exact and approximated option pricing in a stochastic volatility jump

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1、Exactandapproximatedoptionpricinginastochasticvolatilityjump-diffusionmodelFernandaD’Ippoliti,EnricoMoretto,SaraPasquali,andBarbaraTrivellatoAbstract.Weproposeastochasticvolatilityjump-diffusionmodelforoptionpricingwithcontemporaneousjumpsinbothspotreturnandvolatilitydynamics.Themodeladmit

2、s,inthespiritofHeston,aclosed-formsolutionforEuropean-styleoptions.Toevaluatemorecomplexderivativesforwhichthereisnoexplicitpricingexpression,suchasbarrieroptions,anumericalmethodology,basedonan“exactalgorithm”proposedbyBroadieandKaya,isapplied.Thistechniqueiscalledexactasnodiscretisationo

3、fdynamicsisrequired.Weenduptestingthegoodnessofourmethodologyusing,asrealdata,pricesandimpliedvolatilitiesfromtheDJEuroStoxx50marketandprovidingsomenumericalresultsforbarrieroptionsandtheirGreeks.Keywords:stochasticvolatilityjump-diffusionmodels,barrieroptionpricing,rejectionsam-pling1Intr

4、oductionInrecentyears,manyauthorshavetriedtoovercometheHestonsetting[11].Thisisduetothefactthattheabilityofstochasticvolatilitymodelstopriceshort-timeoptionsislimited[1,14].In[2],theauthoradded(proportional)log-normaljumpstothedynamicsofspotreturnsintheHestonmodel(see[10]forlog-uniformjump

5、s)andextendedtheFourierinversionoptionpricingmethodologyof[11,15]forEuropeanandAmericanoptions.Thisfurtherimprovementhasnotbeensufficienttocapturetherapidincreaseofvolatilityexperiencedinfinancialmarkets.OnedocumentedexampleofthisfeatureisgivenbythemarketstressofFall1987,whenthevolatilityjum

6、pedupfromroughly20%toover50%.Tofillthisgap,theintroductionofjumpsinvolatilityhasbeenconsideredthenaturalevolutionoftheexistingdiffusivestochasticvolatilitymodelswithjumpsinreturns.In[9],theauthorsrecognisedthat“althoughthemotivationforjumpsinvolatilitywastoimproveonthedynamicsofvolatility,t

7、heresultsindicatethatjumpsinvolatilityalsohaveanimportantcross-sectionalimpactonoptionprices”.Inthiscontext,weformulateastochasticvolatilityjump-diffusionmodelthat,inthespiritofHeston,admitsaclosed-formsolutionforEuropean-styleoptions.Theevolutionoftheunderlyi