Financial Models with Levy Processes and Volatility Clustering

405 Pages·2011·4.587 MB·English

by  Svetlozar T. Rachev, Young Shin Kim, Michele Leonardo Bianchi, Frank J. Fabozzi(auth.)

Financial Models with
Le´ vy Processes and
Volatility Clustering
TheFrankJ.FabozziSeries
FixedIncomeSecurities,SecondEditionbyFrankJ.Fabozzi
FocusonValue:ACorporateandInvestorGuidetoWealthCreationbyJamesL.GrantandJamesA.Abate
HandbookofGlobalFixedIncomeCalculationsbyDragomirKrgin
ManagingaCorporateBondPortfoliobyLelandE.CrabbeandFrankJ.Fabozzi
RealOptionsandOption-EmbeddedSecuritiesbyWilliamT.Moore
CapitalBudgeting:TheoryandPracticebyPamelaP.PetersonandFrankJ.Fabozzi
TheExchange-TradedFundsManualbyGaryL.Gastineau
ProfessionalPerspectivesonFixedIncomePortfolioManagement,Volume3editedbyFrankJ.Fabozzi
InvestinginEmergingFixedIncomeMarketseditedbyFrankJ.FabozziandEfstathiaPilarinu
HandbookofAlternativeAssetsbyMarkJ.P.Anson
TheGlobalMoneyMarketsbyFrankJ.Fabozzi,StevenV.Mann,andMooradChoudhry
TheHandbookofFinancialInstrumentseditedbyFrankJ.Fabozzi
InterestRate,TermStructure,andValuationModelingeditedbyFrankJ.Fabozzi
InvestmentPerformanceMeasurementbyBruceJ.Feibel
TheHandbookofEquityStyleManagementeditedbyT.DanielCogginandFrankJ.Fabozzi
TheTheoryandPracticeofInvestmentManagementeditedbyFrankJ.FabozziandHarryM.Markowitz
FoundationsofEconomicValueAdded,SecondEditionbyJamesL.Grant
FinancialManagementandAnalysis,SecondEditionbyFrankJ.FabozziandPamelaP.Peterson
MeasuringandControllingInterestRateandCreditRisk,SecondEditionbyFrankJ.Fabozzi,StevenV.Mann,and
MooradChoudhry
ProfessionalPerspectivesonFixedIncomePortfolioManagement,Volume4editedbyFrankJ.Fabozzi
TheHandbookofEuropeanFixedIncomeSecuritieseditedbyFrankJ.FabozziandMooradChoudhry
TheHandbookofEuropeanStructuredFinancialProductseditedbyFrankJ.FabozziandMooradChoudhry
TheMathematicsofFinancialModelingandInvestmentManagementbySergioM.FocardiandFrankJ.Fabozzi
ShortSelling:Strategies,Risks,andRewardseditedbyFrankJ.Fabozzi
TheRealEstateInvestmentHandbookbyG.TimothyHaightandDanielSinger
MarketNeutralStrategieseditedbyBruceI.JacobsandKennethN.Le´vy
SecuritiesFinance:SecuritiesLendingandRepurchaseAgreementseditedbyFrankJ.FabozziandStevenV.Mann
Fat-TailedandSkewedAssetReturnDistributionsbySvetlozarT.Rachev,ChristianMenn,andFrankJ.Fabozzi
FinancialModelingoftheEquityMarket:FromCAPMtoCointegrationbyFrankJ.Fabozzi,SergioM.Focardi,and
PetterN.Kolm
AdvancedBondPortfolioManagement:BestPracticesinModelingandStrategieseditedbyFrankJ.Fabozzi,Lionel
Martellini,andPhilippePriaulet
AnalysisofFinancialStatements,SecondEditionbyPamelaP.PetersonandFrankJ.Fabozzi
CollateralizedDebtObligations:StructuresandAnalysis,SecondEditionbyDouglasJ.Lucas,LaurieS.Goodman,and
FrankJ.Fabozzi
HandbookofAlternativeAssets,SecondEditionbyMarkJ.P.Anson
IntroductiontoStructuredFinancebyFrankJ.Fabozzi,HenryA.Davis,andMooradChoudhry
FinancialEconometricsbySvetlozarT.Rachev,StefanMittnik,FrankJ.Fabozzi,SergioM.Focardi,andTeoJasˇic´
DevelopmentsinCollateralizedDebtObligations:NewProductsandInsightsbyDouglasJ.Lucas,LaurieS.Goodman,
FrankJ.Fabozzi,andRebeccaJ.Manning
RobustPortfolioOptimizationandManagementbyFrankJ.Fabozzi,PetterN.Kolm,DessislavaA.Pachamanova,and
SergioM.Focardi
AdvancedStochasticModels,RiskAssessment,andPortfolioOptimizationsbySvetlozarT.Rachev,StoganV.Stoyanov,
andFrankJ.Fabozzi
HowtoSelectInvestmentManagersandEvaluatePerformancebyG.TimothyHaight,StephenO.Morrell,andGlenn
E.Ross
BayesianMethodsinFinancebySvetlozarT.Rachev,JohnS.J.Hsu,BilianaS.Bagasheva,andFrankJ.Fabozzi
TheHandbookofMunicipalBondseditedbySylvanG.FeldsteinandFrankJ.Fabozzi
SubprimeMortgageCreditDerivativesbyLaurieS.Goodman,ShuminLi,DouglasJ.Lucas,ThomasA.Zimmerman,
andFrankJ.Fabozzi
IntroductiontoSecuritizationbyFrankJ.FabozziandVinodKothari
StructuredProductsandRelatedCreditDerivativeseditedbyBrianP.Lancaster,GlennM.Schultz,andFrankJ.Fabozzi
HandbookofFinance:VolumeI:FinancialMarketsandInstrumentseditedbyFrankJ.Fabozzi
HandbookofFinance:VolumeII:FinancialManagementandAssetManagementeditedbyFrankJ.Fabozzi
HandbookofFinance:VolumeIII:Valuation,FinancialModeling,andQuantitativeToolseditedbyFrankJ.Fabozzi
Finance:CapitalMarkets,FinancialManagement,andInvestmentManagementbyFrankJ.FabozziandPamelaPeterson-
Drake
ActivePrivateEquityRealEstateStrategyeditedbyDavidJ.Lynn
FoundationsandApplicationsoftheTimeValueofMoneybyPamelaPeterson-DrakeandFrankJ.Fabozzi
LeveragedFinance:Concepts,Methods,andTradingofHigh-YieldBonds,Loans,andDerivativesbyStephenAntczak,
DouglasLucas,andFrankJ.Fabozzi
ModernFinancialSystems:TheoryandApplicationsbyEdwinNeave
InstitutionalInvestmentManagement:EquityandBondPortfolioStrategiesandApplicationsbyFrankJ.Fabozzi
QuantitativeEquityInvesting:TechniquesandStrategiesbyFrankJ.Fabozzi,SergioM.Focardi,andPetterN.Kolm
SimulationandOptimizationinFinance:ModelingwithMATLAB,@Risk,orVBAbyDessislavaA.Pachamanovaand
FrankJ.Fabozzi
Financial Models with
Le´ vy Processes and
Volatility Clustering
SVETLOZAR T. RACHEV
YOUNG SHIN KIM
MICHELE LEONARDO BIANCHI
FRANK J. FABOZZI
John Wiley & Sons, Inc.
Copyright(cid:2)C 2011byJohnWiley&Sons,Inc.Allrightsreserved.
PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey.
PublishedsimultaneouslyinCanada.
Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedin
anyformorbyanymeans,electronic,mechanical,photocopying,recording,scanning,or
otherwise,exceptaspermittedunderSection107or108ofthe1976UnitedStatesCopyright
Act,withouteitherthepriorwrittenpermissionofthePublisher,orauthorizationthrough
paymentoftheappropriateper-copyfeetotheCopyrightClearanceCenter,Inc.,222
RosewoodDrive,Danvers,MA01923,[phone],fax[phone],orontheWeb
atwww.copyright.com.RequeststothePublisherforpermissionshouldbeaddressedtothe
PermissionsDepartment,JohnWiley&Sons,Inc.,111RiverStreet,Hoboken,NJ07030,
[phone],fax[phone],oronlineathttp://www.wiley.com/go/permissions.
LimitofLiability/DisclaimerofWarranty:Whilethepublisherandauthorhaveusedtheir
besteffortsinpreparingthisbook,theymakenorepresentationsorwarrantieswithrespectto
theaccuracyorcompletenessofthecontentsofthisbookandspecificallydisclaimanyimplied
warrantiesofmerchantabilityorfitnessforaparticularpurpose.Nowarrantymaybecreated
orextendedbysalesrepresentativesorwrittensalesmaterials.Theadviceandstrategies
containedhereinmaynotbesuitableforyoursituation.Youshouldconsultwitha
professionalwhereappropriate.Neitherthepublishernorauthorshallbeliableforanylossof
profitoranyothercommercialdamages,includingbutnotlimitedtospecial,incidental,
consequential,orotherdamages.
Forgeneralinformationonourotherproductsandservicesorfortechnicalsupport,please
contactourCustomerCareDepartmentwithintheUnitedStatesat(800)762-2974,outside
theUnitedStatesat(317)572-3993orfax(317)572-4002.
Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsin
printmaynotbeavailableinelectronicbooks.FormoreinformationaboutWileyproducts,
visitourwebsiteatwww.wiley.com.
LibraryofCongressCataloging-in-PublicationData:
FinancialmodelswithLe´vyprocessesandvolatilityclustering/SvetlozarT.Rachev...[etal.].
p.cm.—(TheFrankJ.Fabozziseries)
Includesindex.
ISBN978-0-470-48235-3(cloth);978-0-470-93716-7(ebk);
978-0-470-93726-6(ebk);978-1-118-00670-2(ebk)
1.Capitalassetspricingmodel. 2.Le´vyprocesses. 3.Finance—Mathematicalmodels.
4.Probabilities. I.Rachev,S.T.(SvetlozarTodorov)
HG4637.F562011
332(cid:3).0415015192—dc22 2010033299
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
STR
To my grandchildren Iliana, Zoya, and Svetlozar
YSK
To my wife Myung-Ja and my son Minseob
MLB
To my wife Giorgia
FJF
To my wife Donna and my children Francesco,
Patricia, and Karly
Contents
Preface xv
AbouttheAuthors xix
CHAPTER1
Introduction 1
1.1 TheNeedforBetterFinancialModelingofAssetPrices 1
1.2 TheFamilyofStableDistributionandItsProperties 5
1.2.1 ParameterizationoftheStableDistribution 5
1.2.2 DesirablePropertiesoftheStableDistributions 7
1.2.3 ConsiderationsintheUseoftheStable
Distribution 8
1.3 OptionPricingwithVolatilityClustering 9
1.3.1 Non-GaussianGARCHModels 11
1.4 ModelDependencies 12
1.5 MonteCarlo 13
1.6 OrganizationoftheBook 14
References 15
CHAPTER2
ProbabilityDistributions 19
2.1 BasicConcepts 19
2.2 DiscreteProbabilityDistributions 20
2.2.1 BernoulliDistribution 21
2.2.2 BinomialDistribution 21
2.2.3 PoissonDistribution 22
2.3 ContinuousProbabilityDistributions 22
2.3.1 ProbabilityDistributionFunction,
ProbabilityDensityFunction,and
CumulativeDistributionFunction 23
2.3.2 NormalDistribution 26
2.3.3 ExponentialDistribution 28
2.3.4 GammaDistribution 28
vii
viii CONTENTS
2.3.5 VarianceGammaDistribution 29
2.3.6 InverseGaussianDistribution 30
2.4 StatisticMomentsandQuantiles 30
2.4.1 Location 31
2.4.2 Dispersion 31
2.4.3 Asymmetry 31
2.4.4 ConcentrationinTails 32
2.4.5 StatisticalMoments 32
2.4.6 Quantiles 34
2.4.7 SampleMoments 35
2.5 CharacteristicFunction 35
2.6 JointProbabilityDistributions 39
2.6.1 ConditionalProbability 39
2.6.2 JointProbabilityDistributionDefined 40
2.6.3 MarginalDistribution 41
2.6.4 DependenceofRandomVariables 41
2.6.5 CovarianceandCorrelation 42
2.6.6 MultivariateNormalDistribution 43
2.6.7 EllipticalDistributions 46
2.6.8 CopulaFunctions 47
2.7 Summary 54
References 54
CHAPTER3
StableandTemperedStableDistributions 57
3.1 α-StableDistribution 58
3.1.1 Definitionofanα-StableRandomVariable 58
3.1.2 UsefulPropertiesofanα-StableRandomVariable 61
3.1.3 SmoothlyTruncatedStableDistribution 63
3.2 TemperedStableDistributions 65
3.2.1 ClassicalTemperedStableDistribution 65
3.2.2 GeneralizedClassicalTemperedStable
Distribution 68
3.2.3 ModifiedTemperedStableDistribution 69
3.2.4 NormalTemperedStableDistribution 70
3.2.5 Kim-RachevTemperedStableDistribution 73
3.2.6 RapidlyDecreasingTemperedStableDistribution 75
3.3 InfinitelyDivisibleDistributions 76
3.3.1 ExponentialMoments 80
3.4 Summary 82
Contents ix
3.5 Appendix 82
3.5.1 TheHypergeometricFunction 83
3.5.2 TheConfluentHypergeometricFunction 83
References 84
CHAPTER4
StochasticProcessesinContinuousTime 87
4.1 SomePreliminaries 88
4.2 PoissonProcess 88
4.2.1 CompoundedPoissonProcess 89
4.3 PureJumpProcess 89
4.3.1 GammaProcess 92
4.3.2 InverseGaussianProcess 92
4.3.3 VarianceGammaProcess 92
4.3.4 α-StableProcess 93
4.3.5 TemperedStableProcess 94
4.4 BrownianMotion 95
4.4.1 ArithmeticBrownianMotion 99
4.4.2 GeometricBrownianMotion 99
4.5 Time-ChangedBrownianMotion 100
4.5.1 VarianceGammaProcess 101
4.5.2 NormalInverseGaussianProcess 102
4.5.3 NormalTemperedStableProcess 103
4.6 Le´vyProcess 104
4.7 Summary 105
References 106
CHAPTER5
ConditionalExpectationandChangeofMeasure 107
5.1 Events,σ-Fields,andFiltration 107
5.2 ConditionalExpectation 109
5.3 ChangeofMeasures 111
5.3.1 EquivalentProbabilityMeasure 111
5.3.2 ChangeofMeasureforContinuous-Time
Processes 113
5.3.3 ChangeofMeasureinTemperedStableProcesses 117
5.4 Summary 121
References 121
CHAPTER6
ExponentialLe´vyModels 123
6.1 ExponentialLe´vyModels 123
x CONTENTS
6.2 Fittingα-StableandTemperedStableDistributions 126
6.2.1 FittingtheCharacteristicFunction 126
6.2.2 MaximumLikelihoodEstimationwith
NumericalApproximationoftheDensityFunction 127
6.2.3 AssessingtheGoodnessofFit 127
6.3 Illustration:ParameterEstimationforTempered
StableDistributions 131
6.4 Summary 135
6.5 Appendix:NumericalApproximationofProbability
DensityandCumulativeDistributionFunctions 135
6.5.1 NumericalMethodfortheFourierTransform 139
References 140
CHAPTER7
OptionPricinginExponentialLe´vyModels 141
7.1 OptionContract 141
7.2 BoundaryConditionsforthePriceofanOption 142
7.3 No-ArbitragePricingandEquivalentMartingaleMeasure 145
7.4 OptionPricingundertheBlack-ScholesModel 148
7.5 EuropeanOptionPricingunderExponential
TemperedStableModels 149
7.5.1 Illustration:ImpliedVolatility 152
7.5.2 Illustration:CalibratingRisk-NeutralParameters 153
7.5.3 Illustration:CalibratingMarketParameters
andRisk-NeutralParametersTogether 161
7.6 SubordinatedStockPriceModel 164
7.6.1 StochasticVolatilityLe´vyProcessModel 166
7.7 Summary 167
References 167
CHAPTER8
Simulation 169
8.1 RandomNumberGenerators 170
8.1.1 UniformDistributions 170
8.1.2 DiscreteDistributions 172
8.1.3 ContinuousNonuniformDistributions 172
8.1.4 SimulationofParticularDistributions 177
8.2 SimulationTechniquesforLe´vyProcesses 182
8.2.1 TakingCareofSmallJumps 183
8.2.2 SeriesRepresentation:AGeneralFramework 186
8.2.3 Rosin´skyRejectionMethod 191
8.2.4 α-StableProcesses 192
Contents xi
8.3 TemperedStableProcesses 193
8.3.1 Kim-RachevTemperedStableCase 196
8.3.2 ClassicalTemperedStableCase 198
8.4 TemperedInfinitelyDivisibleProcesses 199
8.4.1 RapidlyDecreasingTemperedStableCase 201
8.4.2 ModifiedTemperedStableCase 202
8.5 Time-ChangedBrownianMotion 203
8.5.1 ClassicalTemperedStableProcesses 205
8.5.2 VarianceGammaandSkewedVariance
GammaProcesses 206
8.5.3 NormalTemperedStableProcesses 207
8.5.4 NormalInverseGaussianProcesses 208
8.6 MonteCarloMethods 209
8.6.1 VarianceReductionTechniques 210
8.6.2 ANonparametricMonteCarloMethod 214
8.6.3 AMonteCarloExample 216
Appendix 217
References 220
CHAPTER9
Multi-Tailt-Distribution 225
9.1 Introduction 225
9.2 PrincipalComponentAnalysis 227
9.2.1 PrincipalComponentTailFunctions 228
9.2.2 DensityofaMulti-Tailt RandomVariable 231
9.3 EstimatingParameters 232
9.3.1 EstimationoftheDispersionMatrix 233
9.3.2 EstimationoftheParameterSet(cid:4) 233
9.4 EmpiricalResults 237
9.4.1 ComparisontoOtherModels 237
9.4.2 Two-DimensionalAnalysis 238
9.4.3 Multi-Tailt ModelCheckfortheDAX 242
9.5 Summary 244
References 246
CHAPTER10
Non-GaussianPortfolioAllocation 247
10.1 Introduction 247
10.2 MultifactorLinearModel 248
10.3 ModelingDependencies 251
10.4 AverageValue-at-Risk 253
10.5 OptimalPortfolios 255