Problems and Solutions in Mathematical Finance Stochastic Calculus

398 Pages·2014·13.026 MB·English

by  Eric Chin, Dian Nel, Sverrir Olafsson

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Problems and Solutions
in Mathematical Finance
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ForothertitlesintheWileyFinanceSeries
pleaseseewww.wiley.com/finance
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Problems and Solutions
in Mathematical Finance
Volume 1: Stochastic Calculus
Eric Chin, Dian Nel and Sverrir Ólafsson
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Thiseditionfirstpublished2014
©2014JohnWiley&Sons,Ltd
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LibraryofCongressCataloging-in-PublicationData
Chin,Eric,
Problemsandsolutionsinmathematicalfinance:stochasticcalculus/EricChin,DianNelandSverrirÓlafsson.
pagescm
Includesbibliographicalreferencesandindex.
ISBN978-1-119-96583-1(cloth)
1.Finance–Mathematicalmodels.2.Stochasticanalysis.I.Nel,Dian,II.Ólafsson,Sverrir,III.Title.
HG106.C4952014
332.01′51922–dc23
2013043864
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
ISBN978-1-119-96583-1(hardback)ISBN978-1-119-96607-4(ebk)
ISBN978-1-119-96608-1(ebk) ISBN978-1-118-84514-1(ebk)
Coverdesign:Cylinder
Typesetin10/12ptTimesLTStdbyLaserwordsPrivateLimited,Chennai,India
PrintedinGreatBritainbyCPIGroup(UK)Ltd,Croydon,CR04YY
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“thebeginningofataskisthebiggeststep”
Plato,TheRepublic
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Contents
Preface ix
Prologue xi
AbouttheAuthors xv
1 GeneralProbabilityTheory 1
1.1 Introduction 1
1.2 ProblemsandSolutions 4
1.2.1 ProbabilitySpaces 4
1.2.2 DiscreteandContinuousRandomVariables 11
1.2.3 PropertiesofExpectations 41
2 WienerProcess 51
2.1 Introduction 51
2.2 ProblemsandSolutions 55
2.2.1 BasicProperties 55
2.2.2 MarkovProperty 68
2.2.3 MartingaleProperty 71
2.2.4 FirstPassageTime 76
2.2.5 ReflectionPrinciple 84
2.2.6 QuadraticVariation 89
3 StochasticDifferentialEquations 95
3.1 Introduction 95
3.2 ProblemsandSolutions 102
3.2.1 Ito¯ Calculus 102
3.2.2 One-DimensionalDiffusionProcess 123
3.2.3 Multi-DimensionalDiffusionProcess 155
4 ChangeofMeasure 185
4.1 Introduction 185
4.2 ProblemsandSolutions 192
4.2.1 MartingaleRepresentationTheorem 192
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viii Contents
4.2.2 Girsanov’sTheorem 194
4.2.3 Risk-NeutralMeasure 221
5 PoissonProcess 243
5.1 Introduction 243
5.2 ProblemsandSolutions 251
5.2.1 PropertiesofPoissonProcess 251
5.2.2 JumpDiffusionProcess 281
5.2.3 Girsanov’sTheoremforJumpProcesses 298
5.2.4 Risk-NeutralMeasureforJumpProcesses 322
AppendixA MathematicsFormulae 331
AppendixB ProbabilityTheoryFormulae 341
AppendixC DifferentialEquationsFormulae 357
Bibliography 365
Notation 369
Index 373