Pricing Asian Options for Jump Diffusions

arXiv:0707.2432v4[cs.CE]12Jan2008PricingAsianOptionsforJumpDiffusions∗†ErhanBayraktar‡HaoXing§AbstractWeconstructasequenceoffunctionsthatuniformlyconverge(oncompactsets)tothepriceofAsianoption,whichiswrittenonastockwhosedynamicsfollowsajumpdiffusion,exponentiallyfast.Eachoftheelementinthissequencesolvesaparabolicpartialdifferen-tialequation(notanintegro-differentialequation).Asaresultweobtainafastnumericalapproximationschemewhoseaccuracyversusspeedcharacteristicscanbecontrolled.Weanalyzetheperformanceofournumericalalgorithmonseveralexamples.KeyWords.PricingAsianOptions,Jumpdiffusions,anIterativeNumericalScheme.1IntroductionWedevelopanefficientnumericalalgorithmtopriceAsianoptions,whicharederivativeswhosepay-offdependsontheaverageofthestockprice,forjumpdiffusions.Thejumpdiffusionmodelsareheavilyusedintheoptionpricingcontextsincetheycancapturetheexcesskurtosisofthestockpricereturnsandalongwiththetheskewintheimpliedvolatilitysurface(seeContandTankov(2003)).Twowell-knownexamplesofthesemodelsarei)themodelofMerton(1976),inwhichthejumpsizesarelog-normallydistributed,andii)themodelofKou(2002),inwhichthelogarithmofjumpsizeshavethesocalleddoubleexponentialdistribution.Hereweconsideralargeclassofjumpdiffusionmodelsincludingthesetwo.ThepricingofAsianoptionsiscomplicatedbecauseitinvolvessolvingapartialdifferentialequation(PDE)withtwospacedimensions,onevariableaccountingfortheaveragestockprice,theotherforthestockpriceitself.However,Veˇceˇr(2001)andVeˇceˇrandXu(2004)wereabletoreducethedimensionoftheproblembyusingachangeofmeasureargument(alsoseeSection2.1).WhenthestockpriceisageometricBrownianmotionVeˇceˇr(2001)showedthatthepriceoftheAsianoptionattimet=0,whichwewilldenotebyS0→V(S0),satisfiesV(S0)=S0·v(z=z∗,t=0)forasuitableconstantz∗,inwhichthefunctionv(·,·)solvesaonedimensionalparabolicPDE.Whenthestockpriceisajumpdiffusion,thenundertheassumptionsthatvt,vzandvzzarecontinuousVeˇceˇrandXu(2004)(Theorem3.3andCorollary3.4)showedthatthefunctionv(·,·)solvesanintegro-partialdifferentialequation.1∗ThisresearchispartiallysupportedbyNSFResearchGrant,DMS-0604491.†DepartmentofMathematics,UniversityofMichigan,530ChurchStreet,AnnArbor,MI48109.‡e-mail:[e-mail]§e-mail:[e-mail]1Theintegro-differen!tialequationinthestatementsofTheorem3.3andCorollary3.4inVeˇceˇrandXu(2004)haveatypoeach:TheintegrandintheintegralwithrespecttothecompensatorνinTheorem3.3,andtheintegrandintheintegralwithrespecttotheLevymeasureKinCorollary3.4shouldbemultipliedwith1+x.1However,ingeneralitishardtoverifythattheseassumptionsaresatisfied.Inthispaper,weshowthatforthejumpdiffusionmodelstheseassumptionsareindeedsatisfied(seeTheorem2.3).Wedothisbyconstructingasequenceoffunctionsthatconvergetosomefunctionvuniformly(oncompactsets)andexponentiallyfast.Thissequenceisconstructedbyusingasuitablefunctionaloperatorthattakesfunctionswithcertainregularitypropertiesintotheuniquesolutionsofparabolicdifferentialequationsandgivesthemmoreregularity.Weshowthatvisthefixedpointofthefunctionaloperatorandthatitsatisfiesthecertainregularityproperties.ThisprooftechniqueissimilartothatofBayraktar(2007),inwhichtheregularityoftheAmericanputoptionpricesareanalyzed.Inthecurrentpaper,somemajortechnicaldifficultiesarisebecausethepay-offfunctionsweconsiderarenotboundedandalsobecausethesequenceoffunctionsconstructedisnotmonotonous(Bayraktar(2007)wasabletoconstructamonotonoussequencebecauseoftheearlyexercisefeatureoftheAmericanoptions).TheiterativeconstructionofthesequenceoffunctionswhichconvergetotheAsianoptionpriceyieldsanefficientnumericalmethodforcomputingthepriceofAsianoptions.Weprovethattheconstructedsequenceoffunctionsconvergestothefunctionv(·,·)uniformly(oncom-pactsets)andexponentiallyfast.Therefore,afterafewiterationsonecanobtainthefunctionv(·,·)tothedesiredlevelofaccuracy,i.e.theaccuracyversusspeedcharacteristicsofthenu-mericalmethodweproposecanbecontrolled.Ontheotherhand,sinceeachelementoftheapproximatingsequencesolvesaparabolicPDE(notanintegro-differentialequation),wecanuseoneoftheclassicalfinitedifferenceschemestodetermineit.WeanalyzetheperformanceofournumericalmethodinSection3.ForasurveyofothernumericalmethodsforpricingAsianoptionsforjumpdiffusionspleaserefertoVeˇceˇrandXu(2004).Therestofthepaperisorganizedasfollows:InSection2.1,wesummarizethefindingsofVeˇceˇrandXu(2004)inthecontextofjump-diffusionmodels.InSection2.2,weintroduceafunctionaloperatorandanalyzeitsproperties.ThesepropertiesareusedinSection2.3toconstructasequenceoffunctionsthatconvergetothepriceoftheAsianoption.ThemainresultofthissectionisTheorem2.2.Finally,inSection3,weanalyzetheperformanceofthenumericalmethodwepropose.2ApproximatingthePriceofAsianOptions2.1DimensionReductionLet(Ω,F,P)beacompleteprobabilityspacehostingaWienerprocess{Bt;t≥0}andaPoissonrandommeasureN,whosemeanmeasureisλν(dy)dt,independentoftheWienerprocess.Let(Ft)t≥0denotethenaturalfiltrationofBandN.Inthisfilteredprobabilityspace,letusdefineaMarkovprocessS={St;t≥0}viaitsdynamicsasdSt=(r−μ)Stdt+σStdBt+St−ZR+(y−1)N(dt,dy),(1)inwhichristheriskfreerate,μ,λ(ξ−1)withassumptionξ,RR+yν(dy)∞