Quasi-Monte Carlo Methods in Finance : With Application to Optimal Asset Allocation [1 ed.] 9783836616645, 9783836666640

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Mario Rometsch

Quasi-Monte Carlo Methods in Finance

Copyright © 2008. Diplomica Verlag. All rights reserved.

With Application to Optimal Asset Allocation

Diplom.de

Mario Rometsch Quasi-Monte Carlo Methods in Finance With Application to Optimal Asset Allocation ISBN: 978-3-8366-1664-5 Herstellung: Diplomica® Verlag GmbH, Hamburg, 2008

Copyright © 2008. Diplomica Verlag. All rights reserved.

Dieses Werk ist urheberrechtlich geschützt. Die dadurch begründeten Rechte, insbesondere die der Übersetzung, des Nachdrucks, des Vortrags, der Entnahme von Abbildungen und Tabellen, der Funksendung, der Mikroverfilmung oder der Vervielfältigung auf anderen Wegen und der Speicherung in Datenverarbeitungsanlagen, bleiben, auch bei nur auszugsweiser Verwertung, vorbehalten. Eine Vervielfältigung dieses Werkes oder von Teilen dieses Werkes ist auch im Einzelfall nur in den Grenzen der gesetzlichen Bestimmungen des Urheberrechtsgesetzes der Bundesrepublik Deutschland in der jeweils geltenden Fassung zulässig. Sie ist grundsätzlich vergütungspflichtig. Zuwiderhandlungen unterliegen den Strafbestimmungen des Urheberrechtes. Die Wiedergabe von Gebrauchsnamen, Handelsnamen, Warenbezeichnungen usw. in diesem Werk berechtigt auch ohne besondere Kennzeichnung nicht zu der Annahme, dass solche Namen im Sinne der Warenzeichen- und Markenschutz-Gesetzgebung als frei zu betrachten wären und daher von jedermann benutzt werden dürften. Die Informationen in diesem Werk wurden mit Sorgfalt erarbeitet. Dennoch können Fehler nicht vollständig ausgeschlossen werden und der Verlag, die Autoren oder Übersetzer übernehmen keine juristische Verantwortung oder irgendeine Haftung für evtl. verbliebene fehlerhafte Angaben und deren Folgen. © Diplomica Verlag GmbH http://www.diplomica.de, Hamburg 2008

Abstract

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We introduce some quasi-Monte Carlo methods and show how to apply them to problems emerging from mathematical finance. Next, the Malliavin derivative is defined and the Clark-Ocone formula is derived. Then, following a work of Detemple, Garcia and Rindisbacher [DGR05] from 2005, the problem of optimal portfolio allocation and consumption is solved, where the optimal portfolio shares are given as expectations of certain random variables and their derivatives. We show how quasi-Monte Carlo methods can be applied to the computation of portfolio weights.

Acknowledgment I am deeply indebted to both my advisors, Prof. Dr. R¨ udiger Kiesel and Prof. Dr. Karsten Urban, for their constant support. Without their help and inspiration, this work would not have been possible. I would also like to thank Prof. Dr. Alexander Keller for providing me his valuable insights on the quasi-Monte Carlo theory. Special thanks go to my parents, Inge and Werner Rometsch for putting me through university, and of course I also thank my sister, Jana Rometsch, for being such an important person for me. I would like to express my gratitude to my scholarship sponsor, the “Deutsche Forschungsgemeinschaft“, for allowing me to work as a research student for over one and a half year now in the “Graduiertenkolleg 1100” here at the university of Ulm. I am grateful to Daniel Bauer for helping me out with some questions that emerged during this work. I would also like to thank Michael Lehn and Alexander Stippler for providing me FLENS, the Flexible Library for Efficient Numerical Solutions. With such an exceptional helpful linear algebra software package for C++, I saved much implementation time while using the full spectrum of computer vector arithmetic. Furthermore, I thank the “Institute for Numerical Mathematics” for letting me use their computer pool.

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I am very thankful to my colleagues Wolfgang H¨ ogerle, Dennis Sch¨ atz and Jens Stephan for proofreading this book numerous times. I would also like to thank the open source community for providing such excellent quality software products such as (in no particular order of preference) Ubuntu Linux, GCC, eclipse, QuantLib, Gnuplot, unison, LATEX and Gnome. Finally, I would like to express my deepest gratitude for the constant support, understanding and love that I received from my beloved mate Sabrina. You make the sun shine everyday.

Contents

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Introduction

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1 Monte Carlo and quasi-Monte Carlo methods 1.1 Numerical integration . . . . . . . . . . . . . . . . 1.2 Evaluation of integrals with Monte Carlo methods 1.3 Quasi-Monte Carlo methods . . . . . . . . . . . . . 1.3.1 Introduction . . . . . . . . . . . . . . . . . 1.3.2 Discrepancy . . . . . . . . . . . . . . . . . . 1.3.3 The Koksma-Hlawka inequality . . . . . . . 1.4 Classical constructions . . . . . . . . . . . . . . . . 1.4.1 One-dimensional sequences . . . . . . . . . 1.4.2 Multi-dimensional sequences . . . . . . . . 1.5 (t,m,s)-nets and (t,s)-sequences . . . . . . . . . . . 1.5.1 Variance reduction . . . . . . . . . . . . . . 1.5.2 Nets and sequences . . . . . . . . . . . . . . 1.5.3 Two constructions for (t,s)-sequences . . . . 1.6 Digital nets and sequences . . . . . . . . . . . . . . 1.7 Lattice rules . . . . . . . . . . . . . . . . . . . . . . 1.8 The curse of dimension revisited . . . . . . . . . . 1.8.1 Padding techniques . . . . . . . . . . . . . . 1.8.2 Latin Supercube sampling . . . . . . . . . . 1.9 Time consumption of the various point generators 1.10 quasi-Monte Carlo methods in Finance . . . . . . . 1.10.1 Example: Arithmetic option . . . . . . . . . 1.10.2 Path generation . . . . . . . . . . . . . . . . 1.10.3 Sampling size . . . . . . . . . . . . . . . . . 1.10.4 Results . . . . . . . . . . . . . . . . . . . .

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1 1 2 3 3 3 5 6 6 7 10 10 11 13 20 21 22 23 23 25 26 26 27 34 36

2 Malliavin Calculus 2.1 Wiener-Itˆ o chaos expansion . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Skorohod integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Differentiation of random variables . . . . . . . . . . . . . . . . . . . . .

40 40 46 50

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2.4 2.5 2.6 2.7

Examples of Malliavin derivatives . . The Clark-Ocone formula . . . . . . The generalized Clark-Ocone formula Multivariate Malliavin Calculus . . .

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3 Asset Allocation 3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Financial market model . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Wealth process . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Expected utility . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Portfolio problem . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Equivalent static problem . . . . . . . . . . . . . . . . . . . . . 3.1.6 Optimal portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Solution of the portfolio problem . . . . . . . . . . . . . . . . . . . . . 3.2.1 Optimal portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Optimal portfolio with constant relative risk aversion (CRRA)

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4 Implementation 4.1 A single state variable model with explicit solution . 4.2 Simulation-based approach . . . . . . . . . . . . . . 4.3 SDE system as multidimensional SDE . . . . . . . . 4.4 Error analysis . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Discretisation error . . . . . . . . . . . . . . . 4.4.2 Conditional expectation approximation error 4.5 Numerical results . . . . . . . . . . . . . . . . . . . . 4.5.1 One year time horizon . . . . . . . . . . . . . 4.5.2 Two year time horizon . . . . . . . . . . . . . 4.5.3 Five year time horizon . . . . . . . . . . . . . 4.5.4 Experiments with a small time horizon . . . .

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97 97 100 101 102 103 104 106 108 111 114 117

Conclusion

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Summary

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List of Figures 1.1

1.14

The quasi-random sequence fills the unit square more equally than the pseudo-random sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . First 1000 points of the 30-dimensional Halton sequence. . . . . . . . . . An example of stratified sampling. . . . . . . . . . . . . . . . . . . . . . An example of Latin hypercube sampling. . . . . . . . . . . . . . . . . . Example of elementary intervals in base 2, with volume 1/16. . . . . . . A (0, 4, 2)-net in base 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . First 1024 points of the 100-dimensional Sobol’ sequence . . . . . . . . . Six steps of a Brownian Bridge construction. . . . . . . . . . . . . . . . 3D view of a Brownian Bridge construction. . . . . . . . . . . . . . . . . Monte Carlo convergence order is as expected. . . . . . . . . . . . . . . The Sobol’ sequence without integrand transformation does not achieve a higher convergence order. . . . . . . . . . . . . . . . . . . . . . . . . . The Sobol’ sequence with a Brownian bridge construction gives better results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Sobol’ sequence with Principal Component Analysis attains optimal convergence order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The rank-1 lattice also attains optimal convergence order. . . . . . . . .

2.1

A sample path of the Wiener process, and the path perturbed at t = 0.3. 58

4.1

A sample path of the market price of risk and the corresponding sample path of the stock. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The dimensionality resulting from the number of simulated trajectories. One year time horizon. Two Monte Carlo computations with various proportions of paths and time discretisation steps. . . . . . . . . . . . . One year time horizon. Three quasi-Monte Carlo computations with various proportions of paths and time discretisation steps. . . . . . . . . One year time horizon. Computation with the Sobol’ sequence with various proportions of paths and time discretisation steps. . . . . . . . . One year time horizon. Computation with the Sobol’ sequence in combination with the Milshtein approximation. . . . . . . . . . . . . . . . .

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13

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4.2 4.3 4.4 4.5 4.6

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8 9 10 11 12 13 30 30 31 37 37 38 38 39

97 105 108 108 109 109

4.7 4.8 4.9 4.10

4.11 4.12 4.13 4.14 4.15 4.16 4.17

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4.18 4.19

One year time horizon. Direct comparison of two quasi-Monte Carlo and a Monte Carlo scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Two year time horizon. The Monte Carlo scheme converges as expected. 111 Two year time horizon. The Monte Carlo method is again outclassed by the Sobol’ sequence in combination with a Brownian Bridge. . . . . . . 111 Two year time horizon. The Sobol’ sequence in combination with a Brownian Bridge construction converges faster than the Sobol’ sequence with a PCA construction. . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Two year time horizon. Computation with the Sobol’ sequence in combination with the Milshtein scheme. . . . . . . . . . . . . . . . . . . . . 112 Two year time horizon. The Latin Supercube sampling from a Sobol’ sequence shows no advantage compared to Monte Carlo sampling. . . . 113 Five year time horizon. The Monte Carlo scheme converges as expected. 114 Five year time horizon. The Monte Carlo method is again outclassed by the Sobol’ sequence in combination with a Brownian Bridge construction. 114 Five year time horizon. Again, the PCA construction does not result in a higher convergence order. . . . . . . . . . . . . . . . . . . . . . . . . . 115 Five year time horizon. Computation with the Sobol’ sequence and the Milshtein scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Five year time horizon. The Latin Supercube sampling gives no advantage over Monte Carlo integration. . . . . . . . . . . . . . . . . . . . . . 116 Small time horizon. The Monte Carlo scheme keeps its convergence order.117 Small time horizon. The Sobol’ sequence with the PCA construction achieves nearly optimal convergence. . . . . . . . . . . . . . . . . . . . . 118

iv

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Introduction Portfolio optimization is a widely studied problem in finance. The common question is, how a small investor should invest his wealth in the market to attain certain goals, like a desired payoff or some insurance against unwished events. The starting point for the mathematical treatment of this is the work of Harry Markowitz in the 1950s. His idea was to set up a relation between the mean return of a portfolio and its variance. In his terminology, an efficient portfolio has minimal variance of return among others with the same mean rate of return. Furthermore, if linear combinations of efficient portfolios and a riskless asset are allowed, this leads to the market portfolio, so that a linear combination of the risk-free asset and the market portfolio dominates any other portfolio in the mean-variance sense. Later, this theory was extended resulting in the CAPM, or capital asset pricing model, which was independently introduced by Treynor, Sharpe, Lintner and Mossin in the 1960s. In this model, every risky asset has a mean rate of return that exceeds the risk-free rate by a specific risk premium, which depends on a certain attribute of the asset, namely its β. The so-called β in turn is the covariance of the asset return normalized by the variance of the market portfolio. The problem of the CAPM is its static nature, investments are made once and then the state of the model changes. Due to this and other simplifications, this model was and is often not found to be realistic. An impact to this research field were the two papers of Robert Merton in 1969 [Mer69] and 1971 [Mer71]. He applied the theory of Itˆ o calculus and stochastic optimal control and solved the corresponding Hamilton-Jacobi-Bellman equation. For his multiperiod model, he assumed constant coefficients and an investor with power utility. Extending the mean-variance analysis, he found that a long-term investor would prefer a portfolio that includes hedging components to protect against fluctuations in the market. Again this approach was generalized by numerous researchers and results in the problem of solving a nonlinear partial differential equation. The next milestone in this series is the work by Cox and Huang [CH89] from 1989, where they solve for “Optimal Consumption and Portfolio Policies when Asset Prices Follow a Diffusion Process”. They apply the martingale technique to get rid of the nonlinear PDE and rather solve a linear PDE. This, with several refinements, is nowadays a standard method for asset allocation in a complete market. Approximately at the same time in 1991, Ocone and Karatzas published two pa-

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pers, [OK91] and [KOL91], in which they extend the Clark-Haussmann representation formula to Wiener functionals under an equivalent measure. They use the martingale method for selecting the optimal terminal wealth resp. optimal consumption and then apply the representation theorem to derive the optimal portfolio strategy. This expression involves expectations of random variables depending on the interest rate, the market prices of risk and unspecified derivatives of these state variables. Because of this unspecification, the hedging terms of the resulting portfolio do not have an explicit form and are pretty difficult to evaluate numerically. Therefore, the recent literature focused more on models with state variable specifications, for which closed-form solutions are available, for example [Wac02], or on specifications which, as mentioned above, rely on dynamic programming.

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One main part of this book is now to present the latest work of Detemple, Garcia and Rindisbacher, [DGR03a] and [DGR05], in which they tie up the ideas from Ocone and Karatzas. They solve for optimal portfolio allocation in a complete market model and derive explicit expressions for the hedging terms. The optimal portfolio shares are found to be expectations of random variables, which allow a simulation-based approach. In opposite to the dynamic programming approach, this method is capable of handling complex, nonlinear dynamics for a large number of state variables. When then the optimal portfolio shares need to be calculated, this basically reduces to a high-dimensional integration over the unit hypercube. Perhaps because of its easy and straightforward implementation, this is most often done with Monte Carlo methods. Niederreiter [Nie92] notes that the official emergence of the Monte Carlo method was a paper of Nicholas Metropolis and Stanislaw Ulam in 1949, but at that time, pseudo-random algorithms had been used for years in secret projects of the U.S. Department of Defense, like the Manhattan project. Since then, these methods have become the most widely used tools for computing high-dimensional integrals. Although they have appealing features like weak regularity conditions, all these Monte Carlo or pseudo-random methods suffer from one severe shortcoming: their convergence order stems from the central limit theorem, hence they converge very slowly. The quasi-Monte Carlo method now tries to overcome this hitch while keeping the applicability to high-dimensional integration at the same time. This is achieved by sampling from very careful chosen deterministic points. The methods and algorithms then are not called pseudo-random but quasi-random. The term “quasi-Monte Carlo” method appeared the first time in a technical report from 1951, and Niederreiters outstanding monograph [Nie92] cites many works and applications to problems like the numerical solution of integral equations or integro-differential equations. Quasi-Monte

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Carlo applications to problems from the finance setting came up in the nineties with ´ work from Paskov, L’Ecuyer et al. This topic composes the second part of this work, in which we will present some concepts from the quasi-Monte Carlo theory to problems that emerge in mathematical finance. In this book, we derive the optimal portfolio formula on the basis of the proofs in [DGR03a] resp. [DGR05]. The main contribution is then the application of some quasiMonte Carlo methods for its computation. Using a simple model with exact solution, different schemes were tested and showed, that the substitution of the pseudo-random number generator for the quasi-random number generator requires special care. This book is organized as follows. In Chapter 1, we present some concepts from the quasi-Monte Carlo theory and show, how these methods can be applied to price a simple financial derivative, an arithmetic call option. In Chapter 2 we follow the lecture notes [Øks97] and present an introduction to the “stochastic calculus of variations“ or Malliavin Calculus, which will culminate in the definition of the Malliavin derivative and the derivation of the Clark-Ocone formula.

Chapter 1: quasi-Monte Carlo methods

Chapter 2: Malliavin Calculus

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Chapter 3: Asset Allocation



Chapter 4: Implementation



The reader wishing a quick start to the asset allocation problem may thus jump directly to Chapter 3. There, we will introduce the financial market model and formulate the optimization problem. We will follow the work of Detemple, Garcia and Rindisbacher [DGR03a] resp. [DGR05] and derive the optimal portfolio formula. However, for this we will need the calculation rules from Chapter 2, so if skipped, the optimal portfolio formula has to be taken as granted. The implementation in Chapter 4 will then apply these formulas to a simple model with explicit solution, and we show, how the quasiMonte Carlo methods from Chapter 1 can be used to improve the efficiency over plain Monte Carlo methods.

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1 Monte Carlo and quasi-Monte Carlo methods To get a feeling for the integration techniques now commonly known as “quasi-Monte Carlo methods”, we will first introduce some basics from numerical integration and classical Monte Carlo methods. After that, we will illustrate the idea behind quasirandom algorithms and present some integration rules. At the end of this chapter, we show how a simple financial derivative can be priced with these techniques. We won’t go too much into theoretical details here, the reader wishing some more thorough overview is referred to the great book of Harald Niederreiter [Nie92]. We will follow this book in our introduction.

1.1 Numerical integration The evaluation of integrals in dimension s is a standard problem of numerical analysis. For s = 1 there are some classical numerical methods as, for example, the trapezoidal rule:   
n 1  : i = 0 or i = n i 2n with ωi = (1.1) f (x) dx ≈ ωi f 1 n : 0