Numerical Valuation of European and American Optio

NumericalValuationofEuropeanandAmericanOptionsunderKou’sJump-DiffusionModel∗JariToivanen†AbstractNumericalmethodsaredevelopedforpricingEuropeanandAmericanop-tionsunderKou’sjump-diffusionmodelwhichassumesthepriceoftheun-derlyingassettobehavelikeageometricalBrownianmotionwithadriftandjumpswhosesizeislog-double-exponentiallydistributed.ThepriceofaEu-ropeanoptionisgivenbyapartialintegro-differentialequation(PIDE)whileAmericanoptionsleadtoalinearcomplementarityproblem(LCP)withthesameoperator.Spatialdifferentialoperatorsarediscretizedusingfinitediffer-encesonnonuniformgridsandtimesteppingisperformedusingtheimplicitRannacherscheme.Fortheevaluationoftheintegraltermeasytoimplementrecursionformulasarederivedwhichhaveoptimalcomputationalcost.WhenpricingEuropeanoptionstheresultingdenselinearsystemsaresolvedusingastationaryiteration.ForAmericanoptionstwowaystosolvetheLCPsarede-scribed:anoperatorslittingmethodandapenaltymethod.Numericalexperi-mentsconfirmthatthedevelopedmethodsareveryefficientasfairlyaccurateoptionpricescanbecomputedinafewmillisecondsonaPC.Keywords:optionpricing,jump-diffusionmodel,partialintegro-differentialequa-tion,linearcomplementarityproblem,finitedifferencemethod,operatorsplittingmethod,penaltymethod1IntroductionTheamountoffinancialoptiontradinghasgrowntoenormousscalesincethepio-neeringworkbyBlackandScholes[7]andMerton[33]onthepricingofoptionsin1973.DuringthelasttwodecadesithasbecomeevidentthattheirassumptionthatthepriceofunderlyingassetbehaveslikeageometricBrownianmotionwithadriftandaconstantvolatilitycannotexplainthemarketpricesofoptionswithvarious∗ThisresearchwassupportedbytheAcademyofFinland,grant#207089†DepartmentofMathematicalInformationTechnology,UniversityofJyv¨askyl¨a,POBox35(Agora),FI-40014UniversityofJyv¨askyl¨a,Finland,[e-mail]1976Mertonproposedtoaddjumpswhichhavenormallydistributedsizetothebehaviorofassetprices[34].Duringthelasttenyearstheresearchonmodelswithjumpshavebecomeveryactive.Alargenum-berofsuchmodelshavebeenproposed;see[12]andreferencestherein.ThemodelproposedbyKouin[27]assumesthedistributionofjumpsizestobealog-double-exponentialfunction;seealso[28].InamorerichCGMYmodel[9]theassetpriceisaL´evyprocesswithpossiblyinfinitejumpactivity.Anothergeneralizationistoassumethevolatilitytobestochasticlikein[20],forexample.Apopularapproachistouseadeterministicvolatilitymodelwherethevolatilityisafunctionoftimeandthepriceoftheunderlyingasset[17].ThisapproachwascombinedwithMerton’sjump-diffusionmodelin[4].Themostgenericmodelsassumestochasticvolatilitywithjumpsintheassetpriceandpossiblyalsoinvolatilityliketheonesin[6],[16].Themostbasicoptionsarecallandputoptionswhichgivetherighttobuyandsell,respectively,theunderlyingassetwithastrikeprice.TheseoptionscanbeEuropeanorAmericanwhichmeansthattheycanbeexercisedonlyattheexpirydateoranytimebeforeit,respectively.UndermanymodelsitispossibletoderiveformulasforthepriceofEuropeanoptions,butAmericanoptionshavetobepricedusuallynumerically.Thereisalargenumberofmethodstopriceoptions.Inthispaperweconsiderapproachwhichsolvesnumericallyapartialintegro-differentialequation(PIDE)orinequalityderivedforthepriceoftheoption.TheEuropeanoptionsleadtoequationswhileAmericanoptionsleadinequalitieswhichhaveaformofalinearcomplementarityproblem(LCP).Inoptionpricingcontext,mostcommonwaytodiscretizedthedifferentialoperatorshavebeenfinitedifferencemethods;see[1]and[37],forexample.Thetreatmentoftheintegraltermassociatedtojumpsinmodelsismorechallenging.Theirdiscretizationleadstofullmatrices.Themultiplicationofavectorbysuchan×nmatrixrequiresO(n2)operationswhenimplementedinastraightforwardmanner.Withanimplicittimediscretizationitisnecessarytosolveaproblemwithafullmatrixwhichiscomputationallymoreexpensivethanamultiplication.Herewereviewearlierworksonpricingoptionsunderjumpmodelsusingfinitedifference/elementmethods.Zhangin[39]pricedAmericanoptionswithMerton’smodel.Theintegraltermwastreatedexplicitlyintimewhilethedifferentialtermsweretreatedimplicitly.Thisleadtoafirst-orderaccuratemethodintimeandastabilityrestrictionforthesizeoftimestep.TavellaandRandallin[37]consideredpricingEuropeanoptions.Theyusedanimplicittimediscretizationwhichleadstothesolutionofproblemswithafullmatrix.Theyproposedastationaryiterativemethodtosolvethesedenseproblemswhichconvergesfairlyrapidly.Eachiterationrequiresasolutionwithatridiagonalmatrixandamultiplicationofavectorbyafullmatrix.AndersenandAndreasenin[4]proposedforEuropeanoptionsanADI-typeoperatorsplittingmethodwithtwofractionalsteps.Inthefirststepthespatialdifferentialoperatoristreatedimplicitlyandtheintegraloperatoristreatedexplicitlyandinthesecondsteptheirrolesareexchanged.Thismethodissecond-orderaccurateintimeandunconditionallystable.Furthermore,theyusedawell-knownfasttechniqueforevaluatingconvolutionintegrals.ItisusesthefastFourier2transform(FFT)anditrequiresO(nlogn)operations.Thus,theirapproachleadtoanessentialreductionofcomputationalcost.In[14]d’Halluin,Forsyth,andLabahnpriceAmericanoptionsusingapenaltymethod.Forth