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Introduction to the Economics and Mathematics of Financial Markets Jakša Cvitanić and Fernando Zapatero The MIT Press Cambridge, Massachusetts London, England c 2004 Massachusetts Institute of Technology  All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. This book was set in 10/13 Times Roman by ICC and was printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Cvitanić, Jakša Introduction to the economics and mathematics of financial markets / Jakša Cvitanić and Fernando Zapatero. p. cm. Includes bibliographical references and index. ISBN 0-262-03320-8 ISBN 0-262-53265-4 (International Student Edition) 1. Finance—Mathematical models—Textbooks. I. Zapatero, Fernando. II. Title. HG106.C86 2004 332.632 01 515—dc22 2003064872 To Vesela, Lucia, Toni and Maitica, Nicolás, Sebastián This page intentionally left blank Contents Preface I 1 1.1 1.2 1.3 1.4 1.5 1.6 2 2.1 THE SETTING: MARKETS, MODELS, INTEREST RATES, UTILITY MAXIMIZATION, RISK xvii 1 Financial Markets Bonds 1.1.1 Types of Bonds 1.1.2 Reasons for Trading Bonds 1.1.3 Risk of Trading Bonds Stocks 1.2.1 How Are Stocks Different from Bonds? 1.2.2 Going Long or Short Derivatives 1.3.1 Futures and Forwards 1.3.2 Marking to Market 1.3.3 Reasons for Trading Futures 1.3.4 Options 1.3.5 Calls and Puts 1.3.6 Option Prices 1.3.7 Reasons for Trading Options 1.3.8 Swaps 1.3.9 Mortgage-Backed Securities; Callable Bonds Organization of Financial Markets 1.4.1 Exchanges 1.4.2 Market Indexes Margins 1.5.1 Trades That Involve Margin Requirements Transaction Costs Summary Problems Further Readings 3 3 5 5 6 7 8 9 9 10 11 12 13 13 15 16 17 19 20 20 21 22 23 24 25 26 29 Interest Rates Computation of Interest Rates 2.1.1 Simple versus Compound Interest; Annualized Rates 2.1.2 Continuous Interest 31 31 32 34 viii 2.2 2.3 3 3.1 3.2 3.3 3.4 3.5 Contents Present Value 2.2.1 Present and Future Values of Cash Flows 2.2.2 Bond Yield 2.2.3 Price-Yield Curves Term Structure of Interest Rates and Forward Rates 2.3.1 Yield Curve 2.3.2 Calculating Spot Rates; Rates Arbitrage 2.3.3 Forward Rates 2.3.4 Term-Structure Theories Summary Problems Further Readings 35 36 39 39 41 41 43 45 47 48 49 51 Models of Securities Prices in Financial Markets Single-Period Models 3.1.1 Asset Dynamics 3.1.2 Portfolio and Wealth Processes 3.1.3 Arrow-Debreu Securities Multiperiod Models 3.2.1 General Model Specifications 3.2.2 Cox-Ross-Rubinstein Binomial Model Continuous-Time Models 3.3.1 Simple Facts about the Merton-Black-Scholes Model 3.3.2 Brownian Motion Process 3.3.3 Diffusion Processes, Stochastic Integrals 3.3.4 Technical Properties of Stochastic Integrals∗ 3.3.5 Itô’s Rule 3.3.6 Merton-Black-Scholes Model 3.3.7 Wealth Process and Portfolio Process Modeling Interest Rates 3.4.1 Discrete-Time Models 3.4.2 Continuous-Time Models Nominal Rates and Real Rates 3.5.1 Discrete-Time Models 3.5.2 Continuous-Time Models 53 54 54 55 57 58 58 60 62 62 63 66 67 69 74 78 79 79 80 81 81 83 Contents 3.6 3.7 4 4.1 4.2 4.3 4.4 4.5 4.6 ix Arbitrage and Market Completeness 3.6.1 Notion of Arbitrage 3.6.2 Arbitrage in Discrete-Time Models 3.6.3 Arbitrage in Continuous-Time Models 3.6.4 Notion of Complete Markets 3.6.5 Complete Markets in Discrete-Time Models 3.6.6 Complete Markets in Continuous-Time Models∗ Appendix 3.7.1 More Details for the Proof of Itô’s Rule 3.7.2 Multidimensional Itô’s Rule Summary Problems Further Readings 83 84 85 86 87 88 92 94 94 97 97 98 101 Optimal Consumption / Portfolio Strategies Preference Relations and Utility Functions 4.1.1 Consumption 4.1.2 Preferences 4.1.3 Concept of Utility Functions 4.1.4 Marginal Utility, Risk Aversion, and Certainty Equivalent 4.1.5 Utility Functions in Multiperiod Discrete-Time Models 4.1.6 Utility Functions in Continuous-Time Models Discrete-Time Utility Maximization 4.2.1 Single Period 4.2.2 Multiperiod Utility Maximization: Dynamic Programming 4.2.3 Optimal Portfolios in the Merton-Black-Scholes Model 4.2.4 Utility from Consumption Utility Maximization in Continuous Time 4.3.1 Hamilton-Jacobi-Bellman PDE Duality/Martingale Approach to Utility Maximization 4.4.1 Martingale Approach in Single-Period Binomial Model 4.4.2 Martingale Approach in Multiperiod Binomial Model 4.4.3 Duality/Martingale Approach in Continuous Time∗ Transaction Costs Incomplete and Asymmetric Information 4.6.1 Single Period 103 103 104 105 107 108 112 112 113 114 116 121 122 122 122 128 128 130 133 138 139 139 x Contents 4.6.2 Incomplete Information in Continuous Time∗ 4.6.3 Power Utility and Normally Distributed Drift∗ Appendix: Proof of Dynamic Programming Principle Summary Problems Further Readings 140 142 145 146 147 150 Risk Risk versus Return: Mean-Variance Analysis 5.1.1 Mean and Variance of a Portfolio 5.1.2 Mean-Variance Efficient Frontier 5.1.3 Computing the Optimal Mean-Variance Portfolio 5.1.4 Computing the Optimal Mutual Fund 5.1.5 Mean-Variance Optimization in Continuous Time∗ VaR: Value at Risk 5.2.1 Definition of VaR 5.2.2 Computing VaR 5.2.3 VaR of a Portfolio of Assets 5.2.4 Alternatives to VaR 5.2.5 The Story of Long-Term Capital Management Summary Problems Further Readings 153 153 154 157 160 163 164 167 167 168 170 171 171 172 172 175 II PRICING AND HEDGING OF DERIVATIVE SECURITIES 177 6 6.1 6.2 Arbitrage and Risk-Neutral Pricing Arbitrage Relationships for Call and Put Options; Put-Call Parity Arbitrage Pricing of Forwards and Futures 6.2.1 Forward Prices 6.2.2 Futures Prices 6.2.3 Futures on Commodities Risk-Neutral Pricing 6.3.1 Martingale Measures; Cox-Ross-Rubinstein (CRR) Model 6.3.2 State Prices in Single-Period Models 6.3.3 No Arbitrage and Risk-Neutral Probabilities 179 179 184 184 186 187 188 188 192 193 4.7 5 5.1 5.2 6.3 Contents 6.4 7 7.1 7.2 7.3 7.4 7.5 xi 6.3.4 Pricing by No Arbitrage 6.3.5 Pricing by Risk-Neutral Expected Values 6.3.6 Martingale Measure for the Merton-Black-Scholes Model 6.3.7 Computing Expectations by the Feynman-Kac PDE 6.3.8 Risk-Neutral Pricing in Continuous Time 6.3.9 Futures and Forwards Revisited∗ Appendix 6.4.1 No Arbitrage Implies Existence of a Risk-Neutral Probability∗ 6.4.2 Completeness and Unique EMM∗ 6.4.3 Another Proof of Theorem 6.4∗ 6.4.4 Proof of Bayes’ Rule∗∗ Summary Problems Further Readings 194 196 197 201 202 203 206 206 207 210 211 211 213 215 Option Pricing Option Pricing in the Binomial Model 7.1.1 Backward Induction and Expectation Formula 7.1.2 Black-Scholes Formula as a Limit of the Binomial Model Formula Option Pricing in the Merton-Black-Scholes Model 7.2.1 Black-Scholes Formula as Expected Value 7.2.2 Black-Scholes Equation 7.2.3 Black-Scholes Formula for the Call Option 7.2.4 Implied Volatility Pricing American Options 7.3.1 Stopping Times and American Options 7.3.2 Binomial Trees and American Options 7.3.3 PDEs and American Options∗ Options on Dividend-Paying Securities 7.4.1 Binomial Model 7.4.2 Merton-Black-Scholes Model Other Types of Options 7.5.1 Currency Options 7.5.2 Futures Options 7.5.3 Exotic Options 217 217 217 220 222 222 222 225 227 228 229 231 233 235 236 238 240 240 242 243 xii 7.6 7.7 7.8 7.9 8 8.1 8.2 8.3 8.4 9 9.1 Contents Pricing in the Presence of Several Random Variables 7.6.1 Options on Two Risky Assets 7.6.2 Quantos 7.6.3 Stochastic Volatility with Complete Markets 7.6.4 Stochastic Volatility with Incomplete Markets; Market Price of Risk∗ 7.6.5 Utility Pricing in Incomplete Markets∗ Merton’s Jump-Diffusion Model∗ Estimation of Variance and ARCH/GARCH Models Appendix: Derivation of the Black-Scholes Formula Summary Problems Further Readings 247 248 252 255 Fixed-Income Market Models and Derivatives Discrete-Time Interest-Rate Modeling 8.1.1 Binomial Tree for the Interest Rate 8.1.2 Black-Derman-Toy Model 8.1.3 Ho-Lee Model Interest-Rate Models in Continuous Time 8.2.1 One-Factor Short-Rate Models 8.2.2 Bond Pricing in Affine Models 8.2.3 HJM Forward-Rate Models 8.2.4 Change of Numeraire∗ 8.2.5 Option Pricing with Random Interest Rate∗ 8.2.6 BGM Market Model∗ Swaps, Caps, and Floors 8.3.1 Interest-Rate Swaps and Swaptions 8.3.2 Caplets, Caps, and Floors Credit/Default Risk Summary Problems Further Readings 275 275 276 279 281 286 287 289 291 295 296 299 301 301 305 306 308 309 312 Hedging Hedging with Futures 9.1.1 Perfect Hedge 313 313 313 256 257 260 262 265 267 268 273 Contents 9.2 9.3 9.4 9.5 9.1.2 Cross-Hedging; Basis Risk 9.1.3 Rolling the Hedge Forward 9.1.4 Quantity Uncertainty Portfolios of Options as Trading Strategies 9.2.1 Covered Calls and Protective Puts 9.2.2 Bull Spreads and Bear Spreads 9.2.3 Butterfly Spreads 9.2.4 Straddles and Strangles Hedging Options Positions; Delta Hedging 9.3.1 Delta Hedging in Discrete-Time Models 9.3.2 Delta-Neutral Strategies 9.3.3 Deltas of Calls and Puts 9.3.4 Example: Hedging a Call Option 9.3.5 Other Greeks 9.3.6 Stochastic Volatility and Interest Rate 9.3.7 Formulas for Greeks 9.3.8 Portfolio Insurance Perfect Hedging in a Multivariable Continuous-Time Model Hedging in Incomplete Markets Summary Problems Further Readings 10 Bond Hedging 10.1 Duration 10.1.1 Definition and Interpretation 10.1.2 Duration and Change in Yield 10.1.3 Duration of a Portfolio of Bonds 10.2 Immunization 10.2.1 Matching Durations 10.2.2 Duration and Immunization in Continuous Time 10.3 Convexity Summary Problems Further Readings xiii 314 316 317 317 318 318 319 321 322 323 325 327 327 330 332 333 333 334 335 336 337 340 341 341 341 345 346 347 347 350 351 352 352 353 xiv Contents 11 Numerical Methods 11.1 Binomial Tree Methods 11.1.1 Computations in the Cox-Ross-Rubinstein Model 11.1.2 Computing Option Sensitivities 11.1.3 Extensions of the Tree Method 11.2 Monte Carlo Simulation 11.2.1 Monte Carlo Basics 11.2.2 Generating Random Numbers 11.2.3 Variance Reduction Techniques 11.2.4 Simulation in a Continuous-Time Multivariable Model 11.2.5 Computation of Hedging Portfolios by Finite Differences 11.2.6 Retrieval of Volatility Method for Hedging and Utility Maximization∗ 11.3 Numerical Solutions of PDEs; Finite-Difference Methods 11.3.1 Implicit Finite-Difference Method 11.3.2 Explicit Finite-Difference Method Summary Problems Further Readings 355 355 355 358 359 361 362 363 364 367 370 III 381 EQUILIBRIUM MODELS 12 Equilibrium Fundamentals 12.1 Concept of Equilibrium 12.1.1 Definition and Single-Period Case 12.1.2 A Two-Period Example 12.1.3 Continuous-Time Equilibrium 12.2 Single-Agent and Multiagent Equilibrium 12.2.1 Representative Agent 12.2.2 Single-Period Aggregation 12.3 Pure Exchange Equilibrium 12.3.1 Basic Idea and Single-Period Case 12.3.2 Multiperiod Discrete-Time Model 12.3.3 Continuous-Time Pure Exchange Equilibrium 12.4 Existence of Equilibrium 12.4.1 Equilibrium Existence in Discrete Time 371 373 374 376 377 378 380 383 383 383 387 389 389 389 389 391 392 394 395 398 399 Contents xv 12.4.2 Equilibrium Existence in Continuous Time 12.4.3 Determining Market Parameters in Equilibrium Summary Problems Further Readings 400 403 406 406 407 13 CAPM 13.1 Basic CAPM 13.1.1 CAPM Equilibrium Argument 13.1.2 Capital Market Line 13.1.3 CAPM formula 13.2 Economic Interpretations 13.2.1 Securities Market Line 13.2.2 Systematic and Nonsystematic Risk 13.2.3 Asset Pricing Implications: Performance Evaluation 13.2.4 Pricing Formulas 13.2.5 Empirical Tests 13.3 Alternative Derivation of the CAPM∗ 13.4 Continuous-Time, Intertemporal CAPM∗ 13.5 Consumption CAPM∗ Summary Problems Further Readings 409 409 409 411 412 413 413 414 416 418 419 420 423 427 430 430 432 14 14.1 14.2 14.3 433 433 436 438 438 439 441 442 445 445 445 Multifactor Models Discrete-Time Multifactor Models Arbitrage Pricing Theory (APT) Multifactor Models in Continuous Time∗ 14.3.1 Model Parameters and Variables 14.3.2 Value Function and Optimal Portfolio 14.3.3 Separation Theorem 14.3.4 Intertemporal Multifactor CAPM Summary Problems Further Readings xvi Contents 15 Other Pure Exchange Equilibria 15.1 Term-Structure Equilibria 15.1.1 Equilibrium Term Structure in Discrete Time 15.1.2 Equilibrium Term Structure in Continuous Time; CIR Model 15.2 Informational Equilibria 15.2.1 Discrete-Time Models with Incomplete Information 15.2.2 Continuous-Time Models with Incomplete Information 15.3 Equilibrium with Heterogeneous Agents 15.3.1 Discrete-Time Equilibrium with Heterogeneous Agents 15.3.2 Continuous-Time Equilibrium with Heterogeneous Agents 15.4 International Equilibrium; Equilibrium with Two Prices 15.4.1 Discrete-Time International Equilibrium 15.4.2 Continuous-Time International Equilibrium Summary Problems Further Readings 447 447 447 449 451 451 454 457 458 459 461 462 463 466 466 467 16 Appendix: Probability Theory Essentials 16.1 Discrete Random Variables 16.1.1 Expectation and Variance 16.2 Continuous Random Variables 16.2.1 Expectation and Variance 16.3 Several Random Variables 16.3.1 Independence 16.3.2 Correlation and Covariance 16.4 Normal Random Variables 16.5 Properties of Conditional Expectations 16.6 Martingale Definition 16.7 Random Walk and Brownian Motion 469 469 469 470 470 471 471 472 472 474 476 476 References Index 479 487 Preface Why We Wrote the Book The subject of financial markets is fascinating to many people: to those who care about money and investments, to those who care about the well-being of modern society, to those who like gambling, to those who like applications of mathematics, and so on. We, the authors of this book, care about many of these things (no, not the gambling), but what we care about most is teaching. The main reason for writing this book has been our belief that we can successfully teach the fundamentals of the economic and mathematical aspects of financial markets to almost everyone (again, we are not sure about gamblers). Why are we in this teaching business instead of following the path of many of our former students, the path of making money by pursuing a career in the financial industry? Well, they don’t have the pleasure of writing a book for the enthusiastic reader like yourself! Prerequisites This text is written in such a way that it can be used at different levels and for different groups of undergraduate and graduate students. After the first, introductory chapter, each chapter starts with sections on the single-period model, goes to multiperiod models, and finishes with continuous-time models. The single-period and multiperiod models require only basic calculus and an elementary introductory probability/statistics course. Those sections can be taught to third- and fourth-year undergraduate students in economics, business, and similar fields. They could be taught to mathematics and engineering students at an even earlier stage. In order to be able to read continuous-time sections, it is helpful to have been exposed to an advanced undergraduate course in probability. Some material needed from such a probability course is briefly reviewed in chapter 16. Who Is It For? The book can also serve as an introductory text for graduate students in finance, financial economics, financial engineering, and mathematical finance. Some material from continuoustime sections is, indeed, usually considered to be graduate material. We try to explain much of that material in an intuitive way, while providing some of the proofs in appendixes to the chapters. The book is not meant to compete with numerous excellent graduate-level books in financial mathematics and financial economics, which are typically written in a mathematically more formal way, using a theorem-proof type of structure. Some of those more advanced books are mentioned in the references, and they present a natural next step in getting to know the subject on a more theoretical and advanced level. xviii Preface Structure of the Book We have divided the book into three parts. Part I goes over the basic securities, organization of financial markets, the concept of interest rates, the main mathematical models, and ways to measure in a quantitative way the risk and the reward of trading in the market. Part II deals with option pricing and hedging, and similar material is present in virtually every recent book on financial markets. We choose to emphasize the so-called martingale, probabilistic approach consistently throughout the book, as opposed to the differentialequations approach or other existing approaches. For example, the one proof of the BlackScholes formula that we provide is done calculating the corresponding expected value. Part III is devoted to one of the favorite subjects of financial economics, the equilibrium approach to asset pricing. This part is often omitted from books in the field of financial mathematics, having fewer direct applications to option pricing and hedging. However, it is this theory that gives a qualitative insight into the behavior of market participants and how the prices are formed in the market. What Can a Course Cover? We have used parts of the material from the book for teaching various courses at the University of Southern California: undergraduate courses in economics and business, a masterslevel course in mathematical finance, and option and investment courses for MBA students. For example, an undergraduate course for economics/business students that emphasizes option pricing could cover the following (in this order): The first three chapters without continuous-time sections; chapter 10 on bond hedging could also be done immediately after chapter 2 on interest rates • The first two chapters of part II on no-arbitrage pricing and option pricing, without most of the continuous-time sections, but including basic Black-Scholes theory • • Chapters on hedging in part II, with or without continuous-time sections The mean-variance section in chapter 5 on risk; chapter 13 on CAPM could also be done immediately after that section • If time remains, or if this is an undergraduate economics course that emphasizes equilibrium/asset pricing as opposed to option pricing, or if this is a two-semester course, one could also cover • discrete-time sections in chapter 4 on utility. • discrete-time sections in part III on equilibrium models. Preface xix Courses aimed at more mathematically oriented students could go very quickly through the discrete-time sections, and instead spend more time on continuous-time sections. A one-semester course would likely have to make a choice: to focus on no-arbitrage option pricing methods in part II or to focus on equilibrium models in part III. Web Page for This Book, Excel Files The web page http://math.usc.edu/∼cvitanic/book.html will be regularly updated with material related to the book, such as corrections of typos. It also contains Microsoft Excel files, with names like ch1.xls. That particular file has all the figures from chapter 1, along with all the computations needed to produce them. We use Excel because we want the reader to be able to reproduce and modify all the figures in the book. A slight disadvantage of this choice is that our figures sometimes look less professional than if they had been done by a specialized drawing software. We use only basic features of Excel, except for Monte Carlo simulation for which we use the Visual Basic programming language, incorporated in Excel. The readers are expected to learn the basic features of Excel on their own, if they are not already familiar with them. At a few places in the book we give “Excel Tips” that point out the trickier commands that have been used for creating a figure. Other, more mathematically oriented software may be more efficient for longer computations such as Monte Carlo, and we leave the choice of the software to be used with some of the homework problems to the instructor or the reader. In particular, we do not use any optimization software or differential equations software, even though the instructor could think of projects using those. Notation Asterisk Sections and problems designated by an asterisk are more sophisticated in mathematical terms, require extensive use of computer software, or are otherwise somewhat unusual and outside of the main thread of the book. These sections and problems could be skipped, although we suggest that students do most of the problems that require use of computers. Dagger End-of-chapter problems that are solved in the student’s manual are preceded by a dagger. Greek Letters We use many letters from the Greek alphabet, sometimes both lowercase and uppercase, and we list them here with appropriate pronunciation: α (alpha), β (beta), γ ,  (gamma), δ,  (delta), ε (epsilon), ζ (zeta), η (eta), θ (theta), λ (lambda), μ (mu), ξ (xi), π,  (pi), ω,  (omega), ρ (rho), σ,  (sigma), τ (tau), ϕ,  (phi). xx Preface Acknowledgments First and foremost, we are immensely grateful to our families for the support they provided us while working on the book. We have received great help and support from the staff of our publisher, MIT Press, and, in particular, we have enjoyed working with Elizabeth Murry, who helped us go through the writing and production process in a smooth and efficient manner. J. C.’s research and the writing of this book have been partially supported by National Science Foundation grant DMS-00-99549. Some of the continuous-time sections in parts I and II originated from the lecture notes prepared in summer 2000 while J. C. was visiting the University of the Witwatersrand in Johannesburg, and he is very thankful to his host, David Rod Taylor, the director of the Mathematical Finance Programme at Wits. Numerous colleagues have made useful comments and suggestions, including Krzysztof Burdzy, Paul Dufresne, Neil Gretzky, Assad Jalali, Dmitry Kramkov, Ali Lazrak, Lionel Martellini, Adam Ostaszewski, Kaushik Ronnie Sircar, Costis Skiadas, Halil Mete Soner, Adam Speight, David Rod Taylor, and Mihail Zervos. In particular, D. Kramkov provided us with proofs in the appendix of chapter 6. Some material on continuous-time utility maximization with incomplete information is taken from a joint work with A. Lazrak and L. Martellini, and on continuous-time mean-variance optimization from a joint work with A. Lazrak. Moreover, the following students provided their comments and pointed out errors in the working manuscript: Paula Guedes, Frank Denis Hiebsch, and Chulhee Lee. Of course, we are solely responsible for any remaining errors. A Prevailing Theme: Pricing by Expected Values Before we start with the book’s material, we would like to give a quick illustration here in the preface of a connection between a price of a security and the optimal trading strategy of an investor investing in that security. We present it in a simple model, but this connection is present in most market models, and, in fact, the resulting pricing formula is of the form that will follow us through all three parts of this book. We will repeat this type of argument later in more detail, and we present it early here only to give the reader a general taste of what the book is about. The reader may want to skip the following derivation, and go directly to equation (0.3). Consider a security S with today’s price S(0), and at a future time 1 its price S(1) either has value s u with probability p, or value s d with probability 1 − p. There is also a risk-free security that returns 1 + r dollars at time 1 for every dollar invested today. We assume that s d < (1 + r )S(0) < s u . Suppose an investor has initial capital x, and has to decide how many shares δ of security S to hold, while depositing the rest of his wealth in the bank Preface xxi account with interest rate r . In other words, his wealth X (1) at time 1 is X (1) = δS(1) + [x − δS(0)](1 + r ) The investor wants to maximize his expected utility E[U (X (1))] = pU (X u ) + (1 − p)U (X d ) where U is a so-called utility function, while X u , X d is his final wealth in the case S(1) = s u , S(1) = s d , respectively. Substituting for these values, taking the derivative with respect to δ and setting it equal to zero, we get pU  (X u )[s u − S(0)(1 + r )] + (1 − p)U  (X d )[s d − S(0)(1 + r )] = 0 The left-hand side can be written as E[U  (X (1)){S(1) − S(0)(1 + r )}], which, when made equal to zero, implies, with arbitrary wealth X replaced by optimal wealth X̂ ,   U  ( X̂ (1)) S(1) (0.1) S(0) = E E(U  [X̂ (1)]) 1 + r If we denote Z (1) := U  (X̂ (1)) E{U  (X̂ (1))} we see that the today’s price of our security S is given by   S(1) S(0) = E Z (1) 1+r (0.2) (0.3) We will see that prices of most securities (with some exceptions, like American options) in the models of this book are of this form: the today’s price S(0) is an expected value of the future price S(1), multiplied (“discounted”) by a certain random factor. Effectively, we get the today’s price as a weighted average of the discounted future price, but with weights that depend on the outcomes of the random variable Z (1). Moreover, in standard optionpricing models (having a so-called completeness property) we will not need to use utility functions, since Z (1) will be independent of the investor’s utility. The random variable Z (1) is sometimes called change of measure, while the ratio Z (1)/(1 + r ) is called state-price density, stochastic discount factor, pricing kernel, or marginal rate of substitution, depending on the context and interpretation. There is another interpretation of this formula, using a new probability; hence the name “change of (probability) measure.” For example, if, as in our preceding example, Z (1) takes two possible values Z u (1) and Z d (1) with xxii Preface probabilities p, 1 − p, respectively, we can define p ∗ := p Z u (1), 1 − p ∗ = (1 − p)Z d (1) The values of Z (1) are such that p ∗ is a probability, and we interpret p ∗ and 1 − p ∗ as modified probabilities of the movements of asset S. Then, we can write equation (0.3) as   S(1) (0.4) S(0) = E ∗ 1+r where E ∗ denotes the expectation under the new probabilities, p ∗ , 1 − p ∗ . Thus the price today is the expected value of the discounted future value, where the expected value is computed under a special, so-called risk-neutral probability, usually different from the real-world probability. Final Word We hope that we have aroused your interest about the subject of this book. If you turn out to be a very careful reader, we would be thankful if you could inform us of any remaining typos and errors that you find by sending an e-mail to our current e-mail addresses. Enjoy the book! Jakša Cvitanić and Fernando Zapatero E-mail addresses: cvitanic@math.usc.edu, zapatero@usc.edu
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