Springer Finance Editorial Board M. Avellaneda G. Barone-Adesi M. Broadie M.H.A. Davis E. Derman C. Klüppelberg E. Kopp W. Schachermayer
Springer Finance Springer Finance is a programme of books aimed at students, academics and practitioners working on increasingly technical approaches to the analysis of financial markets. It aims to cover a variety of topics, not only mathematical finance but foreign exchanges, term structure, risk management, portfolio theory, equity derivatives, and financial economics. M. Ammann, Credit Risk Valuation: Methods, Models, and Application (2001) K. Back, A Course in Derivative Securities: Introduction to Theory and Computation (2005) E. Barucci, Financial Markets Theory. Equilibrium, Efficiency and Information (2003) T.R. Bielecki and M. Rutkowski, Credit Risk: Modeling, Valuation and Hedging (2002) N.H. Bingham and R. Kiesel, Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives (1998, 2nd ed. 2004) D. Brigo and F. Mercurio, Interest Rate Models: Theory and Practice (2001) R. Buff, Uncertain Volatility Models-Theory and Application (2002) R.A. Dana and M. Jeanblanc, Financial Markets in Continuous Time (2002) G. Deboeck and T. Kohonen (Editors), Visual Explorations in Finance with Self-Organizing Maps (1998) R.J. Elliott and P.E. Kopp, Mathematics of Financial Markets (1999, 2nd ed. 2005) H. Geman, D. Madan, S.R. Pliska and T. Vorst (Editors), Mathematical Finance- Bachelier Congress 2000 (2001) M. Gundlach, F. Lehrbass (Editors),CreditRisk + in the Banking Industry (2004) B.P. Kellerhals, Asset Pricing (2004) Y.-K. Kwok, Mathematical Models of Financial Derivatives (1998) M. Külpmann, Irrational Exuberance Reconsidered (2004) P. Malliavin and A. Thalmaier, Stochastic Calculus of Variations in Mathematical Finance (2005) A. Meucci, Risk and Asset Allocation (2005) A. Pelsser, Efficient Methods for Valuing Interest Rate Derivatives (2000) J.-L. Prigent, Weak Convergence of Financial Markets (2003) B. Schmid, Credit Risk Pricing Models (2004) S.E. Shreve, Stochastic Calculus for Finance I (2004) S.E. Shreve, Stochastic Calculus for Finance II (2004) M. Yor, Exponential Functionals of Brownian Motion and Related Processes (2001) R. Zagst, Interest-Rate Management (2002) Y.-L.Zhu,X.Wu,I.-L.Chern, Derivative Securities and Difference Methods (2004) A. Ziegler, Incomplete Information and Heterogeneous Beliefs in Continuous-time Finance (2003) A. Ziegler, A Game Theory Analysis of Options (2004)
Kerry Back A Course in Derivative Securities Introduction to Theory and Computation 123
Kerry Back Department of Finance Mays Business School Texas A&M University 306 Wehner Building College Station, TX 77843-4218 USA e-mail: kback@mays.tamu.edu Mathematics Subject Classification (2000): 91B28, 91B70, 9104, 65C05, 65M06, 60G44, 6004 JEL Classification: G13, C63 Library of Congress Control Number: 2005922929 ISBN-10 3-540-25373-4 Springer-Verlag Berlin Heidelberg New York ISBN-13 978-3-540-25373-0 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com Springer-Verlag Berlin Heidelberg 2005 Printed in The Netherlands MATLAB isatrademarkofthemathworks,inc.andisusedwithpermission.themathworks does not warrant the accuracy of the text or exercises in this book. This book s use or discussion of MATLAB software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB software. Visual Basic (R) is a registered trademark of Microsoft Corporation in the United States and/or other countries. This book is an independent publication and is not affiliated with, nor has it been authorized, sponsored, or otherwise approved by Microsoft Corporation. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: design & production, Heidelberg Typesetting by the author using a Springer L A TEX macro package Printed on acid-free paper 41/3142sz - 5 4 3 2 1 0
To my parents, Roy and Verla.
Preface This book is an outgrowth of notes compiled by the author while teaching courses for undergraduate and masters/mba finance students at Washington University in St. Louis and the Institut für Höhere Studien in Vienna. At one time, a course in Options and Futures was considered an advanced finance elective, but now such a course is nearly mandatory for any finance major and is an elective chosen by many non-finance majors as well. Moreover, students are exposed to derivative securities in courses on Investments, International Finance, Risk Management, Investment Banking, Fixed Income, etc. This expansion of education in derivative securities mirrors the increased importance of derivative securities in corporate finance and investment management. MBA and undergraduate courses typically (and appropriately) focus on the use of derivatives for hedging and speculating. This is sufficient for many students. However, the seller of derivatives, in addition to needing to understand buy-side demands, is confronted with the need to price and hedge. Moreover, the buyer of derivatives, depending on the degree of competition between sellers, may very likely benefit from some knowledge of pricing as well. It is pricing and hedging that is the primary focus of this book. Through learning the fundamentals of pricing and hedging, students also acquire a deeper understanding of the contracts themselves. Hopefully, this book will also be of use to practitioners and for students in Masters of Financial Engineering programs and, to some extent, Ph.D. students in finance. The book is concerned with pricing and hedging derivatives in frictionless markets. By frictionless, I mean that the book ignores transaction costs (commissions, bid-ask spreads and the price impacts of trades), margin (collateral) requirements and any restrictions on short selling. The theory of pricing and hedging in frictionless markets stems of course from the work of Black and Scholes [6] and Merton [51] and is a very well developed theory. It is based on the assumption that there are no arbitrage opportunities in the market. The theory is the foundation for pricing and hedging in markets with frictions (i.e., in real markets!) but practice can differ from theory in important ways if the frictions are significant. For example, an arbitrage opportunity in
VIII Preface a frictionless market often will not be an arbitrage opportunity for a trader who moves the market when he trades, faces collateral requirements, etc. This book has nothing to say about how one should deviate from the benchmark frictionless theory when frictions are important. Another important omission from the book is jump processes the book deals exclusively with binomial and Brownian motion models. The book is intended primarily to be used for advanced courses in derivative securities. It is self-contained, and the first chapter presents the basic financial concepts. However, much material (functioning of security exchanges, payoff diagrams, spread strategies, etc.) that is standard in an introductory book has not been included here. On the other hand, though it is not an introductory book, it is not truly an advanced book on derivatives either. On any of the topics covered in the book, there are more advanced treatments available in book form already. However, the books that I have seen (and there are indeed many) are either too narrow in focus for the courses I taught or not easily accessible to the students I taught or (most commonly) both. If this book is successful, it will be as a bridge between an introductory course in Options and Futures and the more advanced literature. Towards that end, I have included cites to more advanced books in appropriate places throughout. The book includes an introduction to computational methods, and the term introduction is meant quite seriously here. The book was developed for students with no prior experience in programming or numerical analysis, and it only covers the most basic ideas. Nevertheless, I believe that this is an extremely important feature of the book. It is my experience that the theory becomes much more accessible to students when they learn to code a formula or to simulate a process. The book builds up to binomial, Monte Carlo, and finite-difference methods by first developing simple programs for simple computations. These serve two roles: they introduce the student to programming, and they result in tools that enable students to solve real problems, allowing the inclusion of exercises of a practical rather than purely theoretical nature. I have used the book for semester-length courses emphasizing calculation (most of the exercises are of that form) and for short courses covering only the theory. Nearly all of the formulas and procedures described in the book are both derived from first principles and implemented in Excel VBA. The VBA programs are in the text and in an Excel workbook that can be downloaded free of charge at www.kerryback.net. I use a few special features of Excel, in particular the cumulative normal distribution function and the random number generator. Otherwise, the programs can easily be translated into any other language. In particular, it is easy to translate them into MATLAB, which also includes a random number generator and the cumulative normal distribution function (or, rather, the closely related error function ) as part of its basic implementation. I chose VBA because students (finance students, at least) can be expected to already have it on their computers and because Excel is a good environment for many exercises, such as analyzing hedges, that do
Preface IX not require programming. An appendix provides the necessary introduction to VBA programming. Viewed as a math book, this is a book in applied math, not math proper. My goal is to get students as quickly as possible to the point where they can compute things. Many mathematical issues (filtrations, completion of filtrations, formal definitions of expectations and conditional expectations, etc.) are entirely ignored. It would not be unfair to call this a cookbook approach. I try to explain intuitively why the recipes work but do not give proofs or even formal statements of the facts that underlie them. I have naturally taken pains to present the theory in what I think is the simplest possible manner. The book uses almost exclusively the probabilistic/martingale approach, both because it is my preference and because it seems easier than partial differential equations for students in business and the social sciences to grasp. A sampling of some of the more or less distinctive characteristics of the book, in terms of exposition, is: Important theoretical results are highlighted in boxes for easy reference; the derivations that are less important and more technical are presented in smaller type and relegated to the ends of sections. Changes of numeraire are introduced in the first chapter in a one-period binomial model, the probability measure corresponding to the underlying as numeraire being given as much emphasis as the risk-neutral measure. The fundamental result for pricing (asset prices are martingales under changes of numeraire) is presented in the first chapter, because it does not need the machinery of stochastic calculus. The basic ideas in pricing digital and share digitals, and hence in deriving the Black-Scholes formula, are also presented in the first chapter. Digitals and share digitals are priced in Chap. 3 before calls and puts. Brownian motion is introduced by simulating it in discrete time. The quadratic variation property is emphasized, including exercises that contrast Brownian motion with continuously differentiable functions of time, in order to motivate Itô s formula. The distribution of the underlying under different numeraires is derived directly from the fundamental pricing result and Itô s formula, bypassing Girsanov s theorem (which is of course also a consequence of Itô s formula). Substantial emphasis is placed on forwards, synthetic forwards, options on forwards and hedging with forwards because these have many applications in fixed income and elsewhere a simple but characteristic example is valuing a European option on a stock paying a known cash dividend as a European option on the synthetic forward with the same maturity. Following Margrabe [50] (who attributes the idea to S. Ross) the formula for exchange options is derived by a change of numeraire from the Black-Scholes formula. Very simple arguments derive Black s formula for forward and futures options from Margrabe s formula and Merton s formula for stock options in the absence of a constant risk-free rate from
X Preface Black s formula. This demonstrates the equivalence of these important option pricing formulas as follows: Black-Scholes = Margrabe = Black = Merton = Black-Scholes Quanto forwards and options are priced by first finding the portfolio that replicates the value of a foreign security translated at a fixed exchange rate and then viewing quanto forwards and options as standard forwards and options on the replicating portfolio. The market model is presented as an introduction to the pricing of fixedincome derivatives. Forward rates are shown to be martingales under the forward measure by virtue of their being forward prices of portfolios that pay spot rates. In order to illustrate how term structure models are used to price fixedincome derivatives, the Vasicek/Hull-White model is worked out in great detail. Other important term structure models are discussed much more briefly. Of course, none of these items is original, but in conjunction with the computational tools, I believe they make the rocket science of derivative securities accessible to a broader group of students. The book is divided into three parts, labeled Introduction to Option Pricing, Advanced Option Pricing, and Fixed Income. Naturally, many of the chapters build upon one another, but it is possible to read Chaps. 1 3, Sects. 7.1 7.2 (the Margrabe and Black formulas) and then Part III on fixed income. For a more complete coverage, but still omitting two of the more difficult chapters, one could read all of Parts I and II except Chaps. 8 and 10, pausing in Chap. 8 to read the definitions of baskets, spreads, barriers, lookbacks and Asians and in Chap. 10 to read the discussion of the fundamental partial differential equation. I would like to thank Mark Broadie, the series editor, for helpful comments, and especially I want to thank my wife, Diana, without whose encouragement and support I could not have written this. She mowed the lawn and managed everything else while I typed, and that is a great gift. College Station, Texas April, 2005 Kerry Back
Contents Part I Introduction to Option Pricing 1 Asset Pricing Basics... 3 1.1 Fundamental Concepts.................................. 3 1.2 State Prices in a One-Period Binomial Model.............. 11 1.3 Probabilities and Numeraires............................. 14 1.4 Asset Pricing with a Continuum of States................. 17 1.5 IntroductiontoOptionPricing... 21 1.6 AnIncompleteMarketsExample... 24 Problems... 25 2 Continuous-Time Models... 27 2.1 SimulatingaBrownianMotion... 28 2.2 QuadraticVariation... 29 2.3 Itô Processes.... 31 2.4 Itô sformula... 33 2.5 Multiple Itô Processes.... 35 2.6 Examples of Itô s Formula... 37 2.7 Reinvesting Dividends................................... 38 2.8 Geometric Brownian Motion............................. 39 2.9 Numeraires and Probabilities............................. 41 2.10 Tail Probabilities of Geometric Brownian Motions.......... 44 2.11 Volatilities............................................. 46 Problems... 48 3 Black-Scholes... 49 3.1 DigitalOptions... 49 3.2 ShareDigitals... 51 3.3 Puts and Calls......................................... 52 3.4 Greeks... 53 3.5 DeltaHedging... 55
XII Contents 3.6 GammaHedging... 57 3.7 Implied Volatilities..................................... 58 3.8 Term Structure of Volatility.............................. 58 3.9 Smiles and Smirks...................................... 60 3.10 CalculationsinVBA... 60 Problems... 67 4 Estimating and Modelling Volatility... 71 4.1 StatisticsReview... 71 4.2 Estimating a Constant Volatility and Mean................ 73 4.3 Estimating a Changing Volatility......................... 75 4.4 GARCHModels... 77 4.5 Stochastic Volatility Models.............................. 79 4.6 Smiles and Smirks Again................................ 82 4.7 HedgingandMarketCompleteness... 83 Problems... 84 5 Introduction to Monte Carlo and Binomial Models... 87 5.1 IntroductiontoMonteCarlo... 87 5.2 IntroductiontoBinomialModels... 89 5.3 Binomial Models for American Options.................... 91 5.4 BinomialParameters... 92 5.5 BinomialGreeks... 94 5.6 Monte Carlo Greeks I: Difference Ratios................... 96 5.7 MonteCarloGreeksII:PathwiseEstimates... 98 5.8 CalculationsinVBA...101 Problems...107 Part II Advanced Option Pricing 6 Foreign Exchange...111 6.1 CurrencyOptions...112 6.2 Options on Foreign Assets Struck in Foreign Currency....... 112 6.3 Options on Foreign Assets Struck in Domestic Currency..... 112 6.4 CurrencyForwardsandFutures...113 6.5 Quantos...116 6.6 Replicating Quantos.................................... 118 6.7 QuantoForwards...121 6.8 Quanto Options........................................ 121 6.9 ReturnSwaps...123 6.10 Uncovered Interest Parity................................ 125 Problems...125
Contents XIII 7 Forward, Futures, and Exchange Options...129 7.1 Margrabe sformula...130 7.2 Black sformula...132 7.3 Merton sformula...136 7.4 DeferredExchangeOptions...139 7.5 CalculationsinVBA...140 7.6 GreeksandHedging...142 7.7 The Relation of Futures Prices to Forward Prices........... 144 7.8 FuturesOptions...145 7.9 Time-Varying Volatility................................. 147 7.10 HedgingwithForwardsandFutures...147 7.11 MarketCompleteness...150 Problems...152 8 Exotic Options...155 8.1 Forward-Start Options.................................. 155 8.2 Compound Options..................................... 158 8.3 American Calls with Discrete Dividends................... 162 8.4 Choosers...166 8.5 Options on the Max or Min.............................. 168 8.6 Barrier Options........................................ 171 8.7 Lookbacks...175 8.8 BasketandSpreadOptions...176 8.9 Asian Options.......................................... 178 8.10 CalculationsinVBA...183 Problems...193 9 More on Monte Carlo and Binomial Valuation...197 9.1 Monte Carlo Models for Path-Dependent Options........... 197 9.2 Binomial Valuation of Basket and Spread Options.......... 198 9.3 Monte Carlo Valuation of Basket and Spread Options....... 200 9.4 Antithetic Variates in Monte Carlo....................... 202 9.5 ControlVariatesinMonteCarlo...203 9.6 Accelerating Binomial Convergence....................... 205 9.7 CalculationsinVBA...207 Problems...216 10 Finite Difference Methods...219 10.1 Fundamental PDE...................................... 219 10.2 Discretizing the PDE................................... 221 10.3 Explicit and Implicit Methods............................ 222 10.4 Crank-Nicolson...225 10.5 European Options...................................... 226 10.6 American Options...................................... 227 10.7 Barrier Options........................................ 227
XIV Contents 10.8 CalculationsinVBA...227 Problems...233 Part III Fixed Income 11 Fixed Income Concepts...237 11.1 TheYieldCurve...237 11.2 LIBOR...239 11.3 Swaps...240 11.4 Yield to Maturity, Duration, and Convexity................ 242 11.5 Principal Components................................... 245 11.6 HedgingPrincipalComponents...248 Problems...249 12 Introduction to Fixed Income Derivatives...253 12.1 CapsandFloors...253 12.2 ForwardRates...254 12.3 Portfolios that Pay Spot Rates........................... 254 12.4 TheMarketModelforCapsandFloors...255 12.5 TheMarketModelforEuropeanSwaptions...257 12.6 ACommentonConsistency...259 12.7 CapletsasPutsonDiscountBonds...260 12.8 SwaptionsasOptionsonCouponBonds...260 12.9 CalculationsinVBA...261 Problems...262 13 Valuing Derivatives in the Extended Vasicek Model...265 13.1 TheShortRateandDiscountBondPrices...265 13.2 TheVasicekModel...266 13.3 EstimatingtheVasicekModel...269 13.4 Hedging in the Vasicek Model............................ 271 13.5 Extensions of the Vasicek Model.......................... 273 13.6 Fitting Discount Bond Prices and Forward Rates........... 275 13.7 Discount Bond Options, Caps and Floors.................. 277 13.8 CouponBondOptionsandSwaptions...280 13.9 Captions and Floortions................................. 283 13.10 Yields and Yield Volatilities.............................. 284 13.11 TheGeneralHull-WhiteModel...285 13.12 CalculationsinVBA...289 Problems...293
Contents XV 14 A Brief Survey of Term Structure Models...295 14.1 Ho-Lee...295 14.2 Black-Derman-Toy...300 14.3 Black-Karasinski...302 14.4 Cox-Ingersoll-Ross...302 14.5 Longstaff-Schwartz...307 14.6 Heath-Jarrow-Morton...310 14.7 MarketModelsAgain...312 Problems...316 Appendices A Programming in VBA...319 A.1 VBAEditorandModules...319 A.2 Subroutines and Functions............................... 320 A.3 Message Box and Input Box............................. 321 A.4 Writing to and Reading from Cells........................ 322 A.5 VariablesandAssignments...323 A.6 MathematicalOperations...324 A.7 RandomNumbers...324 A.8 ForLoops...325 A.9 While Loops and Logical Expressions..................... 326 A.10 If,Else,andElseIfStatements...326 A.11 VariableDeclarations...327 A.12 VariablePassing...328 A.13 Arrays...329 A.14 Debugging...331 B Miscellaneous Facts about Continuous-Time Models...333 B.1 Girsanov stheorem...333 B.2 The Minimum of a Geometric Brownian Motion............ 336 B.3 Bessel Squared Processes and the CIR Model...340 List of Programs...346 List of Symbols...347 References...349 Index...353
Part I Introduction to Option Pricing