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AfinitedifferenceschemeforoptionpricinginjumpdiffusionandexponentialL´evymodels∗RamaContandEkaterinaVoltchkovaRapportInterneCMAPNum´ero513,Sept.2003.AbstractWepresentafinitedifferencemethodforsolvingparabolicpartialintegro-differentialequationswithpossiblysingularkernelswhichariseinoptionpricingtheorywhentherandomevolutionoftheunderlyingassetisdrivenbyaL´evyprocessor,moregenerally,atime-inhomogeneousjump-diffusionprocess.WediscusslocalizationtoafinitedomainandprovideanestimateforthelocalizationerrorunderanintegrabilityconditionontheL´evymeasure.Weproposeanexplicit-implicittime-steppingschemetosolvetheequationandstudystabilityandconvergenceoftheschemesproposed,usingthenotionofviscositysolution.Numericaltestsareper-formedwithsmoothandnon-smoothinitialconditions.OurschemecanbeusedforEuropeanandbarrieroptions,appliesinthecaseofpure-jumpmodelsordegeneratediffusioncoefficients,andextendstotime-dependentcoefficients.Keywords:parabolicintegro-differentialequations,finitedifferencemethods,L´evyprocess,jump-diffusionmodels,optionpricing,viscositysolutions.∗ThisworkwaspresentedattheBachelierseminar(Paris,2003),WorkshoponL´evypro-cessesandPartialintegro-differentialequations(Palaiseau,2003)andIFIP2003(Nice).WethankYvesAchdou,HuyenPham,ChristophSchwabandPeterTankovforhelpfuldiscus-sions.1Contents1ExponentialL´evymodels41.1L´evyprocesses:definitions......................41.2ExponentialL´evymodels......................52Partialintegro-differentialequationforoptionprices62.1Europeanoptions...........................72.2Barrieroptions............................92.3Viscositysolutions..........................113Anexplicit-implicitfinitedifferencescheme163.1Localizationtoaboundeddomain.................173.2Truncationoftheintegral......................193.3Explicit-implicitscheme:finiteactivitycase............213.4Explicit-implicitscheme:infiniteactivitycase...........224Consistency,stabilityandconvergence234.1Consistency..............................244.2MonotonicityandStability.....................254.3Convergence..............................275Numericalresults306Discussionandextensions34AAppendix382Theshortcomingsofdiffusionmodelsinrepresentingtheriskrelatedtolargemarketmovementshaveledtothedevelopmentofvariousoptionpricingmodelswithjumps,wherelargereturnsarerepresentedasdiscontinuitiesofpricesasafunctionoftime.Modelswithjumpsallowformorerealisticrepresentationofpricedynamicsandagreaterflexibilityinmodellingandhavebeenthefocusofmuchrecentwork.Areviewoffinancialmodellingwithjumpprocessesmaybefoundin[12].ExponentialL´evymodels,wherethemarketpriceofanassetisrepresentedastheexponentialSt=exp(rt+Xt)ofaL´evyprocessXt,offeranalyticallytractableexamplesofpositivejumpprocesseswhicharesimpleenoughtoallowadetailedstudybothintermsofstatisticalpropertiesandasmodelsforrisk-neutraldynamicsi.e.optionpricingmodels.OptionpricingwithexponentialL´evymodelsisdiscussedin[12,15,17,20,27].TheflexibilityofchoiceoftheL´evyprocessXallowstocalibratethemodeltomarketpricesofoptionsandreproduceawidevarietyofimpliedvolatilityskews/smiles.TheMarkovpropertyofthepriceallowsustoexpressoptionpricesassolutionsofpartialintegro-differentialequations(PIDEs)whichinvolve,inadditiontoa(possiblydegenerate)second-orderdifferentialoperator,anon-localintegraltermwhichrequiresspecifictreatmentbothatthetheoreticalandnumericallevel.Inthispaper,wediscussthederivationofsuchPIDEsforEuropeanandbarrieroptionsandproposeafinitedifferenceschemeforsolvingthem.Ournumericalsolutionisbasedonsplittingtheoperatorintoalocalandanon-localpart:wetreatthelocaltermusinganimplicitstepandthenon-localtermusinganexplicitstep.Thisidea,previouslyusedfornon-linearPDEs[3],allowsforanefficientimplementationintermsofspeedandstabilityrestrictions.OurschemeextendstoinfiniteactivityL´evyprocesseswithsingularkernelsanddoesnotrequirethediffusionparttobenon-degenerate.Westudytheconsistencyandstabilityofthisscheme,showitsconvergencetoaviscositysolutionofthepartialintegro-differentialequationandstudyitsnumericalperformanceontwoexamples,theMertonmodelwithGaussianjumpsandtheinfiniteactivityVarianceGammamodel.OurschemecanbeusedforEuropeanandbarrieroptionsandcanalsobeextendedtothecaseofnon-constantcoefficients.Suchpartialintegro-differentialequations(PIDEs)havebeenusedbyotherauthorstopriceoptionsinmodelswithjumps[2,10,22,14]butthederivationoftheequationisoftenomittedorincomplete.Insection2,wediscussthederivationofintegro-differentialequationsforEuropeanandbarrieroptions,firstgivingconditionsunderwhichoptionvaluescanbedescribedintermsofclassicalsolutionsfollowedbyamoregeneralresultusingthenotionofviscositysolution.ThecaseofAmericanoptionshasbeendiscussedin[23].SolvingsuchPIDEsbyfinitedifferencemethodsinvolvesseveralapproxi-mations:localizationoftheequationtoaboundeddomain,treatmentofthesingularityduetosmalljumps,discretizationoftheequationinspaceandit-erationintime.Wediscusslocalizationerrorsinsection3.1andprovideanestimateforthelocalizationerrorunderanintegrabilityconditionontheL´evymeasure.Insection3weproposeanexplicit-implicitfinitedifferenceschemeandstudycon
本文标题:A finite difference scheme for option pricing in j
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