for which Denote by U t there exists some  2 S such that Ut C

u X

k  .Xk  Xk1 /  Hu

for all u  t P -a.s.

(9.28)

kDtC1

" Theorem 9.31. The upper QS -Snell envelope UQ t of H is the minimal element of " Q .H /. More precisely U t " Q " .H /, (a) UQ t 2 U t " Q " .H /. (b) UQ t D ess inf U t

425

Section 9.4 Superhedging and risk measures

Proof. The uniform Doob decomposition in Section 9.2 combined with Theorem 9.24 yields an increasing adapted process B and some  2 S such that " UQ u" D UQ t C

u X

k  .Xk  Xk1 / C B t  Bu

P -a.s. for u  t .

kDtC1

So the fact that UQ " dominates H proves (a). " Q " .H / from (a). For the proof of the As to part (b), we first get UQ t  ess inf U t Q " .H / and choose a predictable process  2 S for converse inequality, take U t 2 U t " which (9.28) holds. We must show that the set B WD ¹UQ t  U t º satisfies P Œ B  D 1. Let " " UO t WD UQ t ^ U t D UQ t  IB C U t  IB c : " " Then UO t  UQ t , and our claim will follow if we can show that U t  UO t . Let  denote the predictable process obtained from the uniform Doob decomposition of the P -supermartingale UQ " , and define ´ s if s  t , Os WD s  IB C s  IB c if s > t .

With this choice, O 2 S by predictable convexity, and UO t satisfies (9.28), i.e., UO t 2 Q " .H /. Let U t s X " VOs WD UQ 0 C Ok  .Xk  Xk1 /: kD1

Then VOs  Hs  0 for all s, and so VO  AQ is a Q-supermartingale for each Q 2 QS by Propositions 9.18 and 9.6. Hence, " Q UQ t D ess sup ess sup EQ ΠH  AQ  C At j Ft  Q2QS

2T t

 h i X Q ˇˇ Ok  .Xk  Xk1 /  AQ  ess sup EQ UO t C C A F t t  Q; 

kDtC1

 UO t : " Q " .H /. This proves UQ t  ess inf U t

For European claims, the upper QS -Snell envelope takes the form " Q Q UQ t D ess sup.EQ ΠH E  AT j F t  C A t /; Q2QS

t D 0; : : : ; T:

426

Chapter 9 Hedging under constraints

By taking t D 0, it follows that " Q UQ 0 D sup .EQ ΠH E   EQ ΠAT /

(9.29)

Q2QS

is the smallest initial investment which suffices for superhedging the claim H E . In fact, the formula above can be regarded as a special case of the representation theorem for convex risk measures in our financial market model. This will be explained next. Let us take L1 WD L1 .; F ; P / as the space of all financial positions. A position Y 2 L1 will be regarded as acceptable if it can be hedged with a strategy in S at no additional cost. Thus, we introduce the acceptance set T ° ± X ˇ AS WD Y 2 L1 ˇ 9  2 S W Y C  t  .X t  X t1 /  0 P -a.s. : tD1

Due to the convexity of S, this set AS is convex, and under the mild condition inf¹m 2 R j m 2 AS º > 1;

(9.30)

AS induces a convex risk measure S WD AS via S .Y / WD inf¹m 2 R j m C Y 2 AS ºI see Section 4.1. Note that condition (9.30) holds in particular if S does not contain arbitrage opportunities. In this case, we have in fact S .0/ D inf¹m 2 R j m 2 AS º D 0; i.e., S is normalized. The main results of this chapter can be restated in terms of S : Corollary 9.32. Under condition (9.3), the following conditions are equivalent: (a) S is sensitive. (b) S contains no arbitrage opportunities. (c) PS ¤ ;. If these equivalent conditions hold, then Q

S .Y / D sup .EQ ΠY   EQ ΠAT /;

Y 2 L1 :

Q2QS

In other words, S can be represented in terms of the penalty function ´ Q EQ Œ AT  if Q 2 QS , ˛.Q/ D C1 otherwise.

(9.31)

Section 9.4 Superhedging and risk measures

427

Proof. That (a) implies (b) is obvious. The equivalence between (b) and (c) was shown in Theorem 9.9. Since both sides of (9.31) are cash invariant, it suffices to prove (9.31) for Y  0. But then the representation for S is just a special case of the superQ X , hedging duality (9.29). Finally, (9.31) and (c) imply that S .X /  supPQ 2PS EΠand the sensitivity of S follows.

Chapter 10

Minimizing the hedging error

In this chapter, we present an alternative approach to the problem of hedging in an incomplete market model. Instead of controlling the downside risk, we simply aim at minimizing the quadratic hedging error. We begin with a local version of the minimization problem, which may be viewed as a sequential regression procedure. Its solution involves an orthogonal decomposition of a given contingent claim; this extends a classical decomposition theorem for martingales known as the Kunita–Watanabe decomposition. Often, the value process generated by a locally risk-minimizing strategy can be described as the martingale of conditional expectations of the given contingent claim for a special choice of an equivalent martingale measure. Such “minimal” martingale measures will be studied in Section 10.2. In Section 10.3, we investigate the connection between local risk minimization and the problem of variance-optimal hedging where one tries to minimize the global quadratic hedging error. The local and the global versions coincide if the underlying measure is itself a martingale measure.

10.1

Local quadratic risk

In this section, we no longer restrict our discussion to strategies which are selffinancing. Instead, we admit the possibility that the value of a position is readjusted at the end of each period by an additional investment in the numéraire asset. This means that, in addition to the initial investment at time t D 0, we allow for a cash flow throughout the trading periods up to the final time T . In particular, it will now be possible to replicate any given European claim, simply by matching the difference between the payoff of the claim and the value generated by the preceding strategy with a final transfer at time T .

Definition 10.1. A generalized trading strategy is a pair of two stochastic process . 0 ; / such that  0 D . t0 / tD0;:::;T is adapted, and such that  D . t / tD1;:::;T is a d -dimensional predictable process. The (discounted) value process V of . 0 ; / is defined as V0 WD 00

and

V t WD  t0 C  t  X t

for t  1.

For such a generalized trading strategy . 0 ; /, the gains and losses accumulated up

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Section 10.1 Local quadratic risk

to time t by investing into the risky assets are given by the sum t X

k  .Xk  Xk1 /:

kD1

The value process V takes the form V0 D

10

C  1  X0

V t D V0 C

and

t X

k  .Xk  Xk1 /;

t D 1; : : : ; T;

kD1

if and only if  D . t0 ;  t / tD1;:::;T is a self-financing trading strategy with initial investment V0 D 00 D 10 C 1  X0 . In this case, . t0 / tD1;:::;T is a predictable process. In general, however, the difference Vt 

t X

k  .Xk  Xk1 /

kD1

is now non-trivial, and it can be interpreted as the cumulative cost up to time t . This motivates the following definition. Definition 10.2. The gains process G of a generalized trading strategy . 0 ; / is given by t X G0 WD 0 and G t WD k  .Xk  Xk1 /; t D 1; : : : ; T: kD1

The cost process C of

. 0 ; /

is defined by the difference

C t WD V t  G t ;

t D 0; : : : ; T;

of the value process V and the gains process G. In this and in the following sections, we will measure the risk of a strategy in terms of quadratic criteria for the hedging error, based on the “objective” measure P . Our aim will be to minimize such criteria within the class of those generalized strategies . 0 ; / which replicate a given discounted European claim H in the sense that their value process satisfies VT D H P -a.s. The claim H will be fixed for the remainder of this section. As usual we assume that the -field F0 is trivial, i.e., F0 D ¹;; º. In contrast to the previous sections of Part II, however, our approach does not exclude a priori the existence of arbitrage opportunities, even though the interesting cases will be those in which there exist equivalent martingale measures. Since our approach is based on L2 -techniques, another set of hypotheses is needed.

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Chapter 10 Minimizing the hedging error

Assumption 10.3. Throughout this section, we assume that the discounted claim H and the discounted price process X of the risky assets are both square-integrable with respect to the objective measure P (a) H 2 L2 .; FT ; P / DW L2 .P /. (b) X t 2 L2 .; F t ; P I Rd / for all t . In addition to these assumptions, the quadratic optimality criteria we have in mind require the following integrability conditions for strategies. Definition 10.4. An L2 -admissible strategy for H is a generalized trading strategy . 0 ; / whose value process V satisfies VT D H P -a.s.

and

V t 2 L2 .P / for each t ,

and whose gains process G is such that G t 2 L2 .P /

for each t .

We can now introduce the local version of a quadratic criterion for the hedging error of an L2 -admissible strategy. Definition 10.5. The local risk process of an L2 -admissible strategy . 0 ; / is the process 0 2 Rloc t . ; / WD EΠ.C tC1  C t / j F t ;

t D 0; : : : ; T  1.

O is called a locally risk-minimizing strategy if, for An L2 -admissible strategy .O 0 ; / all t , loc 0 O0 O Rloc t . ; /  R t . ; / P -a.s. 0 for each L2 -admissible strategy . 0 ; / whose value process satisfies V tC1 D OtC1 C O tC1  X tC1 D VOtC1 .

Remark 10.6. The reason for fixing the value V tC1 D VOtC1 in the preceding definition becomes clear when we try to construct a locally risk-minimizing strategy O backwards in time. At time T , we want to construct O 0 ; O 0 ; OT 1 ; OT as .O 0 ; / T 1 T a minimizer for the local risk RTloc1 . 0 ; /. Since the terminal value of every L2 admissible strategy must be equal to H , this minimization requires the side condition T0 C T  XT D H D VOT . As we will see in the proof of Theorem 10.9 below, miniO completely determines O 0 and OT and VOT 1 , but one is still free mality of RTloc1 .O 0 ; / T 0 to choose OT 1 and OT 1 among all T0 1 ; T 1 with T0 1 C T 1  XT 1 D VOT 1 . In the next step, it is therefore natural to minimize RTloc2 . 0 ; / under the condition that VT 1 is equal to the value VOT 1 obtained from the preceding step. Moreover, the problem will now be of the same type as the previous one. }

431

Section 10.1 Local quadratic risk

Although locally risk-minimizing strategies are generally not self-financing, it will turn out that they are “self-financing on average” in the following sense: Definition 10.7. An L2 -admissible strategy is called mean self-financing if its cost process C is a P -martingale, i.e., if EŒ C tC1  C t j F t  D 0

P -a.s. for all t .

In order to formulate conditions for the existence of a locally risk-minimizing strategy, let us first introduce some notation. The conditional covariance of two random variables W and Z with respect to P is defined as cov.W; Z j F t / WD EΠW Z j F t   EΠW j F t  EΠZ j F t  provided that the conditional expectations and their difference make sense. Similarly, we define the conditional variance of W under P is var. W j F t / D EΠW 2 j F t   EΠW j F t 2 D cov.W; W j F t /I see also Exercise 5.2.6. Definition 10.8. Two adapted processes U and Y are called strongly orthogonal with respect to P if the conditional covariances cov. U tC1  U t ; Y tC1  Y t j F t /;

t D 0; : : : ; T  1,

are well-defined and vanish P -almost surely. When we consider the strong orthogonality of two processes U and Y in the sequel, then usually one of them will be a P -martingale. In this case, their conditional covariance reduces to cov. U tC1  U t ; Y tC1  Y t j F t / D EΠ.U tC1  U t /.Y tC1  Y t / j F t : After these preparations, we are now ready to state our first result, namely the following characterization of locally risk-minimizing strategies. Theorem 10.9. An L2 -admissible strategy is locally risk-minimizing if and only if it is mean self-financing and its cost process is strongly orthogonal to X . Proof. The local risk process of any L2 -admissible strategy . 0 ; / can be expressed as a sum of two non-negative terms 0 2 Rloc t . ; / D var. C tC1  C t j F t / C EΠC tC1  C t j F t  :

432

Chapter 10 Minimizing the hedging error

Since the conditional variance does not change if we add F t -measurable random variables to its argument, the first term on the right-hand side takes the form var. C tC1  C t j F t / D var. V tC1   tC1  .X tC1  X t / j F t /:

(10.1)

The second term satisfies EΠC tC1  C t j F t 2 D .EΠV tC1 j F t    tC1  EΠX tC1  X t j F t   V t /2 : (10.2) In a second step, we fix t and V tC1 , and we consider  tC1 and V t as parameters. 0 Our purpose is to derive necessary conditions for the minimality of Rloc t . ; / with respect to variations of  tC1 and V t . To this end, note first that it is possible to change the parameters  t0 and  t in such a way that V t takes any given value, that the modified strategy is still an L2 -admissible strategy for H , and that the values of  tC1 and V tC1 remain unchanged. In particular, the value in (10.1) is not affected by such a 0 modification, and so it is necessary for the optimality of Rloc t . ; / that V t minimizes (10.2). This is the case if and only if V t D EΠV tC1 j F t    tC1  EΠX tC1  X t j F t :

(10.3)

The value of (10.1) is independent of V t and a quadratic form in terms of the F t measurable random vector  tC1 . Thus, (10.1) is minimal if and only if  tC1 solves the linear equation 0 D cov.V tC1   tC1  . X tC1  X t /; X tC1  X t j F t /:

(10.4)

Note that (10.3) is equivalent to EΠC tC1  C t j F t  D EΠV tC1   tC1  .X tC1  X t / j F t   V t D 0: Moreover, given (10.3), the condition (10.4) holds if and only if EΠ.C tC1  C t /.X tC1  X t / j F t  D 0 where we have used the fact that the conditional covariance in (10.4) is not changed by subtracting the F t -measurable random variable V t from the first argument. Backward induction on t concludes the proof. The previous proof provides a recipe for a recursive construction of a locally riskminimizing strategy: If V tC1 is already given, minimize EΠ.C tC1  C t /2 j F t  D EΠ.V tC1  .V t C  tC1  .X tC1  X t ///2 j F t  with respect to V t and  tC1 . This is just a conditional version of the standard problem of determining the linear regression of V tC1 on the increment X tC1  X t . Let us now consider the case d D 1;

433

Section 10.1 Local quadratic risk

where our market model contains just one risky asset. Then the following recursive scheme yields formally an explicit solution VOT WD H; cov. VOtC1 ; X tC1  X t j F t / OtC1 WD  I¹ t C1 ¤0º ; 2 tC1

(10.5)

VOt WD EΠVOtC1 j F t   OtC1  EΠX tC1  X t j F t : 2 Here tC1 is a shorthand notation for the conditional variance 2 tC1 WD var. X tC1  X t j F t /:

O whose Defining Ot0 WD VOt  Ot  X t , we obtain a generalized trading strategy .O 0 ; / O terminal portfolio value VT coincides with H . However, an extra condition is needed to conclude that this strategy is indeed L2 -admissible. Proposition 10.10. Consider a market model with a single risky asset and assume that there exists a constant C such that .EΠX t  X t1 j F t1 /2  C  t2

P -a.s. for all t .

(10.6)

O Moreover, Then the recursion (10.5) defines a locally risk-minimizing strategy .O 0 ; /. 0 O O any other locally risk-minimizing strategy coincides with . ; / up to modifications of Ot on the set ¹ t2 D 0º. O is L2 -admissible. To this end, observe that the Proof. We have to show that .O 0 ; / recursion (10.5) and the condition (10.6) imply that EŒ .Ot  .X t  X t1 //2    cov. VOt ; X t  X t1 j F t1 /2 2 DE  EŒ .X t  X t1 / j F t1  I¹ 2 ¤0º t t4   cov. VOt ; X t  X t1 j F t1 /2  .1 C C /  E t2  .1 C C /  EŒ var. VOt j F t1 / : The last expectation is finite if VOt is square-integrable. In this case, Ot  .X t  X t1 / 2 O follows by L2 .P / and in turn VOt1 2 L2 .P /. Hence, L2 -admissibility of .O 0 ; / 0 O O backward induction. The claim that . ; / is locally risk-minimizing as well as the uniqueness assertion follow immediately from the construction.

434

Chapter 10 Minimizing the hedging error

Remark 10.11. The predictable process t X .EΠXs  Xs1 j Fs1 /2 ; var. Xs  Xs1 j Fs1 /

t D 1; : : : ; T;

sD1

is called the mean-variance trade-off process of X , and condition (10.6) is known as the assumption of bounded mean-variance trade-off. Intuitively, it states that the forecast EŒ X t  X t1 j F t1  of the price increment X t  X t1 is of the same order as the corresponding standard deviation t . } Remark 10.12. The assumption of bounded mean-variance trade-off is equivalent to the existence of some ı < 1 such that .EŒ X t  X t1 j F t1 /2  ı  EŒ .X t  X t1 /2 j F t1 

P -a.s. for all t . (10.7)

Indeed, with ˛ t D EŒ X t  X t1 j F t1 , the assumption of bounded mean-variance trade-off is equivalent to ˛ 2t  C.EŒ .X t  X t1 /2 j F t1   ˛ 2t /; which is seen to be equivalent to (10.7) by choosing ı D C =.1 C C /.

}

Example 10.13. Let us consider a market model consisting of a single risky asset S 1 and a riskless bond S t0 D .1 C r/t ; t D 0; : : : ; T; with constant return r > 1. We assume that S01 D 1, and that the returns R t WD

1 S t1  S t1 ; 1 S t1

t D 1; : : : ; T;

of the risky asset are independent and identically distributed random variables in L2 .P /. Under these assumptions, the discounted price process X , defined by Xt D

t Y 1 C Rs ; 1Cr

t D 0; : : : ; T;

sD1

is square-integrable. Denoting by m Q the mean of R t and by Q 2 its variance, we get m Q r ; 1Cr Q 2 2 j F t1 / D X t1  : .1 C r/2

EΠX t  X t1 j F t1  D X t1  var. X t  X t1

Thus, the condition of bounded mean-variance trade-off holds without any further assumptions, and a locally risk-minimizing strategy exists. Moreover, P is a martingale measure if and only if m Q D r. }

435

Section 10.1 Local quadratic risk

Let us return to our general market model with an arbitrary number of risky assets X D .X 1 ; : : : ; X d /: The following result characterizes the existence of locally risk-minimizing strategies in terms of a decomposition of the claim H . Corollary 10.14. There exists a locally risk-minimizing strategy if and only if H admits a decomposition H DcC

T X

 t  .X t  X t1 / C LT

P -a.s.,

(10.8)

tD1

where c is a constant,  is a d -dimensional predictable process such that  t  .X t  X t1 / 2 L2 .P /

for all t ,

and where L is a square integrable P -martingale which is strongly orthogonal to X O is given and satisfies L0 D 0. In this case, the locally risk-minimizing strategy .O 0 ; / by O D  and by the adapted process O 0 defined via O 0 D c and 0

Ot0 D c C

t X

s  .Xs  Xs1 / C L t   t  X t ;

t D 1; : : : ; T:

sD1

Moreover, the decomposition (10.8) is unique in the sense that the constant c and the martingale L are uniquely determined. O is a given locally risk-minimizing strategy with cost process CO , then Proof. If .O 0 ; / O O L t WD C t  C0 is a square-integrable P -martingale which is strongly orthogonal to X by Theorem 10.9. Hence, we obtain a decomposition (10.8). Conversely, if such a O has the cost process CO D c C L, and decomposition exists, then the strategy .O 0 ; / O is locally risk-minimizing. Theorem 10.9 implies that .O 0 ; / To show that L is uniquely determined, suppose that there exists another decompoQ and L. Q Then sition of H in terms of c, Q , Qt D N t WD c  cQ C L t  L

t X

.Qs  s /  .Xs  Xs1 /

sD1

is a square-integrable P -martingale which is strongly orthogonal to X and which can be represented as a “stochastic integral” with respect to X . Strong orthogonality means that 0 D EŒ .Qt   t /  .X t  X t1 /.X t  X t1 / j F t1 :

436

Chapter 10 Minimizing the hedging error

Multiplying this identity with Qt   t gives 0 D EŒ ..Qt   t /  .X t  X t1 //2 j F t1 ; and so N t  N t1 D .Qt   t /  .X t  X t1 / D 0 P -almost surely. In view of LQ 0 D L0 D 0, we thus get LQ D L and in turn cQ D c. A decomposition of the form (10.8) will be called the orthogonal decomposition of the contingent claim H with respect to the process X . If X is itself a P -martingale, then the orthogonal decomposition reduces to the Kunita–Watanabe decomposition, which we will explain next. To this end, we will need some preparation. Lemma 10.15. For two square-integrable martingales M and N , the following two conditions are equivalent: (a) M and N are strongly orthogonal. (b) The product MN is a martingale. Proof. The martingale property of M and N gives EŒ .M tC1  M t /.N tC1  N t / j F t  D EŒ M tC1 N tC1 j F t   M t N t ; and this expression vanishes if and only if MN is a martingale. Let H 2 denote the space of all square-integrable P -martingales. Via the identity M t D EŒ MT j F t , each M 2 H 2 can be identified with its terminal value MT 2 L2 .P /. With the standard identification of random variables which coincide P -a.s., H 2 becomes a Hilbert space isomorphic to L2 .P /, if endowed with the inner product .M; N /H 2 WD EŒ MT NT ;

M; N 2 H 2 :

Recall from Definition 6.14 that, for a stopping time  , the stopped process M  is defined as M t WD M^t ; t D 0; : : : ; T: Definition 10.16. A subspace S of H 2 is called stable if M  2 S for each M 2 S and every stopping time  . Proposition 10.17. For a stable subspace S of H 2 and for L 2 H 2 with L0 D 0, the following conditions are equivalent: (a) L is orthogonal to S, i.e., .L; M /H 2 D 0 for all M 2 S.

437

Section 10.1 Local quadratic risk

(b) L is strongly orthogonal to S, i.e., for each M 2 S EŒ .L tC1  L t /.M tC1  M t / j F t  D 0 P -a.s. for all t . (c) The product LM is a martingale for each M 2 S. Proof. The equivalence of (b) and (c) follows from Lemma 10.15. To prove (a) , (c), we will show that LM is a martingale for fixed M 2 S if and only if .L; M  /H 2 D 0 for all stopping times   T . By the stopping theorem in the form of Proposition 6.36, .L; M  /H 2 D EŒ LT M  D EŒL M : Using the fact that L0 M0 D 0 and applying the stopping theorem in the form of Theorem 6.15, we conclude that .L; M  /H 2 D 0 for all stopping times   T if and only if LM is a martingale. After these preparations, we can now state the existence theorem for the discretetime version of the Kunita–Watanabe decomposition. Theorem 10.18. If the process X is a square-integrable martingale under P , then every martingale M 2 H 2 is of the form M t D M0 C

t X

s  .Xs  Xs1 / C L t

sD1

where  is a d -dimensional predictable process such that  t .X t X t1 / 2 L2 .P / for each t , and where L is a square-integrable P -martingale which is strongly orthogonal to X and satisfies L0 D 0. Moreover, this decomposition is unique in the sense that L is uniquely determined. Proof. Denote by X the set of all d -dimensional predictable processes  such that  t  .X t  X t1 / 2 L2 .P / for each t , and denote by G t ./ WD

t X

s  .Xs  Xs1 /;

t D 0; : : : ; T;

sD1

the “stochastic integral” of  2 X with respect to X . Since for  2 X the process G./ is a square-integrable P -martingale, the set G of all those martingales can be regarded as a linear subspace of the Hilbert space H 2 . In fact, G is a closed subspace of H 2 . To prove this claim, note that the martingale property of G./ implies that .G./; G.//H 2 D EŒ .GT .//2  D

T X tD1

EΠ. t  .X t  X t1 //2 :

438

Chapter 10 Minimizing the hedging error .n/

Thus, if  .n/ is such that G. .n/ / is a Cauchy sequence in H 2 , then  t  .X t  X t1 / is a Cauchy sequence in L2 .P / for each t . Since P is a martingale measure, we .n/ may apply Lemma 1.69 to conclude that any limit point of  t  .X t  X t1 / is of the form  t  .X t  X t1 / for some  t 2 L0 .; F t1 ; P I Rd /. Hence, G is closed in H 2 . Moreover, G is stable. Indeed, if  2 X and  is a stopping time, then Q where G t^ ./ D G t ./ Qs WD s  I¹sº ;

s D 1; : : : ; T:

Furthermore, we have Q 2 X since EŒ. Qt  .X t  X t1 //2   EŒ. t  .X t  X t1 //2  < 1: Since G is closed, the orthogonal projection N of M  M0 onto G is well-defined by standard Hilbert space techniques. The martingale N belongs to G , and the difference L WD M M0 N is orthogonal to G . By Proposition 10.17, L is strongly orthogonal to G and hence strongly orthogonal to X . Therefore, M D M0 CN CL is the desired decomposition of M . The uniqueness of L follows as in the proof of Corollary 10.14. Remark 10.19. In dimension d D 1, the assumption of bounded mean-variance trade-off (10.6) is clearly satisfied if X is a square-integrable P -martingale. Combining Proposition 10.10 with Corollary 10.14 then yields an alternative proof of Theorem 10.18. Moreover, the recursion (10.5) identifies the predictable process  appearing in the Kunita–Watanabe decomposition of a martingale M t D

10.2

EŒ .M t  M t1 /.X t  X t1 / j F t1   I¹E Œ .X t X t 1 /2 jF t 1 ¤0º : EŒ .X t  X t1 /2 j F t1 

}

Minimal martingale measures

If P is itself a martingale measure, Theorem 10.18 combined with Corollary 10.14 yields immediately a solution to our original problem of constructing locally riskminimizing strategies. Corollary 10.20. If P is a martingale measure, then there exists a locally riskminimizing strategy. Moreover, this strategy is unique in the sense that its value process VO is uniquely determined as VOt D EΠH j F t ;

t D 0; : : : ; T;

(10.9)

and that its cost process is given by CO t D VO0 C L t ;

t D 0; : : : ; T;

where L is the strongly orthogonal P -martingale arising in the Kunita–Watanabe decomposition of VO .

439

Section 10.2 Minimal martingale measures

The identity (10.9) allows for a time-consistent interpretation of VOt as an arbitragefree price for H at time t . In the general case in which X is not a martingale under P , one may ask whether there exists an equivalent martingale measure PO such that the value process VO of a locally risk-minimizing strategy can be obtained in a similar manner as the martingale O H j F t ; EŒ

t D 0; : : : ; T:

(10.10)

Definition 10.21. An equivalent martingale measure PO 2 P is called a minimal martingale measure if   O 2  dP E < 1; dP and if every P -martingale M 2 H 2 which is strongly orthogonal to X is also a PO -martingale. The following result shows that a minimal martingale measure provides the desired representation (10.10) – if such a minimal martingale measure exists. Theorem 10.22. If PO is a minimal martingale measure, and if VO is the value process of a locally risk-minimizing strategy, then O H j F t ; VOt D EŒ

t D 0; : : : ; T:

Proof. Denote by H DcC

T X

 t  .X t  X t1 / C LT

tD1

an orthogonal decomposition of H as in Corollary 10.14. Then VO is given by VOt D c C

t X

s  .Xs  Xs1 / C L t :

sD1

The process L is a PO -martingale, because it is a square-integrable P -martingale strongly orthogonal to X . Moreover, s  .Xs  Xs1 / 2 L1 .PO /, because both s  .Xs  Xs1 / and d PO =dP are square-integrable with respect to P . It follows that VO is a PO -martingale. In view of VOT D H , the assertion follows. Our next goal is to derive a characterization of a minimal martingale measure and to use it in order to obtain criteria for its existence. To this end, we have to analyze the effect of an equivalent change of measure on the structure of martingales. The results we will obtain in this direction are of independent interest, and their continuous-time analogues have a wide range of applications in stochastic analysis.

440

Chapter 10 Minimizing the hedging error

Lemma 10.23. Let PQ be a probability measure equivalent to P . An adapted process MQ is a PQ -martingale if and only if the process   Q ˇ dP ˇ MQ t  E F ˇ t ; dP

t D 0; : : : ; T;

is a P -martingale. Proof. Let us denote



 d PQ ˇˇ Z t WD E ˇ Ft : dP

Observe that MQ t 2 L1 .PQ / if and only if MQ t Z t 2 L1 .P /. Moreover, the process Z is P -a.s. strictly positive by the equivalence of PQ and P . Hence, Proposition A.12 yields that Q MQ tC1 j F t  D EŒ MQ tC1 Z tC1 j F t ; Z t  EŒ Q MQ tC1 j F t  D MQ t if and only if EŒ MQ tC1 Z tC1 j F t  D and it follows that EŒ Q Mt Zt . The following representation (10.12) of the density process may be viewed as the discrete-time version of the Doléans–Dade stochastic exponential in continuous-time stochastic calculus. Proposition 10.24. If PQ is a probability measure equivalent to P , then there exists a P -martingale ƒ such that ƒ0 D 1

and

ƒ tC1  ƒ t > 1

P -a.s. for all t ,

(10.11)

and such that the martingale  Z t WD E

 d PQ ˇˇ F ˇ t ; dP

t D 0; : : : ; T;

can be represented as Zt D

t Y

.1 C ƒs  ƒs1 /;

t D 0; : : : ; T:

(10.12)

sD1

Conversely, if ƒ is a P -martingale with (10.11) and such that (10.12) defines a P -martingale Z, then d PQ WD ZT dP defines a probability measure PQ  P .

441

Section 10.2 Minimal martingale measures

Proof. For PQ  P given, define ƒ by ƒ0 D 1 and ƒ tC1 WD ƒ t C

Z tC1  Z t ; Zt

t D 0; : : : ; T  1:

Clearly, (10.12) holds with this choice of ƒ. In particular, ƒ satisfies (10.11), because the equivalence of P and PQ implies that Z t is P -a.s. strictly positive for all t . In the next step, we show by induction on t that ƒ t 2 L1 .P /. For t D 0 this holds by definition. Suppose that ƒ t 2 L1 .P /. Since Z is non-negative, the conditional expectation of Z tC1 =Z t is well-defined and satisfies P -a.s.   Z tC1 ˇˇ 1 E  EŒ Z tC1 j F t  D 1: ˇ Ft D Zt Zt It follows that Z tC1 =Z t 2 L1 .P / and in turn that ƒ tC1 D ƒ t  1 C

Z tC1 2 L1 .P /: Zt

Now it is easy to derive the martingale property of ƒ: Since Z t is strictly positive, we may divide both sides of the equation EŒ Z tC1 j F t  D Z t by Z t , and we arrive at EŒ ƒ tC1  ƒ t j F t  D 0. As for the second assertion, it is clear that EŒ Z t  D Z0 D 1 for all t , provided that ƒ is a P -martingale such that (10.11) holds and such that (10.12) defines a strictly positive P -martingale Z. The following theorem shows how a martingale M is affected by an equivalent change of the underlying probability measure P . Typically, M will no longer be a martingale under the new measure PQ , and so a non-trivial predictable process .A t / tD1;:::;T will appear in the Doob decomposition M D MQ C A of M under PQ . Alternatively, A may be viewed as the predictable process arising in the Doob decomposition of the PQ -martingale MQ under the measure P . The following result, a discrete-time version of the Girsanov formula, describes A in terms of the martingale ƒ arising in the representation (10.12) of the successive densities. Theorem 10.25. Let P and PQ be two equivalent probability measures, and let ƒ denote the P -martingale arising in the representation (10.12) of the successive densities Z t WD EŒ d PQ =dP j F t . If MQ is a PQ -martingale such that MQ t 2 L1 .P / for all t , then t X Q EŒ .ƒs  ƒs1 /.MQ s  MQ s1 / j Fs1  M t WD M t C sD1

is a P -martingale.

442

Chapter 10 Minimizing the hedging error

Proof. Note first that .ƒ t  ƒ t1 /.MQ t  MQ t1 / D

1 .Z t .MQ t  MQ t1 //  .MQ t  MQ t1 / : (10.13) Z t1

According to Lemma 10.23, Z t .MQ t  MQ t1 / is a martingale increment, and hence belongs to L1 .P /. If we let n WD inf¹t j Z t < 1=nº ^ T;

n D 2; 3; : : : ;

it follows that .ƒ t  ƒ t1 /.MQ t  MQ t1 / I¹n t º 2 L1 .P / : In particular, the conditional expectations appearing in the statement of the theorem are P -a.s. well-defined. Moreover, the identity (10.13) implies that P -a.s. on ¹n  t º EŒ MQ t  MQ t1 j F t1  1 EŒ Z t .MQ t  MQ t1 / j F t1   EŒ .ƒ t  ƒ t1 /.MQ t  MQ t1 / j F t1  Z t1 D EŒ .ƒ t  ƒ t1 /.MQ t  MQ t1 / j F t1 :

D

Thus, we have identified the Doob decomposition of MQ under P . The preceding theorem allows us to characterize those equivalent measures P   P which are martingale measures. Let X DY CB

(10.14)

denote the Doob decomposition of X under P , where Y is a d -dimensional P martingale, and .B t / tD1;:::;T is a d -dimensional predictable process. Corollary 10.26. Let P   P be such that E  Œ jX t j  < 1 for each t , and denote by ƒ the P -martingale arising in the representation (10.12) of the successive densities Z t WD EŒ dP  =dP j F t . Then P  is an equivalent martingale measure if and only if the predictable process B in the Doob decomposition (10.14) satisfies Bt D 

t X

EŒ .ƒs  ƒs1 /.Ys  Ys1 / j Fs1 

sD1

D

t X sD1

P -a.s. for t D 1; : : : ; T .

EŒ .ƒs  ƒs1 /.Xs  Xs1 / j Fs1 

443

Section 10.2 Minimal martingale measures

Proof. If P  is an equivalent martingale measure, then our formula for B is an immediate consequence of Theorem 10.25. For the proof of the converse direction, we denote by X D Y  C B the Doob decomposition of X under P  . Then Y  is a P  -martingale. Using Theorem 10.25, we see that YQ  WD Y  C BQ  is a P -martingale where BQ t D

t X

 EŒ .ƒs  ƒs1 /.Ys  Ys1 / j Fs  D B t :

sD1

On the other hand, Y D X  B D Y  C .B   B/ is a P -martingale. It follows that the Doob decomposition of Y  under P is given by Y  D Y C .B  B  /. Hence, Y C .B  B  / D Y  D YQ   BQ  D YQ  C B: The uniqueness of the Doob decomposition implies B   0, so X is a P  -martingale.

We can now return to our initial task of characterizing a minimal martingale measure. Theorem 10.27. Let PO 2 P be an equivalent martingale measure whose density d PO =dP is square-integrable. Then PO is a minimal martingale measure if and only if the P -martingale ƒ of (10.12) admits a representation as a “stochastic integral” with respect to the P -martingale Y arising in the Doob decomposition of X ƒt D 1 C

t X

s  .Ys  Ys1 /;

t D 0; : : : ; T;

(10.15)

sD1

for some d -dimensional predictable process . Proof. To prove sufficiency of (10.15), we have to show that if M 2 H 2 is strongly orthogonal to X , then M is a PO -martingale. By Lemma 10.23 this follows if we can show that M Z is a P -martingale where 

 d PO ˇˇ Z t WD E ˇ Ft : dP Clearly, M t Z t 2 L1 .P / since M and Z are both square-integrable. For the next step, we introduce the stopping times n WD inf¹t  0 j j tC1 j > n º:

444

Chapter 10 Minimizing the hedging error

By stopping the martingale ƒ at n , we obtain the P -martingale ƒn . Since X is square-integrable, an application of Jensen’s inequality yields that EŒ jY t j2  < 1 for all t . In particular, Mƒn is integrable. Furthermore, Lemma 10.15 shows that the strong orthogonality of M and Y implies that M Y is a d -dimensional P -martingale. Hence, n EŒ M tC1 .ƒtC1  ƒt n / j F t 

D I¹t C1n º tC1  .EŒ M tC1 Y tC1 j F t   EŒ M tC1 j F t Y t / D 0: Noting that Z tn

D

t Y

n .1 C ƒsn  ƒs1 /;

sD1

and that Z n is square-integrable, we conclude that n n EŒ M tC1 Z tC1 j F t  D Z tn EŒ M tC1 .1 C ƒtC1  ƒt n / j F t  D Z tn M t :

Thus, Z n M is a P -martingale for each n. By Doob’s stopping theorem, the process .Z n M /n D .ZM /n is also a P -martingale. Since n % T P -a.s. and n n jM tC1 Z tC1 j

T X

jMs Zs j 2 L1 .P /;

sD0

we may apply the dominated convergence theorem for conditional expectations to obtain the desired martingale property of M Z n n EΠM tC1 Z tC1 j F t  D lim EΠM tC1 Z tC1 j F t  D lim M tn Z tn D M t Z t : n"1

n"1

Thus, PO is a minimal martingale measure. For the proof of the converse assertion of the theorem, denote by Zt D 1 C

t X

s  .Ys  Ys1 / C L t

sD1

the Kunita–Watanabe decomposition of the density process Z with respect to the measure P and the square-integrable martingale Y , as explained in Theorem 10.18. The process L is a square-integrable P -martingale strongly orthogonal to Y , and hence to X . Thus, the assumption that PO is a minimal martingale measure implies that L is also a PO -martingale. Applying Lemma 10.23, it follows that Lt Zt D Lt C Lt

t X sD1

s  .Ys  Ys1 / C L2t

445

Section 10.2 Minimal martingale measures

is a P -martingale. According to Lemma 10.15, the strong orthogonality of L and Y yields that t X Lt s  .Ys  Ys1 / sD1

is a P -martingale; recall that s  .Ys  Ys1 / 2 L2 .P / for all s. But then .L2t / must also be a martingale. In particular, the expectation of L2t is independent of t and so EŒ L2t  D L20 D 0 from which we get that L vanishes P -almost surely. Hence, Z is equal to the “stochastic integral” of  with respect to Y , and we conclude that ƒ tC1  ƒ t D

Z tC1  Z t 1 D  tC1  .Y tC1  Y t /; Zt Zt

so that (10.15) holds with t WD  t =Z t1 . Corollary 10.28. There exists at most one minimal martingale measure. Proof. Let PO and PO 0 be two minimal martingale measures, and denote the martingales in the representation (10.12) by ƒ and ƒ0 , respectively. On the one hand, it follows from Corollary 10.26 that the martingale N WD ƒƒ0 is strongly orthogonal to Y . On the other hand, Theorem 10.27 implies that N admits a representation as a “stochastic integral” with respect to the P -martingale Y Nt D

t X

. s  0s /  .Ys  Ys1 /;

t D 0; : : : ; T:

sD1

Let n WD inf¹t j j tC1  0tC1 j > nº, so that N n is in L2 .P /. Then it follows as in the proof of Corollary 10.14 that N n vanishes P -almost surely. Hence the densities of PO and PO 0 coincide. Recall that, for d D 1, we denote by t2 D var. X t  X t1 j F t1 / the conditional variance of the increments of X . Corollary 10.29. In dimension d D 1, the following two conditions are implied by the existence of a minimal martingale measure PO : (a) The predictable process arising in the representation formula (10.15) is of the form

t D

EΠX t  X t1 j F t1  t2

P -a.s. on ¹ t2 ¤ 0º:

(10.16)

446

Chapter 10 Minimizing the hedging error

(b) For each t , P -a.s. on ¹ t2 ¤ 0º, .X t  X t1 /  EŒ X t  X t1 j F t1  < EŒ .X t  X t1 /2 j F t1 : Proof. (a): Denote by X D Y C B the Doob decomposition of X with respect to P . According to Corollary 10.26, the P -martingale ƒ arising in the representation (10.12) of the density d PO =dP must satisfy B t  B t1 D EŒ .ƒ t  ƒ t1 /.Y t  Y t1 / j F t1 : Using that t2 D EŒ .Y t Y t1 /2 j F t1  and that B t B t1 D EŒ X t X t1 j F t1  yields our formula for t . (b): By Proposition 10.24, the P -martingale ƒ must be such that ƒ t  ƒ t1 D t  .Y t  Y t1 / > 1

P -a.s. for all t .

Given (a), this condition is equivalent to (b). Note that condition (b) of Corollary 10.29 is rather restrictive as it imposes an almost-sure bound on the F t -measurable increment X t  X t1 in terms of F t1 measurable quantities. Theorem 10.30. Consider a market model with a single risky asset satisfying condition (b) of Corollary 10:29 and the assumption .EΠX t  X t1 j F t1 /2  C  t2 ;

P -a.s. for all t  1,

of bounded mean-variance trade-off. Then there exists a unique minimal martingale measure PO whose density d PO =dP D ZT is given via (10.16), (10.15), and (10.12). Proof. Denote by X D Y CB the Doob decomposition of X under P . For t defined via (10.16), the assumption of bounded mean-variance trade-off yields that EΠ. t  .Y t  Y t1 //2 j F t1   C

P -a.s.

(10.17)

Hence, ƒ t defined according to (10.15) is a square-integrable P -martingale. As observed in the second part of the proof of Corollary 10.29, its condition (b) holds if and only if ƒ t  ƒ t1 > 1 for all t , so that Z defined by Zt D

t Y

.1 C ƒs  ƒs1 / D

sD1

t Y

.1 C s  .Ys  Ys1 //

sD1

is P -a.s. strictly positive. Moreover, the bound (10.17) guarantees that Z is a squareintegrable P -martingale. We may thus conclude from Proposition 10.24 that Z is

447

Section 10.2 Minimal martingale measures

the density process of a probability measure PO  P with a square-integrable density d PO =dP . In particular, X t is PO -integrable for all t . Our choice of implies that EŒ .ƒ t  ƒ t1 /.Y t  Y t1 / j F t1  D EŒ X t  X t1 j F t1  D .B t  B t1 /; and so PO is an equivalent martingale measure by Corollary 10.26. Finally, Theorem 10.27 states that PO is a minimal martingale measure, while uniqueness was already established in Corollary 10.28. Example 10.31. Let us consider again the market model of Example 10.13 with independent and identically distributed returns R t 2 L2 .P /. We have seen that the condition of bounded mean-variance trade-off is satisfied without further assumptions. Let m Q WD EŒ R1  and Q 2 WD var.R1 /. A short calculation using the formulas for EŒ X t  X t1 j F t1  and var. X t  X t1 j F t1 / obtained in Example 10.13 shows that the crucial condition (b) of Corollary 10.29 is equivalent to .m Q  r/R1 < Q 2 C m. Q m Q  r/

P -a.s.

(10.18)

Hence, (10.18) is equivalent to the existence of the minimal martingale measure. For m Q > r the condition (10.18) is an upper bound on R1 , while we obtain a lower bound for m Q < r. In the case m Q D r, the measure P is itself the minimal martingale measure, and the condition (10.18) is void. If the distribution of R1 is given, and if a  R1  b

P -a.s.

for certain constants a > 1 and b < 1, then (10.18) is satisfied for all r in a certain neighborhood of m. Q } Remark 10.32. The purpose of condition (b) of Corollary 10.29 is to ensure that the density Z defined via Zt D

t Y

.1 C s  .Ys  Ys1 //

sD1

is strictly positive. In cases where this condition is violated, Z may still be a squareintegrable P -martingale and can be regarded as the density of a signed measure d PO D ZT dP; which shares some properties with the minimal martingale measure of Definition 10.21; see, e.g., [247]. } In the remainder of this section, we consider briefly another quadratic criterion for the risk of an L2 -admissible strategy.

448

Chapter 10 Minimizing the hedging error

Definition 10.33. The remaining conditional risk of an L2 -admissible strategy . 0 ; / with cost process C is given by the process 0 2 Rrem t . ; / WD EΠ.CT  C t / j F t ;

t D 0; : : : ; T:

We say that an L2 -admissible strategy . 0 ; / minimizes the remaining conditional risk if 0 rem 0 Rrem P -a.s. t . ; /  R t . ; / for all t and for each L2 -admissible strategy .0 ; / which coincides with . 0 ; / up to time t . The next result shows that minimizing the remaining conditional risk for a martingale measure is the same as minimizing the local risk. In this case, Corollary 10.20 yields formulas for the value process and the cost process of a minimizing strategy. Proposition 10.34. Assume d D 1. For P 2 P , an L2 -admissible strategy minimizes the remaining conditional risk if and only if it is locally risk minimizing. O be a locally risk-minimizing strategy, which exists by Corollary Proof. Let .O 0 ; / 10.20, and write VO and CO for its value and cost processes. Take another L2 -admissible strategy .0 ; / whose value and cost processes are denoted by V and C . Since VOT D H D VT , the cost process C satisfies CT  C t D VT  V t 

T X

k  .Xk  Xk1 /

kDtC1

D VOt  V t C

T X

.Ok  k /  .Xk  Xk1 / C CO T  CO t :

kDtC1

Since X and CO are strongly orthogonal martingales, the remaining conditional risk of .0 ; / satisfies 0 Rrem t . ; /

D .VOt  V t /2 CE

T h X

i ˇ .Ok  k /2 .Xk  Xk1 /2 ˇ F t CEŒ .CO T  CO t /2 j F t ;

kDtC1

and this expression is minimal if and only if V t D VOt and k D Ok for all k  t C 1 P -almost surely. In general, however, an L2 -admissible strategy minimizing the remaining conditional risk does not exist, as will be shown in the following Section 10.3.

Section 10.3 Variance-optimal hedging

10.3

449

Variance-optimal hedging

Let H 2 L2 .P / be a square-integrable discounted claim. Throughout this section, we assume that the discounted price process X of the risky asset is square-integrable with respect to P EŒ jX t j2 < 1 for all t . As in the previous section, there is no need to exclude the existence of arbitrage opportunities, even though the cases of interest will of course be arbitrage-free. Informally, the problem of variance-minimal hedging is to minimize the quadratic hedging error defined as the squared L2 .P /-distance k H  VT k22 D EŒ .H  VT /2  between H and the terminal value of the value process V of a self-financing trading strategy. Remark 10.35. Mean-variance hedging is closely related to the discussion in the preceding sections, where we considered the problems of minimizing the local conditional risk or the remaining conditional risk within the class of L2 -admissible strategies for H . To see this, let . 0 ; / be an L2 -admissible strategy for H in the sense of Definition 10.4, and denote by V , G, and C the resulting value, gains, and cost processes. The quantity R. 0 ; / WD EŒ .CT  C0 /2  may be called the “global quadratic risk” of . 0 ; /. It coincides with the initial value of the process R0rem . 0 ; / of the remaining conditional risk introduced in Definition 10.33. Note that R. 0 ; / D EŒ .H  V0  GT /2  is independent of the values of the numéraire component  0 at the times t D 1; : : : ; T . Thus, the global quadratic risk of the generalized trading strategy . 0 ; / coincides with the quadratic hedging error EŒ .H  VQT /2  where VQ is the value process of the self-financing trading strategy arising from the d -dimensional predictable process  and the initial investment VQ0 D V0 D 00 . } Let us rephrase the problem of mean-variance hedging in a form which can be interpreted both within the class of self-financing trading strategies and within the

450

Chapter 10 Minimizing the hedging error

context of Section 10.1. For a d -dimensional predictable process  we denote by G./ the gains process G t ./ D

t X

s  .Xs  Xs1 /;

t D 0; : : : ; T;

sD1

associated with . Let us introduce the class S WD ¹ j  is predictable and G t ./ 2 L2 .P / for all t º: Definition 10.36. A pair .V0 ;   / where V0 2 R and   2 S is called a varianceoptimal strategy for the discounted claim H if EŒ .H  V0  GT .  //2   EŒ .H  V0  GT .//2  for all V0 2 R and all  2 S. Our first result identifies a variance-optimal strategy in the case P 2 P . O be a locally risk-minimizing Proposition 10.37. Assume that P 2 P , and let .O 0 ; / 2 O L -admissible strategy as constructed in Corollary 10:20. Then .V0 ;   / WD .O00 ; / is a variance-optimal strategy. Proof. Recall from Remark 10.35 that if . 0 ; / is an L2 -admissible strategy for H with value process V , then the expression EŒ .H  V0  GT .//2  is equal to the initial value R0rem . 0 ; / of the remaining conditional risk process of O . 0 ; /. But according to Proposition 10.34, R0rem is minimized by .O 0 ; /. The general case where X is not a P -martingale will be studied under the simplifying assumption that the market model contains only one risky asset. We will first derive a general existence result, and then determine an explicit solution in a special setting. The key idea for showing the existence of a variance-optimal strategy is to minimize the functional S 3  7! EŒ .H  V0  GT .//2 ; first for fixed V0 , and then to vary the parameter V0 . The first step will be accomplished by projecting H  V0 onto the space of “stochastic integrals” GT WD ¹GT ./ j  2 S º: Clearly, GT is a linear subspace of L2 .P /. Thus, we can obtain the optimal  D .V0 / by using the orthogonal projection of H  V0 on GT as soon as we know that GT is

451

Section 10.3 Variance-optimal hedging

closed in L2 .P /. In order to formulate a criterion for the closedness on GT , we denote by t2 WD var. X t  X t1 j F t1 /; t D 1; : : : ; T; the conditional variance, and by ˛ t WD EŒ X t  X t1 j F t1 ;

t D 1; : : : ; T;

the conditional mean of the increments of X . Proposition 10.38. Suppose that d D 1, and assume the condition of bounded meanvariance trade-off ˛ 2t  C  t2 P -a.s. for all t . (10.19) Then GT is a closed linear subspace of L2 .P /. Proof. Let X D M C A be the Doob decomposition of X into a P -martingale M and a process A such that A0 D 0 and .A t / tD1;:::;T is predictable. Since T2 D var. XT  XT 1 j FT 1 / D EŒ .MT  MT 1 /2 j FT 1 ; we get for  2 S EŒGT ./2  D EŒ.GT 1 ./ C T  .XT  XT 1 //2  D EŒ.GT 1 ./ C T  .AT  AT 1 //2  C EŒ.T  .MT  MT 1 //2 

(10.20)

 EŒT2  T2 : Suppose now that  n is a sequence in S such that GT . n / converges in L2 .P / to some Y . Applying the inequality (10.20) to GT . n /  GT . m / D GT . n   m /, we find that  n WD Tn  T is a Cauchy sequence in L2 .P /. Denote  WD limn  n , and let  T WD I¹ T >0º : T By using our assumption (10.19), we obtain EŒ .Tn  .XT  XT 1 /  T  .XT  XT 1 //2  D EŒ .Tn  T /2  EŒ .XT  XT 1 /2 j FT 1   D EŒ .Tn  T /2  . T2 C ˛T2 /   .1 C C / EŒ .Tn T  T T /2  D .1 C C / EŒ . n  /2 : Since the latter term converges to 0, it follows that GT 1 . n / D GT . n /  Tn  .XT  XT 1 / ! Y  T  .XT  XT 1 /

452

Chapter 10 Minimizing the hedging error

in L2 .P /. A backward iteration of this argument yields a predictable process  2 S such that Y D GT ./. Hence, GT is closed in L2 .P /. Theorem 10.39. In dimension d D 1, the condition (10.19) of bounded mean-variance trade-off guarantees the existence of a variance-optimal strategy .V0 ;   /. Such a strategy is P -a.s. unique up to modifications of  t on ¹ t D 0º. Proof. Let p W L2 .P / ! GT denote the orthogonal projection onto the closed subspace GT of the Hilbert space L2 .P /, i.e., p W L2 .P / ! GT is a linear operator such that EŒ .Y  p.Y //2  D min EŒ .Y  Z/2  (10.21) Z2GT

for all Y 2 L2 .P /. For any V0 2 R we choose some .V0 / 2 S such that GT ..V0 // D p.H  V0 /. The identity (10.21) shows that .V0 / minimizes the functional EŒ .H  V0  GT .//2  among all  2 S. Note that V0 7! GT ..V0 // D p.H  V0 / D p.H /  V0  p.1/ is an affine mapping. Hence, EŒ .H  V0  GT ..V0 ///2  is a quadratic function of V0 and there exists a minimizer V0 . For any V0 2 R and  2 S we clearly have EŒ .H  V0  GT .//2   EŒ .H  V0  GT ..V0 ///2   EŒ .H  V0  GT ..V0 ///2 : Hence .V0 ;   / WD .V0 ; .V0 // is a variance-optimal strategy. Uniqueness follows from (10.20) and an induction argument. Under the additional assumption that ˛ 2t is deterministic for each t t2

(10.22)

(here we use the convention 00 WD 0), the variance-optimal strategy .  ; V0 / can be determined explicitly. It turns out that .  ; V0 / is closely related to the locally

453

Section 10.3 Variance-optimal hedging

O for the discounted claim H . Recall from Proposirisk-minimizing strategy .O 0 ; / 0 O O tion 10.10 that . ; / and its value process VO are determined by the following recursion: VOT D H; cov. VOtC1 ; X tC1  X t j F t / OtC1 D  I¹ t C1 ¤0º ; 2 tC1 VOt D EŒ VOtC1 j F t   OtC1  EŒ X tC1  X t j F t I the numéraire component O 0 is given by Ot0 WD VOt  Ot  X t . Theorem 10.40. Under condition (10.22), V0 WD VO0 and  t WD Ot C

˛ 2t

˛t .VOt1  VO0  G t1 .  //; C t2

t D 1; : : : ; T;

defines a variance-optimal strategy .  ; V0 /. Moreover, EΠ.H  V0 GT .  //2  D

T X

 t  EΠ.CO t  CO t1 /2 ;

tD1

where CO denotes the cost process of . O 0 ; O /, and  t is given by T Y

k2

kDtC1

k2 C ˛k2

 t WD

:

Proof. We prove the assertion by induction on T . For T D 1 the problem is just a particular case of Proposition 10.10, which yields 1 D O1 and V0 D VO0 . For T > 1 we use the orthogonal decomposition H D VOT D VO0 C

T X

Ot  .X t  X t1 / C CO T  CO 0 ;

(10.23)

tD1

of the discounted claim H as constructed in Corollary 10.14. Suppose that the assertion is proved for T  1. Let us consider the minimization of T 7! EΠ.VOT  VT 1  T  .XT  XT 1 //2 ;

(10.24)

where VT 1 is any random variable in L2 .; FT 1 ; P /. By (10.23) and Theorem 10.9, we may write VOT as VOT D VOT 1 C OT  .XT  XT 1 / C CO T  CO T 1 ;

454

Chapter 10 Minimizing the hedging error

where CO is a P -martingale strongly orthogonal to X . Thus, EŒ .VOT  VT 1  T  .XT  XT 1 //2  D EŒ .VOT 1  VT 1 C . OT  T /  .XT  XT 1 / C CO T  CO T 1 /2 : The expectation conditional on FT 1 of the integrand on the right-hand side is equal to .VOT 1  VT 1 /2 C 2.VOT 1  VT 1 /. OT  T /  ˛T  C . OT  T /2 . T2 C ˛T2 / C E .CO T  CO T 1 /2 j FT 1 :

(10.25)

This expression is minimized by T .VT 1 / WD OT C

˛T2

˛T .VOT 1  VT 1 /; C T2

(10.26)

which must also be the minimizer in (10.24). The minimal value in (10.25) is given by 1 .VOT 1  VT 1 /2 C EŒ .CO T  CO T 1 /2 j FT 1 : 1 C ˛T2 = T2 Using our assumption (10.22) that ˛T2 = T2 is constant, we can compute the expectation of the latter expression, and we arrive at the following identity: EŒ .VOT  VT 1  T .VT 1 /  .XT  XT 1 //2  D EŒ .VOT 1  VT 1 /2  

T2 T2 C ˛T2

C EΠ.CO T  CO T 1 /2 :

(10.27)

So far, we have not specified VT 1 . Let us now consider the minimization of EΠ.VOT  VT 1  T .VT 1 /  .XT  XT 1 //2  with respect to VT 1 , when VT 1 is of the form VT 1 D V0 C GT 1 ./ for  2 S and V0 2 R. According to our identity (10.27), this problem is equivalent to the minimization of .V0 ; / 7! EΠ.HT 1  V0 GT 1 .//2 ; where HT 1 WD VOT 1 . By the induction hypotheses, this problem is solved by V0 and . t / tD1;:::;T 1 as defined in the assertion. Inserting our formula (10.26) for T .V0 C GT 1 .  // completes the induction argument. Remark 10.41. The martingale property of CO implies EΠ.CO T  CO 0 /2  D

T X tD1

EΠ.CO t  CO t1 /2 :

455

Section 10.3 Variance-optimal hedging

If CO ¥ CO 0 and if 1 < 1, then EŒ .CO T  CO 0 /2  must be strictly larger than the minimal global risk EŒ .H  .V0 C GT .  ///2  D

T X

 t  EΠ.CO t  CO t1 /2 :

}

tD1

Remark 10.42. If follows from Theorem 10.40 as well as from the preceding remark that the component   of a variance-optimal strategy .  ; V0 / will differ from the corresponding component O of a locally risk-minimizing strategy when ˛ t does not vanish for all t , i.e., when P is not a martingale measure. This explains why there may be no strategy which minimizes the remaining conditional risk in the sense of Definition 10.33: For the minimality of R0rem . 0 ; / D EŒ .CT  C0 /2  D EŒ .H  00  GT .//2  we need that  D   , while the minimality of 0 2 loc 0 RTrem 1 . ; / D EŒ .CT  CT 1 / j FT 1  D RT 1 . ; /

requires T D OT . Hence, the two minimality requirements are in general incompatible. }

Chapter 11

Dynamic risk measures

In this chapter we return to the quantification of financial risk in terms of monetary risk measures. As we saw in Chapter 4, a monetary risk measure specifies the capital which is needed in order to make a given financial position acceptable. In our present dynamic setting, the capital requirement at a given time will depend on the available information and will thus become a random variable. In Section 11.1 we introduce the notion of a conditional monetary risk measure. In the convex case we prove a conditional version of the robust representation theorem which involves the conditional expected losses under various models Q and a conditional penalization of these models. As time varies, both the capital requirement and the penalization of a given probabilistic model become stochastic processes. The structure of these processes depends on how the risk assessments at different times are connected with each other. In Section 11.2 we focus on a strong notion of time consistency which amounts to a recursive property of the successive capital requirements. For a typical model Q, time consistency implies that the corresponding penalization process is a supermartingale under Q, and that its Doob decomposition takes a special form. Time consistency is also characterized by a combined supermartingale property of the capital requirements and the penalization process.

11.1

Conditional risk measures and their robust representation

As in the preceding chapters we fix a filtration .F t / tD0;:::;T on our probability space .; F ; P / such that F0 D ¹;; º and FT D F . As the set X of all financial positions 1 we take the space L1 WD L1 .; F ; P /. The subspace L1 t WD L .; F t ; P / consists of those positions whose outcome only depends on the history up to time t . All inequalities and equalities applied to random variables are meant to hold P -a.s. if not stated otherwise. Definition 11.1. A map  t W L1 ! L1 t will be called a monetary conditional risk measure if it satisfies the following properties for all X; Y 2 L1 : 

Conditional cash invariance:  t .X C X t / D  t .X /  X t for any X t 2 L1 t .



Monotonicity: X  Y )  t .X /   t .Y /.



Normalization:  t .0/ D 0.

Section 11.1 Conditional risk measures and their robust representation

457

A monetary conditional risk measure will be called convex if it satisfies 

Conditional convexity:  t . X C .1  /Y /   t .X / C .1  / t .Y / for

2 L1 t with 0   1.

A convex conditional risk measure will be called coherent if it satisfies in addition 

Conditional positive homogeneity:  t . X / D  t .X / for 2 L1 t such that

 0.

For t D 0 we recover our previous definition of (unconditional and normalized) monetary, convex, and coherent risk measures given in Section 4.1, since the space L1 0 reduces to the real line. To any conditional monetary risk measure  t we associate its acceptance set A t WD ¹X 2 L1 j  t .X /  0º:

(11.1)

One easily checks that A t has the following properties: 

X 2 At ; Y  X ) Y 2 At ;



ess inf¹X 2 L1 t j X 2 A t º D 0, and 0 2 A t .

Note that  t is uniquely determined by its acceptance set since  t .X / D ess inf¹Y 2 L1 t j X C Y 2 A t º:

(11.2)

A conditional monetary risk measure can thus be viewed as the conditional capital requirement needed at time t to make a financial position X acceptable at that time. Moreover,  t is a convex conditional risk measure if and only if A t satisfies the additional property of 

Conditional convexity: X; Y 2 A t and 2 L1 t with 0   1 implies

X C .1  /Y 2 A t .

Conversely, one can use acceptance sets to define conditional monetary risk measures: If a given acceptance set A t L1 satisfies the above conditions then the functional  t W L1 ! L1 t defined via (11.2) is a conditional monetary risk measure. Exercise 11.1.1. Let  t W L1 ! L1 t be a conditional monetary risk measure. For X 2 L1 we define kX kF t WD ess inf¹m t 2 L1 t j jX j  m t P -a.s.º: Show that for X; Y 2 L1 j t .X /   t .Y /j  kX  Y kF t :

}

458

Chapter 11 Dynamic risk measures

Exercise 11.1.2. Show that every conditional monetary risk measure  t W L1 ! L1 t has the following property:  t .IA X / D IA  t .X /

for X 2 L1 and A 2 F t .

(11.3)

Then show that (11.3) is equivalent to the following local property: when X; Y 2 L1 and A 2 F t , then  t .IA X C IAc Y / D IA  t .X / C IAc  t .Y /: }

Hint: It can be helpful to use Exercise 11.1.1.

By M1 .P / we denote the set of all probability measures on .; F / that are absolutely continuous with respect to P . As we have shown in Theorem 4.33, an unconditional convex risk measure that has the Fatou property admits a robust representation of the form .X / D sup .EQ Œ X   ˛.Q// (11.4) Q2M1 .P /

with some penalty function ˛ W M1 .P / ! R [ ¹C1º. In fact we can take the minimal penalty function ˛ min .Q/ D

sup

EQ ΠX :

(11.5)

X2L1 ; .X/0

In this section we are going to prove a robust representation for conditional convex risk measures which is analogous to (11.4). Here the penalty function will depend on the history up to time t , and so it will be given by a map ˛ t from (a subset of) M1 .P / to the set of F t -measurable random variables with values in R [ ¹C1º. In analogy to (11.5), we are going to take the minimal penalty function, defined as the worst conditional loss over all positions which are acceptable at time t : ˛ min t .Q/ D ess sup EQ Œ X j F t :

(11.6)

X2A t

The conditional expectations on the right are a priori defined only Q-a.s., but they also need to be well-defined under the reference measure P . This is the case when Q belongs to the set Q t WD ¹Q 2 M1 .P / j Q  P on F t º: Theorem 11.2. For a convex conditional risk measure  t the following properties are equivalent: (a)  t has the representation  t .X / D ess sup.EQ Œ X j F t   ˛ min t .Q//; Q2Q t

(11.7)

Section 11.1 Conditional risk measures and their robust representation

459

where for Q 2 Q t the penalty function ˛ min t .Q/ is given by (11.6). In fact the representation (11.7) also holds if we replace Q t in (11.7) by the smaller set P t WD ¹Q 2 Q t j Q D P on F t º: (b)  t has the following Fatou property:  t .X /  lim inf  t .Xn / n!1

for any bounded sequence .Xn / L1 which converges P -a.s. to X 2 L1 . (c)  t is continuous from above, i.e., Xn & X

H)

 t .Xn / %  t .X /

for any sequence .Xn / L1 and X 2 L1 . Proof. (a) ) (b): Lebesgue’s dominated convergence theorem for conditional expectations yields EQ Œ Xn j F t  ! EQ Œ X j F t  Q-a.s., and hence P -a.s., for each Q 2 Q t . Thus, P -a.s., min EQ Œ X j F t   ˛ min t .Q/  lim inf ess sup.EQ Œ Xn j F t   ˛ t .Q// n"1

Q2Q t

D lim inf  t .Xn /; n"1

and so the representation (11.7) implies  t .X /  lim infn  t .Xn /. (b) ) (c): The Fatou property yields lim infn  t .Xn /   t .X /. On the other hand we have lim supn  t .Xn /   t .X / by monotonicity, and this implies  t .Xn / !  t .X /. (c) ) (a): Since X C  t .X / 2 A t for any X 2 L1 by conditional cash invariance, the definition (11.6) of ˛ min implies t ˛ min t .Q/  EQ Œ X j F t    t .X /; hence  t .X /  EQ Œ X j F t   ˛ min t .Q/ for any Q 2 Q t . This yields the inequality  t .X /  ess sup.EQ Œ X j F t   ˛ min t .Q//:

(11.8)

Q2Q t

In order to prove the equality in (11.7), both for Q t and for the smaller set P t , it is therefore enough to show that  EP Œ  t .X /   EP ess sup.EQ Œ X j F t   ˛ min (11.9) t .Q// ; 

Q2P t

460 where

Chapter 11 Dynamic risk measures



P t WD ¹Q 2 P t j EP Œ ˛ min t .Q/  < 1º  P t  Q t :

(11.10)

To this end, note first that min EP Œ ˛ min t .Q/  D EQ Œ ˛ t .Q/ 

for Q 2 P t .

(11.11)

Next, we consider the map  W L1 ! R defined by .X / WD EP Π t .X / : It is easy to check that  is a convex risk measure. Property (c) implies that  is continuous from above; equivalently, it has the Fatou property. Thus, Theorem 4.33 states that  has the robust representation .X / D

sup

.EQ Œ X   ˛.Q//;

Q2M1 .P /

where the penalty function ˛.Q/ is given by ˛.Q/ D sup EQ Œ X :

(11.12)

.X/0

Next we prove that Q D P on F t , and hence Q 2 P t , if ˛.Q/ < 1. Indeed, take A 2 F t and > 0. Then  P Œ A  D EP Œ  t . IA /  D . IA /  EQ Œ  IA   ˛.Q/; hence

1 ˛.Q/

for any > 0: Thus ˛.Q/ < 1 implies P Œ A   QŒ A  for any A 2 F t , and hence P D Q on F t . Furthermore, (11.13) EP Œ ˛ min t .Q/   ˛.Q/ P Œ A   QŒ A  C

holds for every Q 2 P t . Indeed, EQ Œ ˛ min t .Q/  D sup EQ Œ Y  Y 2A t

holds by equation (11.16) in Lemma 11.3 below, and so (11.12) and (11.11) imply (11.13) since .X /  0 for all X 2 A t . Thus we have shown ˛.Q/ < 1

H)



Q 2 Pt :

(11.14)

Section 11.1 Conditional risk measures and their robust representation

461

We can now conclude the proof of inequality (11.9) as follows, using first (11.14) and then (11.13) EP Π t .X /  D .X / D

sup

.EQ Œ X   ˛.Q//

Q2M1 .P /

D sup .EQ Œ X   ˛.Q// 

Q2P t

 sup EP Œ EQ Œ X j F t   ˛ min t .Q/  

Q2P t

  EP ess sup.EQ Œ X j F t   ˛ min t .Q// : 

Q2P t

This shows that the identity (11.7) is valid if the essential supremum is only taken over the set P t . In view of inequality (11.8), this coincides with the essential supremum over all Q 2 Q t and in turn over all Q 2 P t . The following lemma was used in the preceding proof. Lemma 11.3. For Q 2 Q t and 0  s  t , EQ Œ ˛ min t .Q/ j Fs  D ess sup EQ Œ Y j Fs ;

(11.15)

EQ Œ ˛ min t .Q/  D sup EQ Œ Y :

(11.16)

Y 2A t

and in particular Y 2A t

Proof. First we show that the family ¹EQ Œ X j F t  j X 2 A t º is directed upward for any Q 2 Q t (see Appendix A.5). Indeed, for X; Y 2 A t we can take Z WD X IA C Y IAc , where A WD ¹EQ Œ X j F t   EQ Œ Y j F t º 2 F t . Conditional convexity of  t implies Z 2 A t , and clearly we have EQ Œ Z j F t  D EQ Œ X j F t  _ EQ Œ Y j F t : Thus the family is directed upward. Hence, Theorem A.33 implies that its essential supremum can be computed as the supremum of an increasing sequence within this Q family, i.e., there exists a sequence .Xn / in A t such that Q ˛ min t .Q/ D lim EQ Œ Xn j F t : n"1

462

Chapter 11 Dynamic risk measures

By monotone convergence we get Q EQ Œ ˛ min t .Q/ j Fs  D lim EQ Œ EQ Œ Xn j F t  j Fs  n"1

D lim EQ ΠXnQ j Fs   ess sup EQ ΠY j Fs : n"1

Y 2A t

The converse inequality follows directly from the definition of ˛ min t .Q/. This shows (11.15), and for s D 0 we get (11.16). Remark 11.4. The proof of the implication “(c) ) (a)” in Theorem 11.2 shows that the representation (11.7) also holds if we replace Q t in (11.7) not only by P t but by the even smaller set

P t D ¹Q 2 P t j EP Œ ˛ min t .Q/  < 1º:

(11.17)

In view of (11.8) and (11.9), we can conclude that the representation (11.7) holds for any set Q such that P t  Q  Q t . } Example 11.5. Suppose that preferences are characterized by an exponential utility function u.x/ D 1  exp.ˇx/ with ˇ > 0. At time t the conditional expected utility of a financial position X 2 L1 is then given by the F t -measurable random variable U t .X / D EP Œ 1  e ˇX j F t : The set A t WD ¹X 2 L1 j U t .X /  U t .0/º D ¹X 2 L1 j EP Œ e ˇX j F t   1º satisfies the conditions required from an acceptance set. In analogy to Example 4.34, the induced convex conditional risk measure  t is given by  t .X / D ess inf¹ Y 2 L1 t j X C Y 2 At º ˇX j F t   e ˇY º; D ess inf¹ Y 2 L1 t j EP Œ e

i.e.,  t .X / D

1 log EP Œ e ˇX j F t : ˇ

Clearly,  t is a monetary conditional risk measure. It will be shown in Exercise 11.1.3 that it is also convex and admits a robust representation with minimal penalty function ˛ min t .Q/ D

1b H t .QjP /; ˇ

Q 2 Pt ;

(11.18)

Section 11.1 Conditional risk measures and their robust representation

463

where h h ˇ i ˇ i b t .QjP / WD EQ log dQ ˇ F t D EP dQ log dQ ˇ F t : H (11.19) dP dP dP denotes the conditional relative entropy of Q 2 P t with respect to P , given F t . For this reason,  t is called the conditional entropic risk measure. } Exercise 11.1.3. Argue as in Lemma 3.29 to show that for any Q 2 P t the conditional relative entropy (11.19) satisfies the following variational identity: b t .QjP / D sup .EQ Œ Z j F t   log EP Œ e Z j F t / H Z2L1

D sup¹EQ Œ Z j F t   log EP Œ e Z j F t  j e Z 2 L1 .P /º; where the second supremum is attained by Z WD log dQ . Then proceed as in ExdP ample 4.34 to conclude that the conditional entropic risk measure is convex and that (11.18) describes indeed its minimal penalty function for Q 2 P t . } In the coherent case we obtain the following representation result: Corollary 11.6. Let  t be a coherent conditional risk measure that has the Fatou property. Then  t has the representation  t .X / D ess sup EQ Œ X j F t ;

(11.20)



Q2P t

where the set P t is given by

P t D ¹Q 2 P t j ˛ min t .Q/ D 0 P -a.s.º and coincides with the set in (11.17). Proof. In the coherent case the penalty function ˛ min t .Q/ can only take the values 0 or 1 for Q 2 Q t . Indeed, for A WD ¹˛ min t .Q/ > 0º; X 2 A t , and any > 0 we have

IA X 2 A t due to conditional positive homogeneity of  t , hence ˛ min t .Q/ D ess sup EQ Œ X j F t  X2A t

 ess sup EQ Π IA X j F t  X2A t

D IA ˛ min t .Q/; and the lower bound converges to infinity on A as " 1. Thus ˛ min t .Q/ D 1 on A. Next, let P t be as in (11.17). Then EP Œ ˛ min t .Q/  < 1 for Q 2 P t , and this implies P Œ A  D 0. Thus P t coincides with the set of all Q 2 Q t for which ˛ min t .Q/ D 0 P -a.s., and so the representation (11.7), where we are free to take P t instead of Q t by Remark 11.4, reduces to (11.20).

464

Chapter 11 Dynamic risk measures

Example 11.7. For any 2 .0; 1/, the acceptance set A t D ¹X 2 L1 j P Œ X < 0 j F t   º defines a monetary conditional risk measure, which we call conditional Value at Risk at level V @R .X jF t / D ess inf¹m t 2 L1 t j P Œ X C m t < 0 j F t   º:

(11.21)

The conditional risk measure V@R .  jF t / is conditionally positively homogenous, but it is not conditionally convex. While the terminology ‘conditional Value at Risk’ seems quite natural, one should be aware of a possible confusion because the unconditional risk measure AV@R, which was introduced in Definition 4.48, is sometimes also called Conditional Value at Risk and denoted by CV@R. } Definition 11.8. For 2 .0; 1 let Q t denote the set of all measures Q 2 P t whose density dQ=dP is P -a.s. bounded by 1= . The resulting coherent conditional risk measure (11.22) AV @R˛ .X jF t / D ess sup EQ Œ X j F t  Q2Q t

is called conditional Average Value at Risk at level . We have the following representation result for the coherent conditional risk measure AV@R .  jF t /. Proposition 11.9. For X 2 L1 , the essential supremum in (11.22) is attained for any QX 2 Q t whose density satisfies ´ 1 dQX on ¹X > V@R .X jF t /º, (11.23) D

dP 0 on ¹X < V@R .X jF t /º, and the set of such maximizers is nonempty. In particular, AV @R˛ .X jF t / D

1 EP Π.X C V@R .X jF t //  C V@R .X jF t /:

(11.24)

Proof. Fix X 2 L1 and note that P Œ X > V@R .X jF t / j F t  D P Œ X C V@R .X jF t / < 0 j F t  h ˇ i 1 D lim P X C V@R .X jF t / C < 0 ˇ F t n n"1  ;

(11.25)

465

Section 11.2 Time consistency

where we have used (11.21) in the last step. On the other hand, (11.21) also implies that h ˇ i 1 P X C V@R .X jF t /  < 0 ˇ F t > n for each n, and hence that P Œ X  V@R .X jF t / j F t  D P Œ X C V@R .X jF t /  0 j F t  h ˇ i 1 D lim P X C V@R .X jF t /  < 0 ˇ F t n n"1  :

(11.26)

The two inequalities (11.25) and (11.26) imply that the set of measures QX 2 Q t satisfying (11.23) is nonempty. Now let Z X D dQX =dP be the density of such a measure, and let Z be the density of any other Q 2 Q t . We furthermore define XQ WD X C V@R .X jF t /. Then V@R .XQ jF t / D 0 and, arguing as in the proof of the general Neyman–Pearson lemma in the form of Theorem A.31,  1  Z I¹XQ 0º  0; .XQ /Z X  .XQ /Z D .XQ /

with P -a.s. equality if and only if Z is also of the form (11.23). The conditional cash invariance of AV@R .jF t / thus yields the first part of the assertion. The identity (11.24) now follows from the following chain of identities: AV @R˛ .X jF t /  V@R .X jF t / D AV@R .XQ jF t / D EP Œ .XQ /Z X  D

11.2

1 EP Π.XQ / :

Time consistency

In this section we consider a sequence of convex conditional risk measures  t W L1 ! L1 t ;

t D 0; 1; : : : ; T:

In such a dynamic setting the key question is how the risk assessments of a financial position at different times are connected to each other. Definition 11.10. A sequence of conditional risk measures . t / tD0;1;:::;T is called (strongly) time-consistent if for any X; Y 2 L1 and for all t  0 the following condition holds:  tC1 .X /   tC1 .Y /

H)

 t .X /   t .Y /:

466

Chapter 11 Dynamic risk measures

Lemma 11.11. Time consistency is equivalent to each of the following two properties: (a)  tC1 .X / D  tC1 .Y /

H)

 t .X / D  t .Y /

for t D 0; 1; : : : T  1:

(b) Recursiveness:  t D  t . tC1 / for t D 0; 1; : : : ; T  1. Proof. Time consistency clearly implies (a). (a) ) (b): Note that  tC1 . tC1 .X // D  tC1 .X /, due to conditional cashinvariance and normalization. Applying (a) with Y WD  tC1 .X / we thus obtain  t .X / D  t . tC1 .X //. (b) implies time consistency: If  tC1 .X /   tC1 .Y / then  t . tC1 .X //   t . tC1 .Y // by monotonicity, and so  t .X /   t .Y / follows from (b). Example 11.12. For fixed ˇ > 0, the sequence of conditional entropic risk measures  t .X / D

1 log EP Œ e ˇX j F t ; ˇ

t D 0; : : : ; T;

is time-consistent. Let us check recursiveness  ˇ i h   1 1  t . tC1 .X // D log EP exp  ˇ  log EP Œ exp.ˇX / j F tC1  ˇ F t ˇ ˇ 1 D log EP Œ EP Œ e ˇX j F tC1  j F t  ˇ D  t .X /: Note, however, that time consistency will be lost if the constant risk aversion parameter ˇ is replaced by an adapted process .ˇ t /. Moreover, under a suitable condition of dynamic law-invariance, the conditional entropic risk measure is in fact the only timeconsistent dynamic convex risk measure; see [190], where this result is reduced to an application of Proposition 2.46. Here we include the ordinary conditional expectation with respect to P as the limiting case ˇ D 0. } Example 11.13. The sequence of coherent conditional risk measures  t .X / D AV @R˛ .X jF t /;

t D 0; : : : ; T;

given by conditional Average Value at Risk at some level 2 .0; 1/ is not timeconsistent, and the same is true for conditional Value at Risk and for conditional meanstandard deviation risk measures defined in terms of Sharpe ratios. To see this, note first that all these risk measures are well defined on L2 .P /, and that they are of the form q  t .X / D EP ΠX j F t  C  EP Π.X  EP ΠX j F t /2  if X has a conditional Gaussian distribution with respect to F t (compare Examples 4.11, 4.12 and Exercise 4.4.2 (b)). Now consider a position of the form X D X1 C X2

467

Section 11.2 Time consistency

for two independent Gaussian random variables Xi with distribution N.0; i2 / and assume that F1 is the -field generated by X1 . Then 1 .X / D X1 C 1 .X2 / D X1 C  2 ; hence 0 .1 .X // D 0 .X1 / C  2 D . 1 C 2 /: On the other hand,

1

0 .X / D . 12 C 22 / 2 ; and so we have 0 .1 .X // > 0 .X / unless X1 or X2 are constant and therefore } 1 D 0 or 2 D 0. Exercise 11.2.1. Show that recursiveness as defined in part (b) of Lemma 11.11 is equivalent to the following condition: for 0  s < t  T and X 2 L1 , s .X / D s . t .X //:

}

The next exercise relies on and generalizes the preceding one. Exercise 11.2.2. For a stopping time  W  ! ¹0; : : : ; T º we define  .X / WD

T X

I¹Dt º  t .X /:

tD0

Show that recursiveness is equivalent to the following property: When is another stopping time such that   , then  .X / D  . .X //: Hint: Use Exercise 11.2.1 and (11.3). Exercise 11.2.3. Let  t , t D 0; : : : ; T , be any sequence of convex conditional risk measures. Show that the recursive definition QT WD T

and

Q t WD  t .Q tC1 /

for t D 0; : : : ; T  1,

yields a time-consistent sequence of convex conditional risk measures.

}

We are now going to characterize time consistency of the sequence . t / tD0;1;:::;T both in terms of the corresponding acceptance sets .A t / tD0;1;:::;T and in terms of the penalty processes .˛ min t .Q// tD0;1;:::;T . Our main goal is to prove a supermartingale criterion which characterizes time consistency in terms of the joint behavior of the stochastic processes . t .X // tD0;1;:::;T and .˛ min t .Q// tD0;1;:::;T .

468

Chapter 11 Dynamic risk measures

To this end we introduce some notation. Suppose that we look just one step ahead and assess the risk only for those positions whose outcome will be known by the end of the next period. This means that we restrict the conditional convex risk measure  t to the space L1 tC1 . The corresponding one-step acceptance set is given by A t;tC1 WD ¹ X 2 L1 tC1 j  t .X /  0 º; and ˛ min t;tC1 .Q/ WD ess sup EQ Œ X j F t  X2A t;t C1

is the resulting one-step penalty function. The following lemma holds for any sequence of monetary conditional risk measures. The equivalences between set inclusions for the acceptance sets and inequalities for the risk measures can be used as starting points for various departures from the strong notion of time consistency which we are considering here; see, e.g., [223], [260], and [263]. For two sets A; B  L1 we will use the following notation: A C B WD ¹X C Y j X 2 A; Y 2 Bº: Lemma 11.14. Let . t / tD0;1;:::;T be a sequence of monetary conditional risk measures. Then the following equivalences hold for all t D 0; : : : ; T 1 and all X 2 L1 : (a) X 2 A t;tC1 C A tC1 ”  tC1 .X / 2 A t;tC1 . (b) A t A t;tC1 C A tC1 ”  t . tC1 /   t . (c) A t  A t;tC1 C A tC1 ”  t . tC1 /   t . Proof. (a) To prove “)” take X D X t;tC1 C X tC1 with X t;tC1 2 A t;tC1 and X tC1 2 A tC1 . Then  tC1 .X / D  tC1 .X tC1 /  X t;tC1  X t;tC1 by cash invariance, and monotonicity implies  t . tC1 .X //   t .X t;tC1 /  0; hence  tC1 .X / 2 A tC1 . The converse direction follows immediately from the decomposition X D X C  tC1 .X /   tC1 .X /, since X C  tC1 .X / 2 A tC1 for all X 2 L1 and  tC1 .X / 2 A t;tC1 by assumption. (b) In order to show “)”, take X 2 L1 . Since X C  t .X / 2 A t A t;tC1 C A tC1 , we obtain  tC1 .X / C  t .X / D  tC1 .X C  t .X // 2 A t;tC1 ;

469

Section 11.2 Time consistency

by (a) and by cash invariance. This implies  t . tC1 .X //   t .X / D  t .. tC1 .X /   t .X ///  0: To prove “(” take X 2 A t . Then  tC1 .X / 2 A t;tC1 by the right hand side of (b), and hence X 2 A t;tC1 C A tC1 by (a). (c) Take X 2 L1 and assume A t  A t;tC1 C A tC1 . Then  t . tC1 .X // C X D  t . tC1 .X //   tC1 .X / C  tC1 .X / C X 2 A t;tC1 C A tC1 belongs to A t . This implies  t .X /   t . tC1 .X // D  t .X C  t . tC1 .X ///  0 by cash invariance, and so we have shown “)”. For the converse direction take X 2 A t;tC1 C A tC1 . Since  tC1 .X / 2 A t;tC1 by (a), we obtain  t .X /   t . tC1 .X //  0; hence X 2 A t . The preceding lemma implies immediately the following result. Proposition 11.15. Let . t / tD0;1;:::;T be a sequence of convex conditional risk measures such that each  t has the Fatou property. Then the following conditions are equivalent: (a) . t / tD0;1;:::;T is time-consistent. (b) A t D A t;tC1 C A tC1 for t D 0; : : : ; T  1. In the sequel, we will investigate the time consistency of a sequence . t / tD0;1;:::;T of convex conditional risk measures in terms of the dynamics of their penalty functions. In doing this, we need to assume that every element of the sequence can be represented in terms of the same set Q of probability measures:  t .X / D ess sup.EQ Œ X j F t   ˛ min t .Q//;

for t D 0; : : : ; T .

(11.27)

Q2Q

These representations are only well-defined if Q  Q t for all t . Since T < 1, every Q 2 Q must be equivalent to P on FT D F , and so we may as well assume Q D ¹ Q 2 M1 .P / j Q  P º:

(11.28)

Definition 11.16. A sequence . t / tD0;1;:::;T of convex conditional risk measures is called sensitive when the representation (11.27) holds in terms of the set Q in (11.28).

470

Chapter 11 Dynamic risk measures

Sensitivity is also called relevance. It generalizes the concept of sensitivity introduced in Section 4.3 to the case of conditional risk measures, and it can be characterized along the lines of Theorem 4.43. In fact, it can be shown that a time-consistent sequence . t / tD0;1;:::;T of conditional convex risk measures is sensitive as soon as 0 is sensitive in the sense of Definition 4.42; see [124]. The following theorem, and in particular the equivalence of its conditions (a) and (c), is the main result of this section. Theorem 11.17. Let . t / tD0;1;:::;T be a sensitive sequence of convex conditional risk measures. Then the following conditions are equivalent: (a) . t / tD0;1;:::;T is time-consistent. (b) For any Q 2 Q, min min ˛ min t .Q/ D ˛ t;tC1 .Q/ C EQ Œ ˛ tC1 .Q/ j F t 

for t D 0; 1; : : : ; T  1:

(c) For any Q 2 Q and any X 2 L1 , the process Q;X

Ut satisfies

WD  t .X / C ˛ min t .Q/; Q;X

t D 0; 1; : : : ; T;

Q;X

EQ ΠU tC1 j F t   U t and is a Q-supermartingale for any Q with

Q-a.s.,

˛0min .Q/

(11.29)

< 1.

We prepare the proof with the following two lemmas. Lemma 11.18. For Q0 ; Q1 2 Q and t 2 ¹0; : : : ; T º there exists Q 2 Q such that Q D Q0 on F t and EQ Œ Y j F t  D EQ1 Œ Y j F t  P -a.s. for all Y 2 L1 .

(11.30)

In particular we have min ˛ min t .Q/ D ˛ t .Q1 /:

Proof. When setting Z WD

(11.31)

 dQ ˇ 1 dQ 1ˇ 1 ˇ dQ0 F t dQ0

then EQ0 Œ Z j F t  D 1, and so dQ D Z dQ0 defines a measure Q 2 Q that coincides with Q0 on F t . Moreover, Proposition A.12 yields (11.30). The identity (11.31) now follows from the definition of ˛ min t . Lemma 11.19. The family ¹EQ Œ X j F t   ˛ min t .Q/ j Q 2 Qº is directed upward.

471

Section 11.2 Time consistency

Proof. By applying Lemma 11.18 with Q0 WD P we see that it is enough to show that the family ¹EQ Œ X j F t   ˛ min t .Q/ j Q 2 Q \ P t º is directed upward. To this end, fix Q1 ; Q2 2 Q \ P t and let Yi WD EQi Œ X j F t   ˛ min t .Qi / and A WD ¹Y2  Y1 º. We then define Z WD

dQ2 dQ1  IA C  I c: dP dP A

This random variable is P -a.s. strictly positive and satisfies h dQ ˇ i h dQ ˇ i 2 ˇ 1 ˇ EP Œ Z j F t  D IA EP F t C IAc EP F t D IA C IAc D 1: dP dP It follows that Z is the density of a probability measure Q 2 Q \ P t . Under Q, we have EQ Œ X j F t  D IA EQ2 Œ X j F t  C IAc EQ1 Œ X j F t : Similarly, min min ˛ min t .Q/ D IA ˛ t .Q2 / C IAc ˛ t .Q1 /:

Therefore, EQ Œ X j F t   ˛ min t .Q/ D IA Y2 C IAc Y1 D Y0 _ Y1 ; which concludes the proof. Proof of Theorem 11:17. (a) ) (b): Since any X 2 A t can be written as the sum of some X t;tC1 2 A t;tC1 and some X tC1 2 A tC1 , we obtain ˛ min t .Q/ D ess sup EQ Œ X j F t  X2A t

D

EQ ΠX t;tC1 j F t  C ess sup EQ ΠX tC1 j F t 

ess sup X t;t C1 2A t;t C1

D

˛ min t;tC1 .Q/

C

X t C1 2A t C1

EQ Œ ˛ min tC1 .Q/ j F t



for any Q 2 Q, using Lemma 11.3 in the last step. (b) ) (c): Fix X 2 L1 . By Lemma 11.19 and Theorem A.33 we can find a sequence Qn 2 Q such that  tC1 .X / can be identified as the limit of an increasing sequence: .EQn Œ X j F tC1   ˛ min (11.32) tC1 .Qn // %  tC1 .X / P -a.s. Q;X

Now take Q2Q and write U t WD U t Using property (b) we see that

. We have to show that U t  EQ ΠU tC1 j F t .

EQ Œ U tC1 j F t  D EQ Œ  tC1 .X / C ˛ min tC1 .Q/ j F t  min D EQ Œ  tC1 .X / j F t   ˛ min t;tC1 .Q/ C ˛ t .Q/:

(11.33)

472

Chapter 11 Dynamic risk measures

Now we use the sequence of measures Qn appearing in (11.32). By Lemma 11.18, we are free to assume that Qn D Q on F tC1 , and this implies min ˛ min t;tC1 .Qn / D ˛ t;tC1 .Q/:

(11.34)

Using the approximation (11.32) and monotone convergence for conditional expectations, and applying property (b) together with equation (11.34) to each Qn , we obtain min EQ Œ  tC1 .X / j F t  D lim .EQn Œ X j F t  C ˛ min t;tC1 .Qn /  ˛ t .Qn // n"1

min D lim .EQn Œ X j F t   ˛ min t .Qn // C ˛ t;tC1 .Q/ n"1

  t .X / C ˛ min t;tC1 .Q/: Together with (11.33) this yields EQ Œ U tC1 j F t    t .X / C ˛ min t .Q/ D U t : When in addition ˛0 .Q/ < 1 then (b) implies EQ Œ ˛ min t .Q/   ˛0 .Q/ < 1. Moreover,  t .X / is bounded by kX k1 , and so U t is integrable and hence a supermartingale. (c) ) (a): Take X; Y 2 L1 such that  tC1 .X /   tC1 .Y /; we have to show  t .X /   t .Y /. For each Q 2 Q, min  t .Y / C ˛ min t .Q/  EQ Œ  tC1 .Y / C ˛ tC1 .Q/ j F t 

 EQ Œ  tC1 .X / C ˛ min tC1 .Q/ j F t   EQ Œ EQ Œ X j F tC1  j F t  D EQ Œ X j F t : The first inequality follows from the supermartingale property in (c), and in the third step we have used the inequality  tC1 .X / C ˛ min tC1 .Q/  EQ Œ X j F tC1  which is valid for any Q 2 Q; see (11.8). Thus  t .Y /  EQ Œ X j F t   ˛ min t .Q/ for all Q 2 Q, and this implies the desired inequality  t .Y /   t .X / due to our assumption that  t admits the representation (11.27). Remark 11.20. It follows from property (b) of Theorem 11.17 that min EQ Œ ˛ min tC1 .Q/ j F t   ˛ t .Q/

for any Q 2 Q:

(11.35)

473

Section 11.2 Time consistency

min This in turn implies EQ Œ ˛ min t .Q/  < 1 for all t  0 if ˛0 .Q/ < 1. Thus, the process .˛ min t .Q// tD0;1;:::;T is a Q-supermartingale for any such Q. This can be seen as a built-in learning effect: If the dynamics are in fact driven by the probability measure Q then the penalization of this measures decreases “on average” in the sense that it is a supermartingale under Q. Note that property (b) provides more information than just the supermartingale property: It actually yields an explicit description of the predictable increasing process in the Doob decomposition of the supermartingale min .˛ min t .Q// tD0;1;:::;T in terms of the “one-step” penalty functions ˛ t;tC1 .Q/. More precisely, t1 X Q min ˛ min .Q/ D M  ˛k;kC1 .Q/; t t kD0

where M Q is a martingale under Q.

}

Exercise 11.2.4. Show that the cost of superhedging defines a time-consistent sequence of conditional convex risk measures. More precisely, in the situation and with the notation of Chapter 9, let T ° ± X ˇ ˇ 9  2 S W mt C Y C St .Y / WD inf m t 2 L1   .X  X /  0 P -a.s. k k k1 t kDt

for Y 2 L1 . Show that .St / tD0;:::;T is a time-consistent sequence of conditional convex risk measures when the conditions of Corollary 9.32 are satisfied. } Remark 11.21. The weaker version (11.35) of property (b) is in fact equivalent to a weaker notion of time consistency, called weak time consistency, that is defined by the implication  tC1 .X /  0 H)  t .X /  0; and which is equivalent to the following weaker version A tC1 A t ;

t D 0; 1; : : : ; T  1:

of property (b) in Proposition 11.15; see Exercise 11.2.5.

}

Exercise 11.2.5. Let . t / tD0;1;:::;T be a sensitive sequence of convex conditional risk measures. Show that the following conditions are equivalent: (a) . t / tD0;1;:::;T is weakly time consistent in the sense that  tC1 .X /  0 implies  t .X /  0. (b) The acceptance sets satisfy the inclusion A tC1 A t for t D 0; : : : ; T  1. min (c) The penalty functions satisfy EQ Œ ˛ min tC1 .Q/ j F t   ˛ t .Q/ for Q 2 Q.

}

474

Chapter 11 Dynamic risk measures

We will now consider the coherent case in Theorem 11.17. An initial discussion of this case was already given at the end of Section 6.5 in connection with our analysis of stability under pasting. Recall from Definition 6.38 that the pasting of two equivalent probability measures Q1 and Q2 in a stopping time  T is the probability measure Q A  WD EQ1 Œ Q2 Œ A j F  ; QŒ

A 2 FT :

According to Definition 6.41, a set Q of equivalent probability measures is called stable if, for any Q1 ; Q2 2 Q and 2 T , also their pasting in is belongs to Q. We can now state the following characterization of time consistency for coherent dynamic risk measures. It extends, and provides a converse of, Theorem 6.51. Theorem 11.22. Let . t / tD0;1;:::;T be a sensitive sequence of convex conditional risk measures, and assume that 0 is coherent. Then the following conditions are equivalent: (a) . t / tD0;1;:::;T is time-consistent. (b) There exists a stable set Q  Q such that  t .X / D ess sup EQ ΠX j F t 

for t D 0; : : : ; T and X 2 L1 .

(11.36)

Q2Q

In particular, each  t is coherent. Proof. It was shown in Theorem 6.51 that (b) implies recursiveness and in turn (a). To show the implication “(a) ) (b)”, let

Q WD Q \ P0 D Q \ ¹Q 2 M1 .P / j ˛0min .Q/ < 1º: Then ˛ min t .Q/ is a nonnegative Q-supermartingale for any Q 2 Q by Theorem min min 11.17. Since 0 is coherent, we have ˛0 .Q/ D 0, and hence ˛ t .Q/ D 0 Q-a.s. (and hence P -a.s.) for each t , due to the supermartingale property (compare Exercise 6.1.2). Therefore the representation 11.36 follows with Remark 11.4. In particular, each  t is coherent. To show the stability of Q , we recall from Proposition 6.43 that the stability of Q is equivalent to the following fact: For Q1 ; Q2 2 Q , t 2 ¹0; : : : ; T º, and B 2 F t , also the probability measure

QΠA  WD EQ1 ΠQ2 ΠA j F t   IB C IA\B c ;

A 2 FT ;

belongs to Q . Clearly, this measure Q is equivalent to P and hence belongs to Q. To prove that ˛0min .Q/ < 1, and in turn Q 2 Q , we show first that ˛ min t .Q/ D 0. Indeed, for X 2 A t , EQ Œ X j F t  D IB EQ2 Œ X j F t  C IB c EQ1 Œ X j F t   0;

475

Section 11.2 Time consistency

because for i D 1; 2 0 D ˛ min t .Qi / D ess sup EQi Œ X j F t : X2A t

This shows our claim ˛ min t .Q/ D 0. Next, part (b) of Theorem 11.17 yields that min min ˛ min t1 .Q/ D ˛ t1;t .Q/ C EQ Œ ˛ t .Q/ j F t1 

D ˛ min t1;t .Q/ D

ess sup X2A t 1 \L1 t

D

ess sup X2A t 1 \L1 t

EQ ΠX j F t  EQ1 ΠX j F t 

 ˛ min t1 .Q1 / D 0; where we have used the fact that Q D Q1 on F t in the fourth identity. An iteration of this argument shows ˛0min .Q/ D 0. Remark 11.23. As in the other chapters of Part II, we have limited the discussion to a finite horizon T < 1. If we pass to an infinite horizon, the characterization of time consistency in terms of supermartingale properties in Theorem 11.17 yields convergence results for risk measures which can be seen as nonlinear extensions of martingale convergence; cf. [124]. }

Appendix

A.1

Convexity

This section contains a few basic facts on convex functions and on convex sets in Euclidean space. Denote by p jxj WD x  x the Euclidean norm of x 2 Rn . Proposition A.1. Suppose that C  Rn is a non-empty convex set with 0 … C . Then there exists  2 Rn with   x  0 for all x 2 C , and with   x0 > 0 for at least one x0 2 C . Moreover, if infx2C jxj > 0, then one can find  2 Rn with infx2C   x > 0. Proof. First we consider the case in which infx2C jxj > 0. This infimum is attained by some y in the closure C of C . Since the set C is also convex, jyC˛.xy/j2  jyj2 for each x 2 C and all ˛ 2 Œ0; 1. Thus 2˛ y  .x  y/ C ˛ 2 jx  yj2  0: For ˛ # 0 we obtain y  x  y  y > 0, and so we can take  WD y. Now let C be any non-empty convex subset of Rn such that 0 … C . In a first step, we will show that C is a proper subset of Rn . To this end, let ¹e1 ; : : : ; ek º be a maximal collection of linearly independent vectors in C . Then each x 2 C can be expressed as a linear combination of e1 ; : : : ; ek . We claim that

z WD 

k X

ei

iD1

is not contained in C . We assume by way of contradiction that z 2 C . Then there are P zn 2 C converging to z. If we write zn D kiD1 in ei , then zn ! z is equivalent to the convergence in ! 1 for all i . It follows that for some n0 2 N all coefficients

in0 are strictly negative. Let ˛0 WD

1

1 Pk

i iD1 n0

j

and

˛j WD

1

 n0 Pk

i iD1 n0

for j D 1; : : : ; k:

477

Section A.1 Convexity

Then the ˛j ’s are non-negative and sum up to 1. Thus, the convexity of C implies that 0 D ˛0 zn0 C ˛1 e1 C    C ˛k ek 2 C ; which is a contradiction. Hence, z is not contained in C. Now we are in a position to prove the existence of a separating  in the case in which 0 is a boundary point of C , and thus infx2C jxj D 0. We may assume without loss of generality that the linear hull of C is the full space Rn . Since we already know that C is not dense in Rn , we may choose a sequence .zm /  Rn such that infx2C jx  zm j > 0 and zm ! 0. Then Cm WD C  zm satisfies infx2Cm jxj > 0, and the first part of the proof yields corresponding vectors m . We may assume that jm j D 1 for all n. By compactness of the .n  1/-dimensional unit sphere, there exists a convergent subsequence .mk / with limit , which satisfies   x D lim mk  x D lim mk  .x  zmk /  0 k"1

k"1

for all x 2 C . Since  is also a unit vector and C is not contained in a proper linear subspace of Rn , the case   x D 0 for all x 2 C cannot occur, and so there must be some x0 2 C with   x0 > 0. Definition A.2. Let A be any subset of a linear space E. The convex hull of A is defined as conv A D

n °X iD1

n ± X ˇ ˛i xi ˇ xi 2 A; ˛i  0; ˛i D 1; n 2 N : iD1

It is straightforward to check that conv A is the smallest convex set containing A. Let us now turn to convex functions on R. Definition A.3. A function f W R ! R [ ¹C1º is called a proper convex function if f .x/ < 1 for some x 2 R and if f .˛x C .1  ˛/y/  ˛ f .x/ C .1  ˛/ f .y/ for x; y 2 R and ˛ 2 Œ0; 1. The effective domain of f , denoted by dom f , consists of all x 2 R such that f .x/ < 1. Clearly, the effective domain of a proper convex function f is a real interval S D dom f . If considered as a function f W S ! R, the function f is convex in the usual sense. Conversely, any convex function f W S ! R defined on some non-empty interval S may be viewed as a proper convex function defined by f .x/ WD C1 for x 2 RnS. The following proposition summarizes continuity and differentiability properties of a proper convex function on its effective domain.

478

Appendix

Proposition A.4. Let f be a proper convex function, and denote by D the interior of dom f . (a) f is upper semicontinuous on dom f and locally Lipschitz continuous on D. (b) f admits left- and right-hand derivatives f0 .y/ WD lim

x"y

f .x/  f .y/ xy

and

fC0 .y/ WD lim

z#y

f .z/  f .y/ zy

at each y 2 D. Both fC0 and f0 are increasing functions and satisfy f0  fC0 . (c) The right-hand derivative fC0 is right-continuous, the left-hand derivative f0 is left-continuous. (d) f is differentiable a.e. in D, and for any x0 2 D Z x Z f .x/ D f .x0 / C fC0 .y/ dy D f .x0 / C x0

x

x0

f0 .y/ dy;

x 2 D:

Proof. We first prove part (b). For x; y; z 2 D with x < y < z, we take ˛ 2 .0; 1/ such that y D ˛z C .1  ˛/x. Using the convexity of f , one gets f .y/  f .x/ f .z/  f .x/ f .z/  f .y/   : yx zx zy

(A.1)

Thus, the difference quotient f .x/  f .y/ xy is an increasing function of x, which shows the existence of the left- and right-hand derivatives. Moreover, we get f0 .y/  fC0 .y/  f0 .z/ for y < z. (a): Let z 2 dom f , and take a sequence .xn /  dom f such that xn ! z. Without loss of generality, we may assume that xn # z or xn " z. In either case, xn D ın x1 C .1  ın /z, where ın # 0. Convexity of f yields lim sup f .xn /  lim sup.ın f .x1 / C .1  ın / f .z// D f .z/; n"1

n"1

and so f is upper semicontinuous. To prove local Lipschitz continuity, take a  x < y  b such that Œa; b  D. We get from part (b) that fC0 .a/  fC0 .x/ 

f .x/  f .y/  f0 .y/  fC0 .b/: xy

Hence, f is a Lipschitz continuous function on Œa; b with Lipschitz constant L WD jfC0 .a/j _ jf0 .b/j.

479

Section A.1 Convexity

(c): Continuity of f shows that for x < z f .z/  f .y/ f .z/  f .x/ D lim  lim sup fC0 .y/: zx z  y y#x y#x Taking z # x yields fC0 .x/  lim supy#x fC0 .y/. Since fC0 is increasing, we must in fact have fC0 .y/ ! fC0 .x/ as y # x. In the same way, one shows left-continuity of f0 . (d): Since the function f is Lipschitz continuous, it is absolutely continuous. By Lebesgue’s differentiation theorem, f is hence a.e. differentiable and equal to the integral of its derivative, which is equal to f0 .x/ D fC0 .x/ for a.e. x 2 D. Exercise A.1.1. Let f W .0; 1/ ! R be a function that is bounded from above and satisfies x C y  1 1 (A.2) f  f .x/ C f .y/ 2 2 2 for x; y 2 .0; 1/. Prove that f is continuous and conclude then that f is convex. Note: See [144, p. 96] for the construction of an unbounded real-valued function f that satisfies f .x C y/ D f .x/ C f .y/, and hence (A.2), but that is neither convex nor continuous. } Definition A.5. The Fenchel–Legendre transform of a function f W R ! R [ ¹C1º is defined as f  .y/ WD sup .y x  f .x//; y 2 R: x2R

If f ¥ C1, then f  is a convex and lower semicontinuous as the supremum of the affine functions y 7! y x  f .x/. In particular, f  is a proper convex function which is continuous on its effective domain. If f is itself a proper convex function, then f  is also called the conjugate function of f . Proposition A.6. Let f be a proper convex function. (a) For all x; y 2 R, xy  f .x/ C f  .y/

(A.3)

with equality if x belongs to the interior of dom f and if y 2 Œf0 .x/; fC0 .x/. (b) If f is lower semicontinuous, then f  D f , i.e., f .x/ D sup .x y  f  .y//;

x 2 R:

y2R

Proof. (a): The inequality (A.3) is obvious. Now suppose that x0 belongs to the interior of dom f . Proposition A.4 yields f .x/  f .x0 / C f˙0 .x0 /.x  x0 / for all x in the interior of dom f and, by upper semi-continuity, for all x 2 R. Hence,

480

Appendix

f .x/  f .x0 / C y0 .x  x0 / whenever y0 2 Œf0 .x0 /; fC0 .x0 /. This shows that x y0  f .x/  x0 y0  f .x0 / for all x 2 R, i.e., x0 y0  f .x0 / D sup .x y0  f .x// D f  .y0 /: x2R

(b): We first show the following auxiliary claim: If ˇ < f .x0 /, then there exists an affine function h such that h.x0 / D ˇ and h.x/ < f .x/ for all x. For the proof of this claim let C WD ¹.x; a/ 2 R2 j f .x/  aº: C is usually called the epigraph of f . It is nonempty since f is proper, convex, and closed due to the lower semicontinuity of f . The point .x0 ; ˇ/ does not belong to C , and Proposition A.1 thus yields some  D .1 ; 2 / 2 R2 such that inf .1 x C 2 f .x//  ı WD

x2dom f

inf .1 x C 2 a/ > 1 x0 C 2 ˇ:

.x;a/2C

If f .x0 / < 1, we get 1 x0 C 2 f .x0 / > 1 x0 C 2 ˇ. Hence 2 > 0, and one checks that 1 h.x/ WD  .x  x0 / C ˇ 2 is as desired. If f .x0 / D 1 and 2 > 0, then the same definition works. Now Q Q 0/ > 0 assume that f .x0 / D 1 and 2 D 0. Letting h.x/ WD ı  1 x we have h.x Q and h.x/  0 for x 2 dom f . Since f is proper, the first step of the proof of our claim allows us to construct an affine function g with g < f . If g.x0 /  ˇ, then Q h WD g C ˇ  g.x0 / is as desired. Otherwise, we let h.x/ WD g.x/ C h.x/ for Q

WD .ˇ  g.x0 //=h.x0 /. This concludes the proof of our auxiliary claim. Now we can prove part (b) of the assertion. It is clear from the definition that f  f  . Suppose there exists a point x0 such that f .x0 / > f  .x0 /. Take ˇ strictly between f  .x0 / and f .x0 /. By the auxiliary claim, there exists an affine function h < f such that h.x0 / D ˇ. Let us write h.x/ D y0 x C ˛. Then it follows that f  .y0 / < ˛ and hence f  .x0 /  y0 x0  f  .y0 / > h.x0 / D ˇ; which is a contradiction.

A.2

Absolutely continuous probability measures

Suppose that P and Q are two probability measures on a measurable space .; F /. Definition A.7. Q is said to be absolutely continuous with respect to P on the algebra F , and we write Q P , if for all A 2 F , PŒA D 0

H)

QΠA  D 0:

481

Section A.2 Absolutely continuous probability measures

If both Q P and P Q hold, we will say that Q and P are equivalent, and we will write Q  P . The following characterization of absolute continuity is known as the Radon–Nikodym theorem: Theorem A.8 (Radon–Nikodym). Q is absolutely continuous with respect to P on F if and only if there exists an F -measurable function '  0 such that Z Z F dQ D F ' dP for all F -measurable functions F  0. (A.4) Proof. See, e.g., § 17 of [21]. The function ' is called the density or Radon–Nikodym derivative of Q with respect to P , and we will write dQ WD ': dP Clearly, the Radon–Nikodym derivative is uniquely determined through (A.4). Corollary A.9. If Q P on F , then QP

dQ >0 dP

”

P -a.s.

In this case, the density of P with respect to Q is given by dP D dQ



dQ dP

1 :

Proof. Suppose that Q P , let ' WD dQ=dP . Take an F -measurable function F  0. Then Z Z Z F dQ D F ' dP D F dQ: ¹'>0º

¹'>0º

In particular, QŒ ' D 0  D 0. Replacing F with F ' 1 yields Z Z Z F ' 1 dQ D F ' 1 dQ D F ' 1 ' dP: ¹'>0º

Note that the term on the right-hand side equals P Π' D 0  D 0. This proves the result.

¹'>0º

R

F dP for all F if and only if

The preceding result admits the following generalization.

482

Appendix

Exercise A.2.1. Let P , Q, and R be three probability measures on .; F / such that R Q and Q P . Show that R P and that the corresponding Radon–Nikodym derivative can be obtained in the following way: dR dR dQ D  : dP dQ dP Note that we can obtain Corollary A.9 as a special case by choosing R WD P .

}

Remark A.10. Let us stress that absolute continuity depends on the underlying field F . For example, let P be the Lebesgue measure on  WD Œ0; 1/. Then every probability measure Q is absolutely continuous with respect to P on a -algebra F0 which is generated by finitely many intervals Œai1 ; ai / with 0 D a0 < a1 <    < an D 1. However, if we take for F the Borel -algebra on , then for instance a Dirac point mass Q D ıx is clearly not absolutely continuous with respect to P on F . } While the preceding example shows that, in general, absolute continuity is not preserved under an enlargement of the underlying -algebra, the next proposition states that it is safe to take smaller -algebras. This proposition involves the notion of a conditional expectation EŒ F j F0  of an F -measurable function F  0 with respect to a probability measure P and a -algebra F0  F . Recall that EŒ F j F0  may be defined as the P -a.s. unique F0 -measurable random variable F0 such that EŒ F I A0  D EŒ F0 I A0 

for all A0 2 F0 ;

(A.5)

see, e.g., § 15 of [20]. Note also our shorthand convention of writing EŒ F I A0  WD EŒ F IA0 : Clearly, we can replace in (A.5) the class of all indicator functions of sets in F0 by the class of all bounded F0 -measurable functions or by the class of all non-negative F0 -measurable functions. Proposition A.11. Suppose that Q and P are two probability measures on the measurable space .; F / and that Q P on F with density '. If F0 is a -algebra contained in F , then Q P on F0 , and the corresponding density is given by dQ ˇˇ ˇ D EŒ ' j F0  dP F0

P -a.s.

Section A.2 Absolutely continuous probability measures

483

Proof. Q P on F0 follows immediately from the definition of absolute continuity. Since ' is the density on F  F0 , it follows for A 2 F0 that Z Z QΠA  D ' dP D EΠ' j F0  dP: A

A

Therefore the F0 -measurable random variable EΠ' j F0  must coincide with the density on F0 . Now we prove a formula for computing a conditional expectation EQ ΠF j F0  under a measure Q in terms of conditional expectations with respect to another measure P with Q P . Proposition A.12. Suppose that Q P on F with density ', and that F0 F is another -algebra. Then, for any F -measurable F  0, EQ ΠF j F0  D

1  EΠF ' j F0  Q-a.s. EΠ' j F0 

Proof. Suppose that G0  0 is F0 -measurable. Then EQ Œ G0 F  D EŒ G0 F '  D EŒ G0  EŒ F ' jF0  : Let '0 WD EŒ' j F0 . Proposition A.11 implies that '0 > 0 Q-almost surely. Hence, we may assume that G0 D 0 P -a.s. on ¹'0 D 0º, and Corollary A.9 yields  EŒ G0  EŒ F ' jF0   D EQ G0 

 1 EΠF ' j F0  : EΠ' j F0 

This proves the assertion. If neither Q P nor P Q holds, one can use the following Lebesgue decomposition of Q with respect to P . Theorem A.13. For any two probability measures Q and P on .; F /, there exists a set N 2 F with QΠN  D 0 and a F -measurable function '  0 such that Z P ΠA  D P ΠA \ N  C ' dQ for all A 2 F . A

One writes

´ dP ' WD dQ C1

on N c , on N .

484

Appendix

Proof. Let R WD 12 .Q C P /. Then both Q and P are absolutely continuous with respect to R with respective densities dQ=dR and dP =dR. Let ² N WD

³ dQ D0 : dR

Then QŒ N  D 0. We define ´ dP dP  . dQ /1 WD ' WD dR dR dQ C1

on N c , on N .

Then, for F -measurable f  0, Z

Z

Z

dP dR dR N Nc   Z Z dQ 1 dP D  f dP C f dQ dR dR N Nc Z Z f dP C f ' dQ; D

f dP D

f dP C

f

N

where we have used the fact that QΠN  D 0 in the last step.

A.3

Quantile functions

Suppose that F W .a; b/ ! R is an increasing function which is not necessarily strictly increasing. Let c WD lim F .x/ and d WD lim F .x/: x#a

x"b

Definition A.14. A function q W .c; d / ! .a; b/ is called an inverse function for F if F .q.s//  s  F .q.s/C/

for all s 2 .c; d /.

The functions q  .s/ WD sup¹x 2 R j F .x/ < s º and

q C .s/ WD inf¹x 2 R j F .x/ > sº

are called the left- and right-continuous inverse functions. The following lemma explains the reason for calling q  and q C the left- and rightcontinuous inverse functions of F .

485

Section A.3 Quantile functions

Lemma A.15. A function q W .c; d / ! .a; b/ is an inverse function for F if and only if q  .s/  q.s/  q C .s/ for all s 2 .c; d /. In particular, q  and q C are inverse functions. Moreover, q  is left-continuous, q C is right-continuous, and every inverse function q is increasing and satisfies q.s/ D q  .s/ and q.sC/ D q C .s/ for all s 2 .c; d /. In particular, any two inverse functions coincide a.e. on .c; d /. Proof. We have q   q C , and any inverse function q satisfies q   q  q C , due to the definitions of q  and q C . Hence, the first part of the assertion follows if we can show that F .q C .s//  s  F .q  .s/C/ for all s. But x < q C .s/ implies F .x/  s and y > q  .s/ implies F .y/  s, which gives the result. Next, the set ¹x j F .x/ > sº is the union of the sets ¹x j F .x/ > s C "º for " < 0, and so q C is right-continuous. An analogous argument shows the left-continuity of q  . It is clear that both q  and q C are increasing, so that the second part of the assertion follows. Remark A.16. The left- and right-continuous inverse functions can also be represented as q  .s/ D inf¹x 2 R j F .x/  s º

and

q C .s/ D sup¹x 2 R j F .x/  sº:

To see this, note first that q  .s/ is clearly dominated by the infimum on the right. On the other hand, y > q  .s/ implies F .y/  s, and we get q  .s/  inf¹x 2 R j F .x/  sº. The proof for q C is analogous. } Lemma A.17. Let q be an inverse function for F . Then F is an inverse function for q. In particular, F .xC/ D inf¹s 2 .c; d / j q.s/ > xº for x with F .x/ < d .

(A.6)

Proof. If s > F .x/ then q.s/  q  .s/  x, and hence q.F .x/C/  x. Conversely, s < F .x/ implies q.s/  q C .s/  x, and thus q.F .x//  x. This proves that F is an inverse function for q. Remark A.18. By defining q.d / WD b we can extend (A.6) to F .xC/ D inf¹s 2 .c; d  j q.s/ > xº for all x 2 .a; b/.

}

From now on we will assume that F W R ! Œ0; 1 is increasing and right-continuous and that F is normalized in the sense that c D 0 and d D 1. This assumption always holds if F is the distribution function of a random variable X on some probability

486

Appendix

space .; F ; P /, i.e., F is given by F .x/ D P Œ X  x . The following lemma shows in particular that also the converse is true: any normalized increasing rightcontinuous functions F W R ! Œ0; 1 is the distribution function of some random variable. By considering the laws of random variables, we also obtain the one-to-one correspondence F .x/ D ..1; x/ between all Borel probability measures on R and all normalized increasing right-continuous functions F W R ! Œ0; 1. Lemma A.19. Let U be a random variable on a probability space .; F ; P / with a uniform distribution on .0; 1/, i.e., P Œ U  s  D s for all s 2 .0; 1/. If q is an inverse function of a normalized increasing right-continuous function F W R ! Œ0; 1, then X.!/ WD q.U.!// has the distribution function F . Proof. First note that any inverse function for F is measurable because it coincides with the measurable function q C outside the countable set ¹s 2 .0; 1/ j q  .s/ < q C .s/º. Since q.F .x//  x, we have q.s/  x for s < F .x/. Moreover, Lemma A.17 shows that q.s/  x implies F .x/  F .q.s// D F .q.s/C/  s. It follows that .0; F .x// ¹s 2 .0; 1/ j q.s/  xº .0; F .x/: Hence, F .x/ D P Œ U 2 .0; F .x//   P Œ U 2 ¹s j q.s/  xº   P Œ U 2 .0; F .x/  D F .x/: The assertion now follows from the identity P Œ U 2 ¹s j q.s/  xº  D P Œ X  x . Definition A.20. An inverse function q W .0; 1/ ! R of a distribution function F is called a quantile function. That is, q is a function with F .q.s//  s  F .q.s//

for all s 2 .0; 1/.

The left- and rightcontinuous inverses, q  .s/ D sup¹x 2 R j F .x/ < sº

and

q C .s/ D inf¹x 2 R j F .x/ > sº;

are called the lower and upper quantile functions. We will often use the generic notation FX for the distribution function of a random variable X . When the emphasis is on the law of X , we will also write F . In the same manner, we will write qX or q for the corresponding quantile functions. The value qX . / of a quantile function at a given level 2 .0; 1/ is often called a

-quantile of X . The following result complements Lemma A.19. It implies that a probability space supports a random variable with uniform distribution on .0; 1/ if and only if it supports any non-constant random variable X with a continuous distribution.

487

Section A.3 Quantile functions

Lemma A.21. Let X be a random variable with a continuous distribution function FX and with quantile function qX . Then U WD FX .X / is uniformly distributed on .0; 1/, and X D qX .U / P -almost surely. Q FQ ; PQ / be a probability space that supports a random variable UQ with a Proof. Let .; uniform distribution on .0; 1/. Then XQ WD qX .UQ / has the same distribution as X due to Lemma A.19. Hence, FX .X / and FX .XQ / also have the same distribution. On the other hand, if FX is continuous, then FX .qX .s// D s and thus FX .XQ / D UQ . To show that X D qX .U / P -a.s., note first that qXC .FX .t //  t and hence qX .U / D C qX .U /  X P -almost surely. Now let f W R ! .0; 1/ be a strictly increasing function. Since qX .U / and X have the same law, we have EŒ f .qX .U //  D EŒ f .X /  and get P Œ qX .U / > X  D 0. There are several possibilities how the preceding lemma can be generalized to the case of discontinuous distribution functions FX . A first possibility is provided in the following exercise, which is taken from Rüschendorf [228]; see also Ferguson [111]. The second possibility will be given in Lemma A.28. Exercise A.3.1. Let X be a random variable with distribution function FX . The modified distribution function of X is defined by FX .x; / WD P Œ X < x  C P Œ X D x ;

x 2 R; 2 Œ0; 1:

Suppose that UQ is a random variable that is independent of X and uniformly distributed on .0; 1/. Show that U WD FX .X; UQ / is uniformly distributed on .0; 1/ and that X D qX .U /

P -a.s.

}

The following lemma uses the concept of the Fenchel–Legendre transform of a convex function as introduced in Definition A.5. Lemma A.22. Let X be a random variable with distribution function FX and quantile function qX such that EŒ jX j  < 1. Then the Fenchel–Legendre transform of the convex function Z x

‰.x/ WD

FX .z/ dz D EΠ.x  X /C 

1

is given by ´R y



‰ .y/ D sup .xy  ‰.x// D x2R

qX .t / dt C1 0

if 0  y  1, otherwise.

Moreover, for 0 < y < 1, the supremum above is attained in x if and only if x is a y-quantile of X .

488

Appendix

Proof. Note first that, by Fubini’s theorem and Lemma A.19, Z 1 hZ x i C ‰.x/ D E I¹X zº dz D EŒ .x  X /  D .x  qX .t //C dt: 1

(A.7)

0

It follows that ‰  .y/ D C1 for y < 0, ‰  .0/ D  infx ‰.x/ D 0, Z 1 Z ‰  .1/ D sup .x  ‰.x// D lim x  .x  qX .t //C dt D x"1 0

x2R

1

qX .t / dt; 0

and ‰  .y/ D 1 for y > 1. To prove our formula for 0 < y < 1, note that the right-hand and left-hand derivatives of the concave function f .x/ D xy  ‰.x/ are given by fC0 .x/ D y  FX .x/ and f0 .x/ D y  FX .x/. A point x is a maximizer of f if fC0 .x/  0 and f0 .x/  0, which is equivalent to x being a y-quantile. Taking x D qX .y/ and using (A.7) gives Z y Z y ‰.x/ D .x  qX .t // dt D xy  qX .t / dt; 0

0

and our formula follows. Lemma A.23. If X D f .Y / for an increasing function f and qY is a quantile function for Y , then f .qY .t // is a quantile function for X . In particular, qX .t / D qf .Y / .t / D f .qY .t //

for a.e. t 2 .0; 1/,

for any quantile function qX of X . If f is decreasing, then f .qY .1  t // is a quantile function for X . In particular, qX .t / D qf .Y / .t / D f .qY .1  t // for a.e. t 2 .0; 1/. Proof. If f is decreasing, then q.t / WD f .qY .1  t // satisfies FX .q.t // D P Œ f .Y /  f .qY .1  t //   P Œ Y  qY .1  t /  t  P Œ Y > qY .1  t /   FX .q.t //; since FY .qY .1  t //  1  t  FY .qY .1  t // by definition. Hence q.t / D f .qY .1  t // is a quantile function. A similar argument applies to an increasing function f . The following theorem is a version of the Hardy–Littlewood inequalities. They estimate the expectation EŒ X Y  in terms of quantile functions qX and qY .

489

Section A.3 Quantile functions

Theorem A.24. Let X and Y be two random variables on .; F ; P / with quantile functions qX and qY . Then, Z 1 Z 1 qX .1  s/qY .s/ ds  EΠX Y   qX .s/qY .s/ ds; 0

0

provided that all integrals are well defined. If X D f .Y / and the lower (upper) bound is finite, then the lower .upper/ bound is attained if and only if f can be chosen as a decreasing .increasing/ function. Proof. We first prove the result for X; Y  0. By Fubini’s theorem, Z 1 i hZ 1 I¹X >xº dx I¹Y >yº dy EŒ X Y  D E 0 1Z 1

Z D

0

P ΠX > x; Y > y  dx dy: 0

0

Since P Œ X > x; Y > y   .P Œ X > x   P Œ Y  y  /C Z 1 D I¹FY .y/sº I¹s1FX .x/º ds; 0

and since C .s/ qZ

Z

1

D sup¹x  0 j FZ .x/  s º D 0

I¹FZ .x/sº dx

for any random variable Z  0, another application of Fubini’s theorem yields Z 1 Z 1 C C qX .1  s/ qY .s/ ds D qX .1  s/ qY .s/ ds: EŒ X Y   0

0

In the same way, the upper estimate follows from the inequality P Œ X > x; Y > y   P Œ X > x  ^ P Œ Y > y  Z 1 I¹FX .x/sº I¹FY .y/sº ds: D 0

For X D f .Y /, Z

1

f .qY .t //qY .t / dt;

EΠX Y  D EΠf .Y /Y  D

(A.8)

0

due to Lemma A.19, and so Lemma A.23 implies that the upper and lower bounds are attained for increasing and decreasing functions, respectively.

490

Appendix

Conversely, assume that X D f .Y /, and that the upper bound is attained and finite Z 1 EΠf .Y /Y  D qX .t /qY .t / dt < 1: (A.9) 0

Our aim is to show that X D f .Y / D fQ.Y / P -a.s.,

(A.10)

where fQ is the increasing function on Œ0; 1/ defined by fQ.x/ WD qX .FY .x// if x is a continuity point of FY , and by fQ.x/ WD

1 FY .x/  FY .x/

Z

FY .x/

qX .t / dt FY .x/

otherwise. Note that fQ.qY / D E ΠqX j qY ;

(A.11)

where E Π j qY  denotes the conditional expectation with respect to qY under the Lebesgue measure on .0; 1/. Therefore, (A.9) implies that Z 1 Z 1 1> f .qY .t //qY .t / dt D EΠf .Y /Y  D qX .t /qY .t / dt 0 0 (A.12) Z 1 Q D f .qY .t //qY .t / dt; 0

where we have used Lemma A.19 in the first identity. After these preparations, we can proceed to proving (A.10). Let  denote the distribution of Y . By introducing the positive measures d D f d and d Q D fQ d, (A.12) can be written as Z 1 Z Z Z 1

.Œy; 1// dy D x .dx/ D x .dx/ Q D

.Œy; Q 1// dy: (A.13) 0

0

On the other hand, with g denoting the increasing function IŒy;1/ , the upper Hardy– Littlewood inequality, Lemma A.23, and (A.11) yield

.Œy; 1// D EŒ g.Y /f .Y /  Z 1  qg.Y / .t /qX .t / dt 0

Z D

1

g.qY .t //fQ.qY .t // dt

0

D .Œy; Q 1//:

491

Section A.3 Quantile functions

In view of (A.13), we obtain D , Q hence f D fQ -a.s. and X D fQ.Y / P -almost surely. An analogous argument applies to the lower bound, and the proof for X; Y  0 is concluded. The result for general X and Y is reduced to the case of non-negative random variables by separately considering the positive and negative parts of X and Y EΠX Y  D EΠX C Y C   EΠX C Y    EΠX  Y C  C EΠX  Y   Z 1 Z 1  qX C .t /qY C .t / dt  qX C .t /qY  .1  t / dt (A.14) 0

0

Z

1

 0

qX  .1  t /qY C .t / dt C

Z

1

qX  .t /qY  .t / dt; 0

where we have used the upper Hardy–Littlewood inequality on the positive terms and the lower one on the negative terms. Since qZ C .t / D .qZ .t //C and qZ  .t / D due to Lemma A.23, one checks that the right.qZ .1  t // for all random variables R1 hand side of (A.14) is equal to 0 qX .t /qY .t / dt , and we obtain the general form of the upper Hardy–Littlewood inequality. The same argument also works for the lower one. Now suppose that X D f .Y /. We first note that (A.8) still holds, and so Lemma A.23 implies that the upper and lower bounds are attained for increasing and decreasing functions, respectively. Conversely, let us assume that the upper Hardy– Littlewood inequality is an identity. Then all four inequalities used in (A.14) must also be equalities. Using the fact that X Y C D f .Y C /Y C and X Y  D f .Y  /Y  , the assertion is reduced to the case of non-negative random variables, and one checks that f can be chosen as an increasing function. The same argument applies if the lower Hardy–Littlewood inequality is attained. Remark A.25. For indicator functions of two sets A and B in F , the Hardy–Littlewood inequalities reduce to the elementary inequalities .P Œ A  C P Œ B   1/C  P Œ A \ B   P Œ A  ^ P Œ B I

(A.15)

note that these estimates were used in the preceding proof. Applied to the sets ¹X  xº and ¹Y  yº, where X and Y are random variables with distribution functions FX and FY and joint distribution function FX;Y defined by FX;Y .x; y/ D P Œ X  x; Y  y , they take the form .FX .x/ C FY .y/  1/C  FX;Y .x; y/  FX .X / ^ FY .y/:

(A.16)

The estimates (A.15) and (A.16) are often called Fréchet bounds, and the Hardy– Littlewood inequalities provide their natural extension from sets to random variables. }

492

Appendix

Definition A.26. A probability space .; :F ; P / is called atomless if it contains no atoms. That is, there is no set A 2 F such that P Œ A  > 0 and P Œ B  D 0 or P Œ B  D P Œ A  whenever B 2 F is a subset of A. Proposition A.27. For any probability space, the following conditions are equivalent: (a) .; F ; P / is atomless. (b) There exists an i.i.d. sequence X1 ; X2 ; : : : of random variables with Bernoulli distribution 1 P Œ X1 D 1  D P Œ X 1 D 0  D : 2 (c) For any 2 M1 .R/ there exist i.i.d. random variables Y1 ; Y2 ; : : : with common distribution . (d) .; F ; P / supports a random variable with a continuous distribution. Proof. (a) ) (b): We need the following intuitive fact from measure theory: If .; F ; P / is atomless, then for every A 2 F and all ı with 0  ı  P Œ A  there exists a measurable set B  A such that P Œ B  D ı; see Theorem 9.51 of [3]. Thus, we may take a set A 2 F such that P Œ A  D 1=2 and define X1 WD 1 on A and X1 WD 0 on Ac . Now suppose that X1 ; : : : ; Xn have already been constructed. Then P Œ X1 D x1 ; : : : ; Xn D xn  D 2n for all x1 ; : : : ; xn 2 ¹0; 1º, and this property is equivalent to X1 ; : : : ; Xn being independent with the desired symmetric Bernoulli distribution. For all x1 ; : : : ; xn 2 ¹0; 1º we may choose a set B  ¹X1 D x1 ; : : : ; Xn D xn º such that P Œ B  D 2.nC1/ and define XnC1 WD 1 on B and XnC1 WD 0 on B c \ ¹X1 D x1 ; : : : ; Xn D xn º. Clearly, the collection X1 ; : : : ; XnC1 is again i.i.d. with a symmetric Bernoulli distribution. (b) ) (c): By relabeling the sequence X1 ; X2 ; : : : , we may obtain a double-indexed sequence .Xi;j /i;j 2N of independent Bernoulli-distributed random variables. If we let 1 X Ui WD 2n Xi;n ; nD1

then it is straightforward to check that Ui has a uniform distribution. Let q be a quantile function for . Lemma A.19 shows that the i.i.d. sequence Yi WD q.Ui /, i D 1; 2; : : : , has common distribution . The proofs of the implications (c) ) (d) and (d) ) (a) are straightforward.

493

Section A.4 The Neyman–Pearson lemma

The following lemma is taken from Ryff [229]. Lemma A.28. If X is a random variable on an atomless probability space, then there exists a random variable with a uniform distribution on .0; 1/ such that X D qX .U / P -almost surely. Proof. Without loss of generality, we may assume that qX D qXC . Then Ix WD ¹t 2 .0; 1/ j qX .t / D xº is a (possibly empty or degenerate) real interval with Lebesgue measure .Ix / D P Œ X D x  for each x 2 R. Consider the set D WD ¹x 2 R j P Œ X D x  > 0º, which is at most countable. For each x 2 D, the probability space .; F ; P Œ  j X D x / is again atomless and hence supports a random variable Ux with a uniform law on Ix . That is, P Œ Ux 2 A j X D x  D .A \ Ix /= .Ix / or, equivalently, P Œ Ux 2 A; X D x  D .A \ Ix /

for all measurable A  .0; 1/.

(A.17)

On D c D .0; 1/nD, qX is one-to-one and hence admits a right-continuous inverse function F (which can actually be taken as FX , but this fact will not be needed here). We let U.!/ WD F .X.!//I¹X.!/…Dº C UX.!/ I¹X.!/2Dº ; which clearly is a measurable random variable. By definition we have qX .U.!// D X.!/ for all !. It remains to show that U has a uniform law. To this end, take a measurable subset A of .0; 1/. Using (A.17) we get X P Œ U 2 A  D P Œ U 2 A; X … D  C P Œ U 2 A; X D x  x2D

D P Œ F .X / 2 A; X … D  C

X

.A \ Ix /:

x2D

Ic

Now let denote the complement of c qX .I /º P -a.s. and hence

S

x2D Ix

in .0; 1/. Then ¹X … Dº D ¹X 2

P Œ F .X / 2 A; X … D  D P Œ X 2 qX .A \ I c /  D .qX 2 qX .A \ I c // D .A \ I c /; where we have used the fact that ı qX1 D P ı X 1 . This proves the result.

A.4

The Neyman–Pearson lemma

Suppose that P and Q are two probability measures on .; F /, and denote by Z dP PŒA D PŒA \ N  C dQ; A 2 F , dQ A

494

Appendix

the Lebesgue decomposition of P with respect to Q as in Theorem A.13. For fixed c  0, we let ³ ² dP A0 WD >c ; dQ where we make use of the convention that dP =dQ D 1 on N . Proposition A.29 (Neyman–Pearson lemma). If A 2 F is such that QŒ A   QŒ A0 , then P Œ A   P Œ A0 . Proof. Let F WD IA0  IA . Then F  0 on N , and F  .dP =dQ  c/  0. Hence Z

0

PŒA   PŒA D

F dP Z

Z

D

F dP C N

Z

c

F

dP dQ dQ

F dQ

D c.QŒ A0   QŒ A /: This proves the proposition. Remark A.30. In statistical test theory, A0 is interpreted as the likelihood quotient test of the null hypothesis Q against the alternative hypothesis P : If the outcome ! of a statistical experiment is in A0 , then the null hypothesis is rejected. There are two possible kinds of error which can occur in such a test. A type 1 error occurs if the null hypotheses is rejected despite the fact that Q is the “true” probability. Similarly, a type 2 error occurs when the null hypothesis is not rejected, although Q is not the “true” probability. The probability of a type 1 error is given by QŒ A0 . This quantity is usually called the size or the significance level of the statistical test A0 . A type 2 error occurs with probability P Œ .A0 /c . The complementary probability P Œ A0  D 1  P Œ .A0 /c  is called the power of the test A0 . In this setting, the set A of Proposition A.29 can be regarded as another statistical test to which our likelihood quotient test is compared. The proposition can thus be restated as follows: A likelihood quotient test has maximal power on its significance level. } Indicator functions of sets take only the values 0 and 1. We now generalize Proposition A.29 by considering F -measurable functions W  ! Œ0; 1; let R denote the set of all such functions.

495

Section A.4 The Neyman–Pearson lemma

Theorem A.31. Let … WD 12 .P C Q/, and define the density ' WD dP =dQ as above. (a) Take c  0, and suppose that 0

Then, for any Z

2 R, Z dQ 

0

0

2 R satisfies …-a.s. ´ 1 on ¹' > cº; D 0 on ¹' < cº: Z

dQ

H)

Z dP 

(A.18)

0

dP:

(A.19)

0 2 R of the form (A.18) such that (b) For R 0any ˛0 2 .0; 1/ there is some dQ D ˛0 . More precisely, if c is an .1  ˛0 /-quantile of ' under Q, we can define 0 by 0 D I¹'>cº C I¹'Dcº ;

where  is defined as ´  WD (c) Any

0

0 ˛0 QŒ '>c  QŒ 'Dc 

if QΠ' D c  D 0, otherwise.

2 R satisfying (A.19) is of the form (A.18) for some c  0.

Proof. (a): Take F WD 0  and repeat the proof of Proposition A.29. (b): Let F denote the distribution function of ' under Q. Then QŒ ' > c  D 1  F .c/  ˛0 and QŒ ' D c  D F .c/  F .c/  F .c/  1 C ˛0 D ˛0  QŒ ' > c : R 0 Hence 0    1 and 0 belongs to R. The fact that dQ D ˛0 is obvious.  (c): Suppose that satisfies Z Z Z Z   dQ H) dP  dP: dQ  R  The cases in which ˛0 WD dQ equals For 0 < ˛0 < 1, we R 0or 1 are Rtrivial. 0 as in part (b). Then ˛ D 0 dQ. One also has that can take dQ D 0 R  R 0 dP D dP , as can be seen by applying (A.19) to both  and 0 with reversed roles. Hence, for f WD 0   and N D ¹' D 1º, Z Z Z Z f dP C f  .'  c/ dQ: 0 D f dP  c f dQ D N

But (A.18) implies that both f  0 P -a.s. on N , and f  .'  c/  0 Q-a.s. Hence f vanishes …-a.s. on ¹' ¤ cº.

496

Appendix

Remark A.32. In the context of Remark A.30, an element of R is interpreted as a randomized statistical test: If ! is the outcome of a statistical experiment and p WD .!/, then the null hypothesis is rejected with probability p, i.e., after performing an independent random coin toss with success probability p. Significance level and power of a randomized test are defined as above, and a test of the form (A.18) is called a generalized likelihood quotient test. Thus, the general Neyman–Pearson lemma in the form of Theorem A.31 can be stated as follows: A randomized test has maximal power on its significance level, if and only if it is a generalized likelihood quotient test. }

A.5

The essential supremum of a family of random variables

In this section, we discuss the essential supremum of an arbitrary family ˆ of random variables on a given probability space .; F ; P /. Consider first the case in which the set ˆ is countable. Then '  .!/ WD sup'2ˆ '.!/ will also be a random variable, i.e., '  is measurable. Measurability of the pointwise supremum, however, is not guaranteed if ˆ is uncountable. Even if the pointwise supremum is measurable, it may not be the right concept, when we focus on almost sure properties. This can be illustrated by taking P as the Lebesgue measure on  WD Œ0; 1 and ˆ WD ¹I¹xº j 0  x  1º. Then sup'2ˆ '.x/  1 whereas ' D 0 P -a.s. for each single ' 2 ˆ. This suggests the following notion of an essential supremum defined in terms of almost sure inequalities. Theorem A.33. Let ˆ be any set of random variables on .; F ; P /. (a) There exists a random variable '  with the following two properties. (i) '   ' P -a.s. for all ' 2 ˆ. (ii) '   P -a.s. for every random variable satisfying  ' P -a.s. for all ' 2 ˆ. (b) Suppose in addition that ˆ is directed upwards, i.e., for '; 'Q 2 ˆ there exists 2 ˆ with  '_ '. Q Then there exists an increasing sequence '1  '2     in ˆ such that '  D limn 'n P -almost surely. Definition A.34. The random variable '  in Theorem A:33 is called the essential supremum of ˆ with respect to P , and we write ess sup ˆ D ess sup ' WD '  : '2ˆ

The essential infimum of ˆ with respect to P is defined as ess inf ˆ D ess inf ' WD  ess sup.'/: '2ˆ

'2ˆ

497

Section A.6 Spaces of measures

Proof of Theorem A.33. Without loss of generality, we may assume that each ' 2 ˆ Q WD ¹f ı ' j ' 2 ˆº with takes values in Œ0; 1; otherwise we may consider ˆ f W R ! Œ0; 1 strictly increasing. If ‰  ˆ is countable, let '‰ .!/ WD sup'2‰ '.!/. Then '‰ is measurable. We claim that the upper bound c WD sup¹EŒ '‰  j ‰  ˆ countableº is attained S by some countable ‰   ˆ. To see this, take ‰n with EŒ '‰n  ! c and  let ‰ WD n ‰n . Then ‰  is countable and EŒ '‰  D c. We now show that '  WD '‰ satisfies (a). Suppose that (a) does not hold. Then there exists ' 2 ˆ such that P Œ ' > '   > 0. Hence ‰ 0 WD ‰  [ ¹'º satisfies EŒ '‰0  > EŒ '‰  D c; in contradiction to the definition of c. Furthermore, if is any other random variable satisfying (a), then obviously  '  . Finally, the construction shows that '‰ can be approximated by an increasing sequence if ‰ is directed upwards. Remark A.35. For a given random variable X let ˆ be the set of all constants c such that P Œ X > c  > 0. The number ess sup X WD sup ˆ is the smallest constant c  C1 such that X  c P -a.s. and called the essential supremum of X with respect to P . The essential infimum of X is defined as ess inf X WD ess sup.X /:

A.6

}

Spaces of measures

Let S be a topological space. S is called metrizable if there exists a metric d on S which generates the topology of S. That is, the open d -balls B" .x/ WD ¹y 2 S j d.x; y/ < "º;

x 2 S; " > 0;

form a base for the topology of S in the sense that a set U  S is open if and only if it can be written as a union of such d -balls. A convenient feature of metrizable spaces is that their topological properties can be characterized via convergent sequences. For instance, a subset A of the metrizable space S is closed if and only if for every convergent sequence in A its limit point is also contained in A. Moreover, a function f W S ! R is continuous at y 2 S if and only if f .yn / converges to f .y/ for

498

Appendix

every sequence .yn / converging to y. We write Cb .S / for the set of all bounded and continuous functions on S . The metrizable space S is called separable if there exists a countable dense subset ¹x1 ; x2 ; : : : º of S . In this case, the Borel -algebra S of S is generated by the open d -balls B" .x/ with radii " > 0, " 2 Q, and centered in x 2 ¹x1 ; x2 ; : : : º. In what follows, we will always assume that S is separable and metrizable. If, moreover, the metric d can be chosen to be complete, i.e., if every Cauchy sequence with respect to d converges to some point in S, then S is called a Polish space. Clearly, Rd with the Euclidean distance is a complete and separable metric space, hence a Polish space. Let us denote by M.S/ WD M.S; S/ the set of all non-negative finite measures on .S; S/. Every 2 M.S / is of the form D ˛  for some factor ˛ 2 Œ0; 1/ and some probability measure  on the measurable space .S; S/. The space of all probability measures on .S; S/ is denoted by M1 .S/ D M1 .S; S/: Definition A.36. The weak topology on M.S/ is the coarsest topology for which all mappings Z M.S/ 3 7!

f d ;

f 2 Cb .S /;

are continuous. It follows from this definition that the sets U" . I f1 ; : : : ; fn / WD

Z Z n ° ˇ ± \ ˇ ˇˇ ˇ  2 M.S/ ˇ ˇ fi d  fi d ˇ < "

(A.20)

iD1

for 2 M.S/, " > 0, n 2 N, and f1 ; : : : ; fn 2 Cb .S / form a base for the weak topology on M.S/; for details see, e.g., Section 2.13 of [3]. Since the constant function 1 is continuous, Z ° ± ˇ M1 .S/ D 2 M.S/ ˇ .S / D 1 d D 1 is a closed subset of M.S/. A well-known example for weak convergence of probability measures is the classical central limit theorem; the following version is needed in Section 5.7. Theorem A.37. Suppose that for each N 2 N we are given N independent random .N / .N / variables Y1 ; : : : ; YN on .N ; FN ; PN / which satisfy the following conditions:

499

Section A.6 Spaces of measures   

.N /

There are constants N such that N ! 0 and jYk j  N PN -a.s. PN .N /  ! m. kD1 EN ΠYk PN .N / / ! 2 , where varN denotes the variance with respect kD1 varN .Yk to PN .

Then the distributions of ZN WD

N X

.N /

Yk

;

N D 1; 2; : : : ;

kD1

converge weakly to the normal distribution with mean m and variance 2 . Proof. See, for instance, the corollary to Theorem 7.1.2 of [55]. The following theorem allows us to examine the weak topology in terms of weakly converging sequences of measures. Theorem A.38. The space M.S/ is separable and metrizable for the weak topology. If S is Polish, then so is M.S/. Moreover, if S0 is a dense subset of S , then the set n °X

± ˇ ˛i ıxi ˇ ˛i 2 QC ; xi 2 S0 ; n 2 N

iD1

of simple measures on S0 with rational weights is dense in M.S / for the weak topology. Proof. In most textbooks on measure theory, the previous result is proved for M1 .S / instead of M.S/; see, e.g., Theorem 14.12 of [3]. The general case requires only minor modifications. It is treated in full generality in Chapter IX, § 5, of [32]. The following characterization of weak convergence in M.S / is known as the “portmanteau theorem”. Theorem A.39. For any sequence ; 1 ; 2 ; : : : of measures in M.S /, the following conditions are equivalent: (a) The sequence . n /n2N converges weakly to . (b) n .S/ ! .S/ and lim sup n .A/  .A/

for every closed set A  S.

n"1

(c) n .S/ ! .S/ and lim inf n .U /  .U / n"1

for every open set U  S .

500

Appendix

(d) n .B/ ! .B/ for every Borel set B whose boundary @B is not charged by

in the sense that .@B/ D 0. R R (e) f d n ! f d for every bounded measurable function f which is -a.e. continuous. R R (f) f d n ! f d for every bounded and uniformly continuous function f . Proof. The result is proved for M1 .S/ in [3], Theorem 14.3. The general case requires only minor modifications; see Chapter IX of [32]. Remark A.40. It follows from the portmanteau theorem that, on S D R, weak convergence of n to is equivalent to the condition F .x/  lim inf Fn .x/  lim sup Fn .x/  F .x/ n"1

n"1

for the corresponding distribution functions .Fn / and F , or to the pointwise convergence of Fn .x/ to F .x/ in any continuity point of F . It is also equivalent to the condition C  C q .t / D q .t /  lim inf qn .t /  lim sup qn .t /  q .t / n"1

n"1

for any choice of the quantile functions qn of n , or to the pointwise convergence of C .t / in any continuity point of q C . . qn .t / to q }  The next theorem can be regarded as a stability result for weak convergence. Theorem A.41 (Slutsky). Suppose that, for n 2 N, Xn and Yn are real-valued random variables on .n ; Fn ; Pn / such that the laws of Xn converge weakly to the law of X , and the laws of Yn converge weakly to ıy for some y 2 R. Then: (a) The laws of Xn C Yn converge weakly to the law of X C y. (b) The laws of Xn  Yn converge weakly to the law of X  y. Proof. See, for instance, Section 8.1 of [54]. We turn now to the fundamental characterization of the relative compact subsets of M.S / known as Prohorov’s theorem. Theorem A.42 (Prohorov). Let S be a Polish space. A subset M of M.S / is relatively compact for the weak topology if and only if sup .S/ < 1 2M

501

Section A.6 Spaces of measures

and if M is tight, i.e., if for every " > 0 there exists a compact subset K of S such that sup .K c /  ": 2M

In particular, M1 .S/ is weakly compact if S is a compact metric space. Proof. For a proof in the context of probability measures, see for instance Theorem 1 in § III.2 of [251]. The general case requires only minor modifications; see Chapter IX of [32]. Example A.43. Take for S the positive half axis Œ0; 1/ and define

n WD

n1 1 ı0 C ın n n

and

WD ı0 ;

where ıx denotes the Dirac point mass in x 2 S , i.e., ıx .A/ D IA .x/. Clearly, Z Z f d n ! f d for all f 2 Cb .S / so that n converges weakly R to . However, if we take the continuous but unbounded function f .x/ D x, then f d n D 1 for all n so that Z Z } lim f d n D 1 ¤ f d : n"1

The preceding example shows that the weak topology is not an appropriate topology for ensuring the convergence of integrals against unbounded test functions. Let us introduce a suitable transformation of the weak topology which will allow us to deal with certain classes of unbounded functions. We fix a continuous function W S 7! Œ1; 1/ which will serve as a gauge function, and we denote by C .S/ the linear space of all continuous functions f on S for which there exists a constant c such that jf .x/j  c  .x/ for all x 2 S . Furthermore, we denote by M .S/ R the set of all measures 2 M.S/ such that d < 1.

502

Appendix

Definition A.44. The all mappings

-weak topology on M .S / is the coarsest topology for which Z M .S/ 3 7!

f d ;

f 2 C .S /;

are continuous. Since the gauge function takes values in Œ1; 1/, every bounded continuous function f belongs to C .S/. It follows that all mappings Z M .S/ 3 7!

f d ;

f 2 Cb .S /;

are continuous. In particular, the set M1 .S/ WD ¹ 2 M .S/ j .S / D 1 º of all Borel probability measures in M .S/ is closed for the -weak topology. As in the case of the weak topology, it follows that the sets U" . I f1 ; : : : ; fn / WD

Z Z n ° ˇ ± \ ˇ ˇˇ ˇ  2 M .S/ ˇ ˇ fi d  fi d ˇ < " iD1

for 2 M .S /, " > 0, n 2 N, and f1 ; : : : ; fn 2 C .S / form a base for the topology on M .S/. Let us define a mapping

-weak

‰ W M.S/ ! M .S / by d ‰. / WD

1

d ;

2 M.S /:

Clearly, ‰ is a bijective mapping between the two sets M.S / and M .S /. Moreover, if we apply ‰ to an open neighborhood for the weak topology as in (A.20), we get ‰.U" . I f1 ; : : : ; fn // D U" .‰. /I f1 ; : : : ; fn /: Since f 2 C .S/ for each bounded and continuous function f , and since every function in C .S/ arises in this way, we conclude that a subset U of M.S / is weakly open if and only if ‰.U / is open for the -weak topology. Hence, ‰ is a homeomorphism. This observation allows us to translate statements for the weak topology into results for the -weak topology:

503

Section A.6 Spaces of measures

Corollary A.45. For separable and metrizable S , the space M .S / is separable and metrizable for the -weak topology. If S is Polish, then so is M .S /. Moreover, if S0 is a dense subset of S, then the set n °X

± ˇ ˛i ıxi ˇ ˛i 2 QC ; xi 2 S0 ; n 2 N

iD1

of simple measures on S0 with rational weights is dense in M .S / for the topology.

-weak

The preceding corollary implies in particular that it suffices to consider -weakly converging sequences when studying the -weak topology. The following corollary is implied by the portmanteau theorem. Corollary A.46. A sequence . n /n2N in M .S / converges only if Z Z f d n ! f d

-weakly to if and

for every measurable function f which is -a.e. continuous and for which exists a constant c such that jf j  c  -almost everywhere. Prohorov’s theorem translates as follows to our present setting: Corollary A.47. Let S be a Polish space and M be a subset of M .S /. The following conditions are equivalent: (a) M is relatively compact for the

-weak topology.

(b) We have

Z d < 1 ;

sup 2M

and for every " > 0 there exists a compact subset K of S such that Z sup d  " : 2M

Kc

(c) There exists a measurable function  W S ! Œ1; 1 such that each set ¹x 2 S j .x/  n .x/º ; is relatively compact in S, and such that Z sup  d < 1 : 2M

n2N;

504

Appendix

Proof. (a) , (b): This follows immediately from Theorem A.42 and the fact that ‰ is a homeomorphism. (b) ) (c): Take an increasing sequence K1  K2     of compact sets in S such that Z sup d  2n ; 2M

Knc

and define  by .x/ WD

.x/ C

1 X

IK c .x/ .x/: n

nD1

Then ¹  n º  Kn . Moreover, Z Z  d  sup sup 2M

(c) ) (b): Since ¹ 

d C 1 < 1:

2M

º is relatively compact, we have that

c WD sup¹ .x/ j x 2 S; .x/  and hence

Z

Z

sup

d  .1 C c/ sup

2M

2M

Moreover, for n  "1 sup2M satisfies Z sup 2M

.x/º < 1;

Kc

R

 d < 1:

 d , the relatively compact set K WD ¹  n º Z 1 sup d   d  "; n 2M Knc

and so condition (b) is satisfied. We turn now to the task of identifying a linear functional on a space of functions as the integral with respect to a suitable measure. Theorem A.48 (Riesz). Let  be a compact metric space and suppose that I is a linear functional on C./ that is non-negative in the sense that f  0 everywhere on  implies I.f /  0. Then there exists a unique positive Borel measure on  such that Z I.f / D f d for all f 2 C./. To state a general version of the preceding theorem, we need the notion of a vector lattice of real-valued functions on an arbitrary set . This is a linear space L that is stable under the operation of taking the pointwise maximum: for f; g 2 L also f _g 2 L. One example is the space of all bounded measurable functions on .; F /. Another one is the space Cb ./ of all bounded continuous functions on a separable

505

Section A.6 Spaces of measures

metric space . In this case, the -algebra .L/ generated by L coincides with the Borel -algebra of the underlying metric space. Note that Theorem A.48 is implied by the following result, together with Dini’s lemma as recalled in Lemma 4.24. Theorem A.49 (Daniell–Stone). Let I be a linear functional on a vector lattice L of functions on  such that L contains the constants and the following conditions hold: (a) I is non-negative in the sense that f  0 everywhere on  implies I.f /  0. (b) If .fn / is a sequence in L such that fn & 0, then I.fn / & 0. Then there exists a unique positive measure on the measurable space .; .L// such that Z I.f / D f d for all f 2 L. Proof. See, e.g., Theorem 4.5.2 of [97] or Satz 40.5 in [19]. Without the continuity assumption (b), the preceding result takes a different form, as we will discuss now. Definition A.50. Let .; F / be a measurable space. A mapping W F ! R is called a finitely additive set function if .;/ D 0, and if for any finite collection A1 ; : : : ; An 2 F of mutually disjoint sets n n [  X

Ai D

.Ai /: iD1

iD1

We denote by M1;f WD M1;f .; F / the set of all those finitely additive set functions

W F ! Œ0; 1 which are normalized to ./ D 1. The total variation of a finitely additive set function is defined as k kvar WD sup

n °X

± ˇ j .Ai /j ˇ A1 ; : : : ; An disjoint sets in F , n 2 N :

iD1

The space of all finitely additive measures whose total variation is finite is denoted by ba.; F /. We will now give a brief outline of the integration theory with respect to a measure

2 ba WD ba.; F /; for details we refer to Chapter III in [101]. The space X of all bounded measurable functions on .; F / is a Banach space if endowed with the supremum norm, kF k WD sup jF .!/j; F 2 X: !2

506

Appendix

Let X0 denote the linear subspace of all finitely valued step functions which can be represented in the form n X ˛i IAi ; F D iD1

for some n 2 N, ˛i 2 R, and disjoint sets A1 ; : : : ; An 2 F . For this F we define Z F d WD

n X

˛i .Ai /;

iD1

and one can check that this definition is independent of the particular representation of F . Moreover, ˇZ ˇ ˇ ˇ (A.21) ˇ F d ˇ  kF k  k kvar : Since X0 is dense in X with respect to k  k, this inequality allows us to define the integral on Rthe full space X as the extension of the continuous linear functional X0 3 F 7! F d . Clearly, M1;f is contained in ba, and we will denote the integral of a function F 2 X with respect to Q 2 M1;f by Z EQ Œ F  WD F dQ: Theorem A.51. The integral Z `.F / D

F d ;

F 2 X;

defines a one-to-one correspondence between continuous linear functionals ` on X and finitely additive set functions 2 ba. Proof. By definition of the integral and by (A.21), it is clear that any 2 ba defines a continuous linear functional on X. Conversely, if a continuous linear functional ` is given, then we can define a finitely additive set function on .; F / by

.A/ WD `.IA /;

A2F:

If L  0 is such that `.F /  L for kF k  1, then k kvar  L, and so 2 ba. One then checks that the integral with respect to coincides with ` on X0 . Since X0 is R dense in X, we see that F d and `.F / coincide for all F 2 X. Remark A.52. Theorem A.51 yields in particular a one-to-one correspondence between set functions Q 2 M1;f and continuous linear functionals ` on X such that `.1/ D 1 and `.X /  0 for X  0. }

507

Section A.7 Some functional analysis

Example A.53. Clearly, the set M1;f coincides with the set M1 WD M1 .; F / of all -additive probability measures if .; F / can be reduced to a finite set, in the sense that F is generated by a finite partition of . Otherwise, M1;f is strictly larger than M1 . Suppose in fact that there are infinitely many disjoint sets A1 ; A2 ; : : : 2 F , take !n 2 An , and define n

`n .X / WD

1X X.!i /; n

n D 1; 2; : : : :

iD1

The continuous linear functionals `n on X belong to the unit ball B1 in the dual Banach space X 0 . By Theorem A.63, there exists a cluster point ` of .`n /. For any X 2 X there is a subsequence .nk / such that `nk .X / ! `.X /. This implies that `.X /  0 for X  0 and `.1/ D 1. Hence, Theorem A.51 allows us to write `.X / D EQ ΠX Sfor some Q 2 M1;f . But Q is not -additive, since QΠAn  D `.IAn / D 0 and QΠn An  D 1. }

A.7

Some functional analysis

Numerous arguments in this book involve infinite-dimensional vector spaces. Typical examples arising in connection with a probability space .; F ; P / are the spaces Lp WD Lp .; F ; P / for 0  p  1, which we will introduce below. To this end, we first take p 2 .0; 1 and denote by Lp .; F ; P / the set of all F -measurable functions Z on .; F ; P / such that kZkp < 1, where ´ if 0 < p < 1, EŒ jZjp 1=p ; (A.22) kZkp WD inf¹c  0 j P Œ jZj > c  D 0 º; if p D 1. Let us also introduce the space L0 .; F ; P /, defined as the set of all P -a.s. finite random variables. If no ambiguity with respect to -algebra and measure can arise, we may sometimes write Lp .P / or just Lp instead of Lp .; F ; P /. For p 2 Œ0; 1, the space Lp .; F ; P /, or just Lp , is obtained from Lp by identifying random variables which coincide up to a P -null set. Thus, Lp consists of all equivalence classes with respect to the equivalence relation Z  ZQ

W”

Z D ZQ

P -a.s.

(A.23)

If p 2 Œ1; 1 then the vector space Lp is a Banach space with respect to the norm kkp defined in (A.22), i.e., every Cauchy sequence with respect to k  kp converges to some element in Lp . In principle, one should distinguish between a random variable Z 2 Lp and its associated equivalence class ŒZ 2 Lp , of which Z is a representative element. In order to keep things simple, we will follow the usual convention of identifying Z with its equivalence class, i.e., we will just write Z 2 Lp .

508

Appendix

On the space L0 , we use the topology of convergence in P -measure. This topology is generated by the metric d.X; Y / WD EΠjX  Y j ^ 1 ;

X; Y 2 L0 :

(A.24)

Note, however, that d is not a norm. Definition A.54. A linear space E which carries a topology is called a topological vector space if every singleton ¹xº for x 2 E is a closed set, and if the vector space operations are continuous in the following sense: .x; y/ 7! x C y is a continuous mapping from E  E into E, and .˛; x/ 7! ˛x is a continuous mapping from R  E into E. Clearly, every Banach space is a topological vector space. The following result is a generalization of the separation argument in Proposition A.1 to an infinite-dimensional setting. Theorem A.55. In a topological vector space E, any two disjoint convex sets B and C , one of which has an interior point, can be separated by a non-zero continuous linear functional ` on E, i.e., `.x/  `.y/

for all x 2 C and all y 2 B.

(A.25)

Proof. See [101], Theorem V.2.8. If one wishes to strictly separate two convex sets by a linear functional in the sense that one has a strict inequality in (A.25), then one needs additional conditions both on the convex sets and on the underlying space E. Definition A.56. A topological vector space E is called a locally convex space if its topology has a base consisting of convex sets. If E is a Banach space with norm k  k, then the open balls ¹y 2 E j ky  xk < rº;

x 2 E; r > 0;

form by definition a base for the topology of E. Since such balls are convex sets, any Banach space is locally convex. The space L0 .; F ; P / with the topology of convergence in P -measure, however, is not locally convex if .; F ; P / has no atoms; see, e.g., Theorem 12.41 of [3]. The following theorem is one variant of the classical Hahn–Banach theorem on the existence of “separating hyperplanes”.

509

Section A.7 Some functional analysis

Theorem A.57 (Hahn–Banach). Suppose that B and C are two non-empty, disjoint, and convex subsets of a locally convex space E. Then, if B is compact and C is closed, there exists a continuous linear functional ` on E such that sup `.x/ < inf `.y/: x2C

y2B

Proof. See, for instance, [236], p. 65, or [101], Theorem V.2.10. One corollary of the preceding result is that, on a locally convex space E, the collection E 0 WD ¹` W E ! R j ` is continuous and linearº separates the points of E, i.e., for any two distinct points x; y 2 E there exists some ` 2 E 0 such that `.x/ ¤ `.y/. The space E 0 is called the dual or the dual space of E. For instance, if p 2 Œ1; 1/ it is well known that the dual of Lp .; F ; P / is given by Lq .; F ; P /, where p1 C q1 D 1. The following definition describes a natural way in which locally convex topologies often arise. Definition A.58. Let E be linear space, and suppose that F is a linear class of linear functionals on E which separates the points of E. The F -topology on E, denoted by .E; F /, is the topology on E which is obtained by taking as a base all sets of the form ¹y 2 E j j`i .y/  `i .x/j < r; i D 1; : : : ; nº; where n 2 N, x 2 E, `i 2 F , and r > 0. If E already carries a locally convex topology, then the E 0 -topology .E; E 0 / is called the weak topology on E. If E is infinite-dimensional, then E is typically not metrizable in the F -topology. In this case, it may not suffice to consider converging sequences when making topological assertions; see, however, Theorem A.66 below. The following proposition summarizes a few elementary properties of the F -topology. Proposition A.59. Consider the situation of the preceding definition. Then: (a) E is a locally convex space for the F -topology. (b) The F -topology is the coarsest topology on E for which every ` 2 F is continuous. (c) The dual of E for the F -topology is equal to F . Proof. See, e.g., Section V.3 of [101]. Theorem A.60. Suppose that E is a locally convex space and that C is a convex subset of E. Then C is weakly closed if and only if C is closed in the original topology of E.

510

Appendix

Proof. If the convex set C is closed in the original topology then, by Theorem A.57, it is the intersection of the halfspaces H D ¹`  cº such that H  C , and thus closed in the weak topology .E; E 0 /. The converse is clear. For a given locally convex space E we can turn things around and consider E as a set of linear functionals on the dual space E 0 by letting x.`/ WD `.x/ for ` 2 E 0 and x 2 E. The E-topology .E 0 ; E/ obtained in this way is called the weak topology on E 0 . According to part (c) of Proposition A.59, E is then the topological dual of .E 0 ; .E 0 ; E//. For example, the Banach space L1 WD L1 .; F ; P / is the dual of L1 , but the converse is generally not true. However, L1 becomes the dual of L1 if we endow L1 with the weak topology .L1 ; L1 /. The mutual duality between E and E 0 allows us to state a general version of part (b) of Proposition A.6. As in the one-dimensional situation of Definition A.3, a convex function f W E ! R [ ¹C1º is called a proper convex function if f .x/ < 1 for some x 2 R. Definition A.61. The Fenchel–Legendre transform of a function f W E ! R[¹C1º is the function f  on E 0 defined by f  .`/ WD sup .`.x/  f .x//: x2E

If f ¥ C1, then f  is a proper convex and lower semicontinuous function as the supremum of affine functions. If f is itself a proper convex function, then f  is also called the conjugate function of f . Theorem A.62. Let f be a proper convex function on a locally convex space E. If f is lower semicontinuous with respect to .E; E 0 /, then f D f  . It is straightforward to adapt the proof we gave in the one-dimensional case of Proposition A.6 to the infinite-dimensional situation of Theorem A.62; all one has to do is to replace the separating hyperplane lemma by the Hahn–Banach separation theorem in the form of Theorem A.57. One of the reasons for considering the weak topology on a Banach space or, more generally, on a locally convex space is that typically more sets are compact for the weak topology than for the original topology. The following result shows that the unit ball in the dual of a Banach space is weak compact. Here we use the fact that a Banach space .E; k  kE / defines the following norm on its dual E 0 : k`kE 0 WD

sup `.x/;

` 2 E 0:

kxkE 1

Theorem A.63 (Banach–Alaoglu). Let E be a Banach space with dual E 0 . Then ¹` 2 E 0 j k`kE 0  rº is weak compact for every r  0.

Section A.7 Some functional analysis

511

Proof. See, e.g., Theorem IV.21 in [219]. Theorem A.64 (Krein–Šmulian). Let E be a Banach space and suppose that C is a convex subset of the dual space E 0 . Then C is weak closed if and only if C \ ¹` 2 E 0 j k`kE 0  rº is weak closed for each r > 0. Proof. See Theorem V.5.7 in [101]. The preceding theorem implies the following characterization of weak closed sets in L1 D L1 .; F ; P / for a given probability space .; F ; P /. Lemma A.65. A convex subset C of L1 is weak closed if for every r > 0 Cr WD C \ ¹X 2 L1 j kX k1  rº is closed in L1 . Proof. Since Cr is convex and closed in L1 , it is weakly closed in L1 by Theorem A.60. Since the natural injection  .L1 ; .L1 ; L1 // ! .L1 ; .L1 ; L1 / is continuous, Cr is .L1 ; L1 /-closed in L1 . Thus, C is weak closed due to the Krein–Šmulian theorem. Finally, we state a few fundamental results on weakly compact sets. Theorem A.66 (Eberlein–Šmulian). For any subset A of a Banach space E, the following conditions are equivalent: (a) A is weakly sequentially compact, i.e., any sequence in A has a subsequence which converges weakly in E. (b) A is weakly relatively compact, i.e., the weak closure of A is weakly compact. Proof. See [101], Theorem V.6.1. The following result characterizes the weakly relatively compact subsets of the Banach space L1 WD L1 .; F ; P /. It implies, in particular, that a set of the form ¹f 2 L1 j jf j  gº with given g 2 L1 is weakly compact in L1 . Theorem A.67 (Dunford–Pettis). A subset A of L1 is weakly relatively compact if and only if it is bounded and uniformly integrable. Proof. See, e.g., Theorem IV.8.9 or Corollary IV.8.11 in [101].

Notes

In these notes, we do not make any attempt to give a systematic account of all the sources which have been relevant for the development of the field. We simply mention a number of references which had a direct influence on our decisions how to present the topics discussed in this book. More comprehensive lists of references can be found, e.g., Delbaen and Schachermayer [85], Jeanblanc, Yor and Chesney [158], Karatzas and Shreve [171], and McNeil, Frey, and Embrechts [201]. Chapter 1: The proof of Theorem 1.7 is based on Dalang, Morton, and Willinger [67]. Remark 1.18 and Example 1.19 are taken from Schachermayer [233]. Section 1.6 is mainly based on [233], with the exception of Lemma 1.64, which is taken from Kabanov and Stricker [164]. Our proof of Lemma 1.68 combines ideas from [164] with the original argument in [233], as suggested to us by Irina Penner. For a historical overview of the development of arbitrage pricing and for an outlook to continuous-time developments, we refer to [85]. For some mathematical connections between superhedging of call options as discussed in Section 1.3 and bounds on stoploss premiums in insurance see Chapter 5 of Goovaerts et al. [141]. Chapter 2: The results on the structure of preferences developed in this chapter are, to a large extent, standard topics in mathematical economics. We refer to textbooks on expected utility theory such as Fishburn [115], [116], Kreps [187], or Savage [232], and to the survey articles in [11], [16]. The ideas and results of Section 2.1 go back to classical references such as Debreu [77], Eilenberg [103], Milgram [204], and Rader [218]. The theory of affine numerical representations in Section 2.2 was initiated by von Neumann and Morgenstern [209] and further developed by Herstein and Milnor [151]. The characterization of certain equivalents with the translation in Proposition 2.46 goes back to de Finetti [114]; see also Kolmogorv [177] and Nagumo [208]. The drastic consequences of the assumption that a favorable bet is rejected at any level of wealth, as explained in Proposition 2.49, were stressed by Rabin [217]. The discussion of the partial orders