Table Of ContentProblems and Solutions
in Mathematical Finance
ForothertitlesintheWileyFinanceseries
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Problems and Solutions
in Mathematical Finance
Volume 2: Equity Derivatives
´
Eric Chin, Dian Nel and Sverrir Olafsson
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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!
