Stochastic Methods in Asset Pricing Andrew Lyasoff
The MIT Press Cambridge, Massachusetts London, England
© 2017 Massachusetts Institute of Technology All rights are reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage or retrieval) without permission in writing from the publisher. This book is typeset by the author with TEX and Emacs Lisp. ♾ Printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Names: Lyasoff, Andrew, author. Title: Stochastic methods in asset pricing / Andrew Lyasoff. Description: Cambridge, MA : MIT Press, [2017] | Includes bibliographical references and index. Identifiers: LCCN 2017000433 | ISBN 9780262036559 (hardcover : alk. paper) Subjects: LCSH: Securities–Prices–Mathematical models. | Stochastic processes. Classification: LCC HG4636 .L93 2017 | DDC 332.63/2–dc23 LC record available at https://lccn.loc.gov/2017000433
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“The special sphere of finance within economics is the study of allocation and deployment of economic resources, both spatially and across time, in an uncertain environment. To capture the influence and interaction of time and uncertainty effectively requires sophisticated mathematical and computational tools. Indeed, mathematical models of modern finance contain some truly elegant applications of probability and optimization theory. These applications challenge the most powerful computational technologies. But, of course, all that is elegant and challenging in science need not also be practical; and surely, not all that is practical in science is elegant and challenging. Here we have both. In the time since publication of our early work on the option-pricing model, the mathematically complex models of finance theory have had a direct and wide-ranging influence on finance practice. This conjoining of intrinsic intellectual interest with extrinsic application is central to research in modern finance.” — Robert C. Merton (from his Nobel Prize lecture, December 9, 1997)
“There’s No Such Thing as a Free Lunch” — Milton Friedman (title of his 1975 book)
CONTENTS
Preface .
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Notation
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1. Probability Spaces and Related Structures .
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2. Integration
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3. Absolute Continuity, Conditioning, and Independence
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4. Convergence of Random Variables
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Preliminaries
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1.1 Randomness in the Financial Markets . . . . . 1.2 A Bird’s-Eye View of the One-Period Binomial Model 1.3 Probability Spaces . . . . . . . . . . . 1.4 Coin Toss Space and Random Walk . . . . . 1.5 Borel 𝜎-Fields and Lebesgue Measure . . . . .
2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2 3.3 3.4 3.5
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Measurable Functions and Random Variables Distribution Laws . . . . . . . . Lebesgue Integral . . . . . . . . Convergence of Integrals . . . . . . Integration Tools . . . . . . . . The Inverse of an Increasing Function . .
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Quasi-invariance of the Gaussian Distribution under Translation . Moment-Generating Functions, Laplace, and Fourier Transforms . Conditioning and Independence . . . . . . . . . . . . Multivariate Gaussian Distribution . . . . . . . . . . . Hermite–Gauss Quadratures . . . . . . . . . . . . . .
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4.1 Types of Convergence for Sequences of Random Variables . . . . . . . 97 4.2 Uniform Integrability . . . . . . . . . . . . . . . . . . . 103 4.3 Sequences of Independent Random Variables and Events . . . . . . . . 105 4.4 Law of Large Numbers and the Central Limit Theorem . . . . . . . . 108
5. The Art of Random Sampling 5.1 5.2 5.3 5.4 5.5
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Motivation . . . . . . . Layer Cake Formulas . . . . The Antithetic Variates Method . The Importance Sampling Method The Acceptance–Rejection Method
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CONTENTS
6. Equilibrium Asset Pricing in Finite Economies 6.1 6.2 6.3 6.4 6.5 6.6
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Information Structure . . . . . . . . . . . . . . . . . . . Risk Preferences . . . . . . . . . . . . . . . . . . . . The Multiperiod Endowment Economy . . . . . . . . . . . . . General Equilibrium . . . . . . . . . . . . . . . . . . . The Two Fundamental Theorems of Asset Pricing . . . . . . . . . . From Stochastic Discount Factors to Equivalent Measures and Local Martingales
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7. Crash Course on Discrete-Time Martingales
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8. Stochastic Processes and Brownian Motion
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9. Crash Course on Continuous-Time Martingales
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7.1 7.2 7.3
8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 9.1 9.2 9.3 9.4 9.5 9.6 9.7
Basic Concepts and Definitions . . . . . . . . . . . . . . . . 161 Predictable Compensators . . . . . . . . . . . . . . . . . 168 Fundamental Inequalities and Convergence . . . . . . . . . . . . 170
General Properties and Definitions . . . . . . Limit of the Binomial Asset Pricing Model . . . Construction of Brownian Motion and First Properties The Wiener Measure . . . . . . . . . . Filtrations, Stopping Times, and Such . . . . . Brownian Filtrations . . . . . . . . . . Total Variation . . . . . . . . . . . . Quadratic Variation . . . . . . . . . . Brownian Sample Paths Are Nowhere Differentiable Some Special Features of Brownian Sample Paths .
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Definitions and First Properties . . . . . . . . Poisson Process and First Encounter with Lévy Processes Regularity of Paths, Optional Stopping, and Convergence Doob–Meyer Decomposition . . . . . . . . Local Martingales and Semimartingales . . . . . The Space of 𝐿2 -Bounded Martingales . . . . . The Binomial Asset Pricing Model Revisited . . .
10. Stochastic Integration .
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10.1 Basic Examples and Intuition . . . . . . . . . . . 10.2 Stochastic Integrals with Respect to Continuous Local Martingales 10.3 Stochastic Integrals with Respect to Continuous Semimartingales 10.4 Itô’s Formula . . . . . . . . . . . . . . . . 10.5 Stochastic Integrals with Respect to Brownian Motion . . . 10.6 Girsanov’s Theorem . . . . . . . . . . . . . . 10.7 Local Times and Tanaka’s Formula . . . . . . . . . 10.8 Reflected Brownian Motion . . . . . . . . . . . .
11. Stochastic Differential Equations 11.1 11.2 11.3 11.4 11.5
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An Example . . . . . . . . . . . . . . Strong and Weak Solutions . . . . . . . . . Existence of Solutions . . . . . . . . . . . Linear Stochastic Differential Equations . . . . . Some Common Diffusion Models Used in Asset Pricing
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CONTENTS
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12. The Connection between SDEs and PDEs . . . . . . . . . . . . . . 325 12.1 Feynman–Kac Formula . . . . . . . . . . . . . . . . . . 325 12.2 Fokker–Planck Equation . . . . . . . . . . . . . . . . . . 331 13. Brief Introduction to Asset Pricing in Continuous Time 13.1 Basic Concepts and Definitions . . . . . . 13.2 Trading Strategy and Wealth Dynamics . . . 13.3 Equivalent Local Martingale Measures . . . 13.4 The Two Fundamental Theorems of Asset Pricing
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14. Replication and Arbitrage . . . . . . . . . 14.1 Résumé of Malliavin Calculus . . . . . . 14.2 European-Style Contingent Claims . . . . . 14.3 The Martingale Solution to Merton’s Problem . 14.4 American-Style Contingent Claims . . . . . 14.5 Put–Call Symmetry and Foreign Exchange Options 14.6 Exchange Options . . . . . . . . . . 14.7 Stochastic Volatility Models . . . . . . . 14.8 Dupire’s Formula . . . . . . . . . .
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15. Résumé of Stochastic Calculus with Discontinuous Processes . . . . . . . . 397 15.1 Martingales, Local Martingales, and Semimartingales with Jumps . . . . . 397 15.2 Stochastic Integrals with Respect to Semimartingales with Jumps . . . . . 409 15.3 Quadratic Variation and Itô’s Formula . . . . . . . . . . . . . . 417 16. Random Measures and Lévy Processes . . . . . 16.1 Poisson Random Measures . . . . . . . 16.2 Lévy Processes . . . . . . . . . . . 16.3 Stochastic Integrals with Respect to Lévy Processes 16.4 Stochastic Exponents . . . . . . . . . 16.5 Change of Measure and Removal of the Drift . 16.6 Lévy–Itô Diffusions . . . . . . . . . 16.7 An Asset Pricing Model with Jumps in the Returns
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17. Résumé of the Theory and Methods of Stochastic Optimal Control . . . . . 463 17.1 The Moon-Landing Problem . . . . . . . . . . . . . . . . . 464 17.2 Principle of Dynamic Programming and the HJB Equation . . . . . . . 465 17.3 Some Variations of the PDP and the HJB Equation . . . . . . . . . . 474 18. Applications to Dynamic Asset Pricing . . . . . . . . . . . . . . . 18.1 Merton’s Problem with Intertemporal Consumption and No Rebalancing Costs . 18.2 Merton’s Problem with Intertemporal Consumption and Rebalancing Costs . . 18.3 Real Options . . . . . . . . . . . . . . . . . . . . . . 18.4 The Exercise Boundary for American Calls and Puts . . . . . . . . . 18.5 Corporate Debt, Equity, Dividend Policy, and the Modigliani–Miller Proposition Appendix A: Résumé of Analysis and Topology
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Appendix B: Computer Code . . . . . . . . . . . . . B.1 Working with Market Data . . . . . . . . . . . . B.2 Simulation of Multivariate Gaussian Laws . . . . . . . B.3 Numerical Program for American-Style Call Options . . . .
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Select Bibliography Index
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Preface There is no doubt that asset pricing is one manifestly important subject, and packing in a single course of study, or a book, most concepts from probability and stochastic processes that asset pricing – especially asset pricing in continuous time – builds upon is one nearly impossible project. Nevertheless, the present book is the outcome from my attempt at one such project. It grew out of lecture notes that I developed over the course of many years and is faithful to its title: in most part it is a book about stochastic methods used in the domain of asset pricing, not a book on asset pricing, despite the fact that some important aspects of this discipline are featured prominently, and not just as an aside. What follows in this space is the material that I typically cover in a twosemester first-year graduate course (approximately 90 lecture hours). The book is therefore meant for readers who are not familiar with measure-theoretic probability, stochastic calculus, and asset pricing, or perhaps readers who are well versed in stochastic calculus, but are not familiar with its connections to continuous-time finance. Compared to other classical texts that pursue similar objectives – and the book (Karatzas and Shreve 1991) is the first to come to mind – the present one starts from a much earlier stage, namely from measure-theoretic probability and integration (and includes also a brief synopsis of analysis and topology). At the same time it is less rigorous, in the sense that many fundamental results are stated without proofs, which made it possible to keep the scope of the material quite broad and to include some advanced topics such as Lévy processes and stochastic calculus with jumps. In doing this, I attempted the impossible task of introducing in a more or less rigorous fashion some advanced features of local martingales and semimartingales, and connecting those features with the principles of asset pricing, while leaving out most of the proofs. My insistence that models involving processes with jumps must be covered made this task all the more difficult. By and large, in the second decade of the twenty-first century probability theory and stochastic processes can be considered as well developed and mature mathematical disciplines. However, while indispensable for modeling and understanding of random phenomena, including those encountered in finance and economics, the language and the tools that these disciplines provide are yet to become commonplace – as, say, calculus, statistics, or linear algebra already are commonplace. In some sense, this book is an attempt to make the theory of stochastic processes a bit less exotic and a bit more accessible. The reason why achieving such an objective is not easy is that the terminology, theory, methods, and ideas, from the general theory of stochastic processes in continuous time – expounded in (Doob 1953), (Meyer 1966), (Itô and McKean 1974), (Dellacherie and Meyer 1980), (Jacod and Shiryaev 1987), and (Revuz and Yor 1999) – is intrinsically complex and, yes, technical and at times overwhelming. Thus it is impossible to develop an introduction to this subject for first-year graduate students without “glossing over” some of the technical concepts involved. As a result, most introductory texts on the subject, including the present one, differ mainly in what is being “glossed over”
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Preface
and how. My particular choice is based on the recognition that there is a point at which playing down too many connections and asking for too much suspension of disbelief could make a learner of the field feel lost. Most of the material covered in the present book can be found – and in a much more rigorous and detailed form – in a number of classical texts (see below). My goal was to compile a coherent single overview, with at least some attempt at pedagogy. By way of an example, discrete-time martingales are first introduced as a consequence of market equilibrium, and are then connected with the stochastic discount factors, before the general definition is given. In addition, there are more than 450 exercises spread throughout the text (many accompanied with hints), which are meant to help the reader develop better understanding and intuition. A very brief synopsis of analysis and topology is included in appendix A, while other preparatory material is included at the very beginning – I hope these additions will make following the main part of the book a bit easier. The recent results on general market equilibrium for incomplete financial markets from (Dumas and Lyasoff 2012), and on the no-arbitrage condition from (Lyasoff 2014) were incorporated in the book as well. Most of the concrete models discussed in the book (e.g., models of stochastic volatility) were borrowed from research papers, which are quoted in the text. While concrete practical applications are not in the focus of the book, a small number of such applications are detailed, and some of the related (working) computer code is included in appendix B. To the best of my knowledge, the numerical method and the program described in section B.3 are new. This book can be used in a number of ways. The obvious one is to cover the entire material in the course of two semesters. For a one-semester course, one could consider covering chapters 1 through 14 – perhaps omitting 5, 7, and possibly 12. But on the other hand, for students who have already studied measure-theoretic probability and integration, one would start from chapter 5 and spend more time on chapters 13 and 14 (perhaps with greater emphasis on practical applications). It is also possible to design a short course in probability theory (with or without applications to finance) based on chapters 1 through 5, or a short course on stochastic calculus for continuous processes based on chapters 8 through 12, or a short course on stochastic calculus with jumps and Lévy processes based on chapters 15 and 16, or a short course on optimal control based on chapters 17 and 18. Finally, one could design a short introductory course in asset pricing, meant for an audience familiar with all mathematical aspects of probability, stochastic processes and optimal control, by following chapters 6, 13, 14, and 18. For readers who intend to enter the field of finance, it would be beneficial to read this book in parallel with, for example, (Cochrane 2009), (Duffie 2010), and (Föllmer and Schied 2011), and then follow up with at least some of the following books: (Aït-Sahalia and Jacod 2014), (Crépey 2012), (Duffie 2011), (Duffie and Singleton 2012), (Glasserman 2004), (Jeanblanc et al. 2009), (Karatzas and Shreve
Preface
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1998), (Musiela and Rutkowski 2005), and (Carmona and Tehranchi 2007) – the choices are many and very much depend on personal preferences and professional objectives. The readers of this book are assumed to be firmly in the know about undergraduate analysis, linear algebra, and differential equations. Familiarity with real analysis at greater depth – say, at the level of (Rudin 1976) – would be very beneficial. Some basic knowledge of finance and economics – say, by studying (Fabozzi et al. 2010), (Kreps 2013), and (Mishkin 2015) – would be beneficial as well. In my work toward this book I have benefited enormously from my professional interactions with (in a somewhat chronological order): Jordan Stoyanov, Svetoslav Gaidow, Mark Davis, Yuri Kabanov, Alexander Novikov, Albert Shiryaev, Paul-André Meyer, Hans Föllmer, Ioannis Karatzas, Martin Goldstein, Daniel W. Stroock, Persi Diaconis, Ofer Zeitouni, Moshe Zakai, James Norris, Marc Yor, Robert C. Merton, Steven Ross, Peter Carr, and Bernard Dumas. I am greatly indebted to all of them, and am also obliged to the following indispensable sources: (Meyer 1966), (Itô and McKean 1974), (Liptser and Shiryaev 1974), (Dellacherie and Meyer 1980), (Stroock 1987), (Jacod and Shiryaev 1987), (Merton 1992), (Karatzas 1996), (Karatzas and Shreve 1998), (Revuz and Yor 1999), (Dudley 2002), (Duffie 2010), (Föllmer and Schied 2011), and not the least to (Knuth 1984). There are many other sources that are just as relevant, and I acknowledge that the ones on which this book builds in terms of content and style – and these are also the sources most often cited throughout the text – reflect my personal bias, which in turn reflects the circumstances under which I was introduced to the subject. In the last several years I was blessed with lots of extraordinarily motivated students. I am grateful to many of them who spotted numerous typos, errors, and omissions in earlier versions of this book, and extend my special thanks to: Xue Bai, Ge (Dovie) Chu, Yi (Frances) Ding, Juntao (Eric) Fang, Xinwei (Richard) Huang, Tong Jin, Quehao (Tony) Li, Cheng Liang, Knut Lindaas, Dhruv Madeka, Wayne Nielsen, Sunjoon Park, Hao (Sophia) Peng, Maneesha Premaratne, Ao (Nicholas) Shen, Yu Shi, Mingyan Wang, Lyan Wong, Weixuan Xia, Qing (Agatha) Xu, and Wenqi Zhang. Special thanks are due to the MIT Press and my editors Virginia Crossman and Emily Taber for their interest in this project, seemingly unlimited patience, and relentless pursuit of perfection. The debt of gratitude that I owe to my wife Hannelore is yet another example of a nonmeasurable set. In the course of many years she made it possible for me to work on my projects with only a minimal distraction from other matters in life. Without her never failing support and encouragement this book would not have been written. Andrew Lyasoff July 31, 2017
