Levy Processes in Finance: Pricing Financial Derivatives

189 Pages·2003·1.933 MB·English

by  Wim Schoutens(auth.), Walter A. Shewhart, Samuel S. Wilks(eds.)

Le´vyProcessesinFinance:PricingFinancialDerivatives.WimSchoutens
Copyright2003JohnWiley&Sons,Ltd.
ISBN:0-470-85156-2
Le´vyProcessesinFinance
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Le´vy Processes in Finance
PricingFinancialDerivatives
WimSchoutens
KatholiekeUniversiteitLeuven,Belgium
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Contents
Preface xi
Acknowledgements xv
1 Introduction 1
1.1 FinancialAssets 1
1.2 DerivativeSecurities 3
1.2.1 Options 3
1.2.2 PricesofOptionsontheS&P500Index 5
1.3 ModellingAssumptions 7
1.4 Arbitrage 9
2 FinancialMathematicsinContinuousTime 11
2.1 StochasticProcessesandFiltrations 11
2.2 ClassesofProcesses 13
2.2.1 MarkovProcesses 13
2.2.2 Martingales 14
2.2.3 Finite-andInfinite-VariationProcesses 14
2.3 CharacteristicFunctions 15
2.4 StochasticIntegralsandSDEs 16
2.5 FinancialMathematicsinContinuousTime 17
2.5.1 EquivalentMartingaleMeasure 17
2.5.2 PricingFormulasforEuropeanOptions 19
2.6 Dividends 21
3 TheBlack–ScholesModel 23
3.1 TheNormalDistribution 23
3.2 BrownianMotion 24
3.2.1 Definition 25
3.2.2 Properties 26
3.3 GeometricBrownianMotion 27
viii CONTENTS
3.4 TheBlack–ScholesOptionPricingModel 28
3.4.1 TheBlack–ScholesMarketModel 29
3.4.2 MarketCompleteness 30
3.4.3 TheRisk-NeutralSetting 30
3.4.4 ThePricingofOptionsundertheBlack–ScholesModel 30
4 ImperfectionsoftheBlack–ScholesModel 33
4.1 TheNon-GaussianCharacter 33
4.1.1 AsymmetryandExcessKurtosis 33
4.1.2 DensityEstimation 35
4.1.3 StatisticalTesting 36
4.2 StochasticVolatility 38
4.3 InconsistencywithMarketOptionPrices 39
5 LévyProcessesandOUProcesses 43
5.1 LévyProcesses 44
5.1.1 Definition 44
5.1.2 Properties 45
5.2 OUProcesses 47
5.2.1 Self-Decomposability 47
5.2.2 OUProcesses 48
5.3 ExamplesofLévyProcesses 50
5.3.1 ThePoissonProcess 50
5.3.2 TheCompoundPoissonProcess 51
5.3.3 TheGammaProcess 52
5.3.4 TheInverseGaussianProcess 53
5.3.5 TheGeneralizedInverseGaussianProcess 54
5.3.6 TheTemperedStableProcess 56
5.3.7 TheVarianceGammaProcess 57
5.3.8 TheNormalInverseGaussianProcess 59
5.3.9 TheCGMYProcess 60
5.3.10 TheMeixnerProcess 62
5.3.11 TheGeneralizedHyperbolicProcess 65
5.4 AddinganAdditionalDriftTerm 67
5.5 ExamplesofOUProcesses 67
5.5.1 TheGamma–OUProcess 68
5.5.2 TheIG–OUProcess 69
5.5.3 OtherExamples 70
6 StockPriceModelsDrivenbyLévyProcesses 73
6.1 StatisticalTesting 73
6.1.1 ParameterEstimation 73
6.1.2 StatisticalTesting 74
CONTENTS ix
6.2 TheLévyMarketModel 76
6.2.1 MarketIncompleteness 77
6.2.2 TheEquivalentMartingaleMeasure 77
6.2.3 PricingFormulasforEuropeanOptions 80
6.3 CalibrationofMarketOptionPrices 82
7 LévyModelswithStochasticVolatility 85
7.1 TheBNSModel 85
7.1.1 TheBNSModelwithGammaSV 87
7.1.2 TheBNSModelwithIGSV 88
7.2 TheStochasticTimeChange 88
7.2.1 TheIntegratedCIRTimeChange 89
7.2.2 TheIntOUTimeChange 90
7.3 TheLévySVMarketModel 91
7.4 CalibrationofMarketOptionPrices 97
7.4.1 CalibrationoftheBNSModels 97
7.4.2 CalibrationoftheLévySVModels 98
7.5 Conclusion 98
8 SimulationTechniques 101
8.1 SimulationofBasicProcesses 101
8.1.1 SimulationofStandardBrownianMotion 101
8.1.2 SimulationofaPoissonProcess 102
8.2 SimulationofaLévyProcess 102
8.2.1 TheCompoundPoissonApproximation 103
8.2.2 OntheChoiceofthePoissonProcesses 105
8.3 SimulationofanOUProcess 107
8.4 SimulationofParticularProcesses 108
8.4.1 TheGammaProcess 108
8.4.2 TheVGProcess 109
8.4.3 TheTSProcess 111
8.4.4 TheIGProcess 111
8.4.5 TheNIGProcess 113
8.4.6 TheGamma–OUProcess 114
8.4.7 TheIG–OUProcess 115
8.4.8 TheCIRProcess 117
8.4.9 BNSModel 117
9 ExoticOptionPricing 119
9.1 BarrierandLookbackOptions 119
9.1.1 Introduction 119
9.1.2 Black–ScholesBarrierandLookbackOptionPrices 121
9.1.3 LookbackandBarrierOptionsinaLévyMarket 123
x CONTENTS
9.2 OtherExoticOptions 125
9.2.1 ThePerpetualAmericanCallandPutOption 125
9.2.2 ThePerpetualRussianOption 126
9.2.3 Touch-and-OutOptions 126
9.3 ExoticOptionPricingbyMonteCarloSimulation 127
9.3.1 Introduction 127
9.3.2 MonteCarloPricing 127
9.3.3 VarianceReductionbyControlVariates 129
9.3.4 NumericalResults 132
9.3.5 Conclusion 134
10 Interest-RateModels 135
10.1 GeneralInterest-RateTheory 135
10.2 TheGaussianHJMModel 138
10.3 TheLévyHJMModel 141
10.4 BondOptionPricing 142
10.5 Multi-FactorModels 144
AppendixA SpecialFunctions 147
A.1 BesselFunctions 147
A.2 ModifiedBesselFunctions 148
A.3 TheGeneralizedHypergeometricSeries 149
A.4 OrthogonalPolynomials 149
A.4.1 Hermitepolynomialswithparameter 149
A.4.2 Meixner–PollaczekPolynomials 150
AppendixB LévyProcesses 151
B.1 CharacteristicFunctions 151
B.1.1 DistributionsontheNonnegativeIntegers 151
B.1.2 DistributionsonthePositiveHalf-Line 151
B.1.3 DistributionsontheRealLine 152
B.2 LévyTriplets 153
B.2.1 γ 153
B.2.2 TheLévyMeasureν(dx) 154
AppendixC S&P500CallOptionPrices 155
References 157
Index 165
Preface
Thestoryofmodellingfinancialmarketswithstochasticprocessesbeganin1900with
thestudyofBachelier(1900).HemodelledstocksasaBrownianmotionwithdrift.
However,themodelhadmanyimperfections,including,forexample,negativestock
prices. It was 65 years before another, more appropriate, model was suggested by
Samuelson(1965):geometricBrownianmotion.EightyearslaterBlackandScholes
(1973)andMerton(1973)demonstratedhowtopriceEuropeanoptionsbasedonthe
geometricBrownianmodel.Thisstock-pricemodelisnowcalledtheBlack–Scholes
model, for which Scholes and Merton received the Nobel Prize for Economics in
1997(Blackhadalreadydied).
It has become clear, however, that this option-pricing model is inconsistent with
optionsdata.Impliedvolatilitymodelscandobetter,but,fundamentally,theseconsist
ofthewrongbuildingblocks.ToimproveontheperformanceoftheBlack–Scholes
model, Lévy models were proposed in the late 1980s and early 1990s, since when
theyhavebeenrefinedtotakeaccountofdifferentstylizedfeaturesofthemarkets.
Thisbookisconcernedwiththepricingofderivativesecuritiesinmarketmodels
basedonLévyprocesses.Financialmathematicshasrecentlyenjoyedconsiderable
prestigeasaresultofitsimpactonthefinanceindustry.ThetheoryofLévyprocesses
has also seen exciting developments in recent years.The fusion of these two fields
ofmathematicshasprovidednewappliedmodellingperspectiveswithinthecontext
offinanceandfurtherstimulusforthestudyofproblemswithinthecontextofLévy
processes.
ThisbookisaimedatpeopleworkingintheareasofmathematicalfinanceandLévy
processes, with the intention of convincing the former that the rich theory of Lévy
processescanleadtotractableandattractivemodelsthatperformsignificantlybetter
thanthestandardBlack–Scholesmodel.ForthoseworkingwithLévyprocesses,we
hopetoshowhowtheobjectstheystudycanbereadilyappliedinpractice.
We have taken great care not to use too much esoteric mathematics, nor to get
too involved in technicalities, nor to give involved proofs. We focus on the ideas,
andtheintuitionbehindthemodellingprocessanditsapplications.Nevertheless,the
processesinvolvedinthemodellingaredescribedveryaccuratelyandingreatdetail.
Theseprocesseslieattheheartofthetheoryanditisveryimportanttohaveaclear
viewoftheirproperties.