Problems and Solutions in Mathematical Finance ForothertitlesintheWileyFinanceseries pleaseseewww.wiley.com/finance Problems and Solutions in Mathematical Finance Volume 2: Equity Derivatives ´ Eric Chin, Dian Nel and Sverrir Olafsson Thiseditionfirstpublished2017 ©2017JohnWiley&Sons,Ltd Registeredoffice JohnWiley&SonsLtd,TheAtrium,SouthernGate,Chichester,WestSussex,PO198SQ,UnitedKingdom Fordetailsofourglobaleditorialoffices,forcustomerservicesandforinformationabouthowtoapplyfor permissiontoreusethecopyrightmaterialinthisbookpleaseseeourwebsiteatwww.wiley.com. Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmitted,inany formorbyanymeans,electronic,mechanical,photocopying,recordingorotherwise,exceptaspermittedbytheUK Copyright,DesignsandPatentsAct1988,withoutthepriorpermissionofthepublisher. 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ISBN978-1-119-96582-4(hardback)ISBN978-1-119-96610-4(ebk) ISBN978-1-119-96611-1(ebk) ISBN978-1-119-19219-0(obk) Coverdesign:Cylinder Coverimage:©Attitude/Shutterstock Setin10/12ptTimesbyAptaraInc.,NewDelhi,India PrintedinGreatBritainbyTJInternationalLtd,Padstow,Cornwall,UK “Bluedyeisderivedfromtheindigoplantandsurpasseditsparentalcolour” Xunzi,AnExhortationtoLearning Contents Preface ix AbouttheAuthors xi 1 BasicEquityDerivativesTheory 1 1.1 Introduction 1 1.2 ProblemsandSolutions 8 1.2.1 ForwardandFuturesContracts 8 1.2.2 OptionsTheory 15 1.2.3 HedgingStrategies 27 2 EuropeanOptions 63 2.1 Introduction 63 2.2 ProblemsandSolutions 74 2.2.1 BasicProperties 74 2.2.2 Black–ScholesModel 89 2.2.3 Tree-BasedMethods 190 2.2.4 TheGreeks 218 3 AmericanOptions 267 3.1 Introduction 267 3.2 ProblemsandSolutions 271 3.2.1 BasicProperties 271 3.2.2 Time-IndependentOptions 292 3.2.3 Time-DependentOptions 305 4 BarrierOptions 351 4.1 Introduction 351 4.2 ProblemsandSolutions 357 4.2.1 ProbabilisticApproach 357 4.2.2 ReflectionPrincipleApproach 386 4.2.3 FurtherBarrier-StyleOptions 408 viii Contents 5 AsianOptions 439 5.1 Introduction 439 5.2 ProblemsandSolutions 443 5.2.1 DiscreteSampling 443 5.2.2 ContinuousSampling 480 6 ExoticOptions 531 6.1 Introduction 531 6.2 ProblemsandSolutions 532 6.2.1 Path-IndependentOptions 532 6.2.2 Path-DependentOptions 586 7 VolatilityModels 647 7.1 Introduction 647 7.2 ProblemsandSolutions 652 7.2.1 HistoricalandImpliedVolatility 652 7.2.2 LocalVolatility 685 7.2.3 StochasticVolatility 710 7.2.4 VolatilityDerivatives 769 A MathematicsFormulae 787 B ProbabilityTheoryFormulae 797 C DifferentialEquationsFormulae 813 Bibliography 821 Notation 825 Index 829 Preface Mathematical finance is a highly challenging and technical discipline. Its fundamentals and applicationsarebestunderstoodbycombiningatheoreticallysolidapproachwithextensive exercises in solving practical problems. That is the philosophy behind all four volumes in thisseriesonmathematicalfinance.ThissecondoffourvolumesintheseriesProblemsand Solutions in Mathematical Finance is devoted to the discussion of equity derivatives. In the firstvolumewedevelopedtheprobabilisticandstochasticmethodsrequiredforthesuccessful studyofadvancedmathematicalfinance,inparticulardifferenttypesofpricingmodels.The techniquesappliedinthisvolumeassumegoodknowledgeofthetopicscoveredinVolume1. Aswebelievethatgoodworkingknowledgeofmathematicalfinanceisbestacquiredthrough thesolutionofpracticalproblems,allthevolumesinthisseriesarebuiltupinawaythatallows readerstocontinuouslytesttheirknowledgeastheyworkthroughthetexts. This second volume starts with the analysis of basic derivatives, such as forwards and futures,swapsandoptions.Theapproachisbottomup,startingwiththeanalysisofsimplecon- tractsandthenmovingontomoreadvancedinstruments.Allthemajorclassesofoptionsare introducedandextensivelystudied,startingwithplainEuropeanandAmericanoptions.The textthenmovesontocovermorecomplexcontractssuchasbarrier,Asianandexoticoptions. Ineachoptionclass,differenttypesofoptionsareconsidered,includingtime-independentand time-dependentoptions,ornon-path-dependentandpath-dependentoptions. Stochasticfinancialmodelsfrequentlyrequirethefixingofdifferentparameters.Somecan beextracteddirectlyfrommarketdata,othersneedtobefixedbymeansofnumericalmethods oroptimisationtechniques.Dependingonthecontext,thisisdoneindifferentways.Intherisk- neutralworld,thedriftparameterforthegeometricBrownianmotion(Black–Scholesmodel) isextractedfromthebondmarket(i.e.,thereturnsonrisk-freedebt).Thevolatilityparameter, in contrast, is generally determined from market prices, as the so-called implied volatility. However, if a stochastic process is to be fitted to known price data, other methods need to beconsulted,suchasmaximum-likelihoodestimation.Thismethodisappliedtoanumberof stochasticprocessesinthechapteronvolatilitymodels. Inalloptionmodels,volatilitypresentsoneofthemostimportantquantitiesthatdetermine thepriceandtheriskofderivativescontracts.Forthisreason,considerableeffortisputinto theirdiscussionintermsofconcepts,suchasimplied,localandstochasticvolatilities,aswell astheimportantvolatilitysurfaces. At the end of this volume, readers will be equipped with all the major tools required for the modelling and the pricing of a whole range of different derivatives contracts. They will x Preface thereforebereadytotacklenewtechniquesandchallengesdiscussedinthenexttwovolumes, includinginterest-ratemodellinginVolume3andforeignexchange/commodityderivativesin Volume4. Asinthefirstvolume,wehavethefollowingnotetothestudent/reader:Pleasetryhardto solvetheproblemsonyourownbeforeyoulookatthesolutions!