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CY260/Baz-FM CY260/Baz 052181510X September16,2003 12:50 CharCount=0 ii CY260/Baz-FM CY260/Baz 052181510X October3,2008 19:51 CharCount=0 FinancialDerivatives Thisbookoffersasuccinctaccountoftheprinciplesoffinancialderiva- tivespricing.Thefirstchapterprovidesreaderswithanintuitiveexpo- sition of basic random calculus. Concepts such as volatility and time, random walks, geometric Brownian motion, and Itoˆ’s lemma are dis- cussedheuristically.Thesecondchapterdevelopsgenericpricingtech- niquesforassetsandderivatives,determiningthenotionofastochastic discount factor or pricing kernel, and then uses this concept to price conventionalandexoticderivatives.Thethirdchapterappliesthepric- ingconceptstothespecialcaseofinterestratemarkets,namely,bonds and swaps, and discusses factor models and term-structure-consistent models.Thefourthchapterdealswithavarietyofmathematicaltopics thatunderliederivativespricingandportfolioallocationdecisions,such asmean-revertingprocessesandjumpprocesses,anddiscussesrelated toolsofstochasticcalculus,suchasKolmogorovequations,martingales techniques,stochasticcontrol,andpartialdifferentialequations. Jamil Baz is the chief investment strategist of GLG, a London-based hedgefund.Priortoholdingthisposition,hewasaportfoliomanager withPIMCOinLondon,amanagingdirectorintheProprietaryTrad- ingGroupofGoldmanSachs,chiefinvestmentstrategistofDeutsche Bank,andexecutivedirectorofLehmanBrothersfixedincomeresearch division.Dr.BazteachesfinancialeconomicsatOxfordUniversity.He hasdegreesfromtheLondonSchoolofEconomics(M.Sc.),MIT(S.M.), andHarvardUniversity(A.M.,Ph.D.). GeorgeChackoischiefinvestmentofficerofAuda,aglobalassetman- agementfirm,inNewYork.HeisalsoaprofessoratSantaClaraUni- versity, California, where he teaches finance. Dr. Chacko previously servedfortenyearsasaprofessoratHarvardBusinessSchoolinthe financedepartment.Dr.Chackoheldmanagingdirectorshipsinfixed incomesalesandtradingatStateStreetBankinBostonandinpension assetmanagementatIFLinNewYork.HeholdsaB.S.fromMIT,an M.B.A.fromtheUniversityofChicago,andanM.A.andPh.D.from HarvardUniversity. i CY260/Baz-FM CY260/Baz 052181510X September16,2003 12:50 CharCount=0 ii CY260/Baz-FM CY260/Baz 052181510X October11,2008 11:49 CharCount=0 Financial Derivatives Pricing, Applications, and Mathematics JAMIL BAZ GLG GEORGE CHACKO Auda iii CAMBRIDGEUNIVERSITYPRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi Cambridge University Press 32 Avenue of the Americas, New York, NY10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9780521815109 ©Jamil Baz and George Chacko 2004 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2004 First paperback edition 2009 Printed in the United States of America Acatalog recordfor this publication is available from the British Library. Library of Congress Cataloging in Publication Data Baz, Jamil Financial derivatives : pricing, applications, and mathematics / Jamil Baz, George Chacko. p. cm. Includes bibliographical references and index. ISBN 0-521-81510-X 1. Derivative securities. I. Chacko, George. II. Title. HG6024.A3B396 2003 332.63'2 – dc21 2002041452 ISBN 978-0-521-81510-9 hardback ISBN 978-0-521-06679-2 paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate. CY260/Baz-FM CY260/Baz 052181510X September16,2003 12:50 CharCount=0 ToMauriceandElena J.B. Tomyparents G.C. v CY260/Baz-FM CY260/Baz 052181510X September16,2003 12:50 CharCount=0 vi CY260/Baz-FM CY260/Baz 052181510X September16,2003 12:50 CharCount=0 Contents Acknowledgments pagexi Introduction 1 1 PreliminaryMathematics 5 1.1 RandomWalk 5 1.2 AnotherTakeonVolatilityandTime 8 1.3 AFirstGlanceatItoˆ’sLemma 9 1.4 ContinuousTime:BrownianMotion;More onItoˆ’sLemma 11 1.5 Two-DimensionalBrownianMotion 14 1.6 BivariateItoˆ’sLemma 15 1.7 ThreeParadoxesofFinance 16 1.7.1 Paradox1:Siegel’sParadox 16 1.7.2 Paradox2:TheStock,Free-LunchParadox 18 1.7.3 Paradox3:TheSkillVersusLuckParadox 19 2 PrinciplesofFinancialValuation 22 2.1 Uncertainty,UtilityTheory,andRisk 22 2.2 RiskandtheEquilibriumPricingofSecurities 28 2.3 TheBinomialOption-PricingModel 41 2.4 LimitingOption-PricingFormula 46 2.5 Continuous-TimeModels 47 2.5.1 TheBlack-Scholes/MertonModel–Pricing KernelApproach 48 2.5.2 TheBlack-Scholes/MertonModel– ProbabilisticApproach 57 2.5.3 TheBlack-Scholes/MertonModel–Hedging Approach 61 vii CY260/Baz-FM CY260/Baz 052181510X September16,2003 12:50 CharCount=0 viii Contents 2.6 ExoticOptions 63 2.6.1 DigitalOptions 64 2.6.2 PowerOptions 65 2.6.3 AsianOptions 67 2.6.4 BarrierOptions 71 3 InterestRateModels 78 3.1 InterestRateDerivatives:NotSoSimple 78 3.2 BondsandYields 80 3.2.1 PricesandYieldstoMaturity 80 3.2.2 DiscountFactors,Zero-CouponRates,and CouponBias 82 3.2.3 ForwardRates 85 3.3 NaiveModelsofInterestRateRisk 88 3.3.1 Duration 88 3.3.2 Convexity 99 3.3.3 TheFreeLunchintheDurationModel 104 3.4 AnOverviewofInterestRateDerivatives 108 3.4.1 BondswithEmbeddedOptions 109 3.4.2 ForwardRateAgreements 110 3.4.3 EurostripFutures 112 3.4.4 TheConvexityAdjustment 113 3.4.5 Swaps 118 3.4.6 CapsandFloors 120 3.4.7 Swaptions 121 3.5 YieldCurveSwaps 122 3.5.1 TheCMSSwap 122 3.5.2 TheQuantoSwap 127 3.6 FactorModels 131 3.6.1 AGeneralSingle-FactorModel 131 3.6.2 TheMertonModel 135 3.6.3 TheVasicekModel 139 3.6.4 TheCox-Ingersoll-RossModel 142 3.6.5 Risk-NeutralValuation 144 3.7 Term-Structure-ConsistentModels 147 3.7.1 “Equilibrium”Versus“Fitting” 147 3.7.2 TheHo-LeeModel 153 3.7.3 TheHo-LeeModelwithTime-Varying Volatility 157 3.7.4 TheBlack-Derman-ToyModel 162 3.8 RiskyBondsandTheirDerivatives 166 3.8.1 TheMertonModel 167 3.8.2 TheJarrow-TurnbullModel 168