51PRICING ASIAN OPTIONS FOR JUMP DIFFUSION

MathematicalFinance,Vol.21,No.1(January2011),117–143PRICINGASIANOPTIONSFORJUMPDIFFUSIONERHANBAYRAKTARUniversityofMichiganHAOXINGLondonSchoolofEconomicsWeconstructasequenceoffunctionsthatuniformlyconverge(oncompactsets)tothepriceofanAsianoption,whichiswrittenonastockwhosedynamicsfollowajumpdiffusion.Theconvergenceisexponentiallyfast.Weshowthateachelementinthissequenceistheuniqueclassicalsolutionofaparabolicpartialdifferentialequation(notanintegro-differentialequation).Asaresultweobtainafastnumericalapproximationschemewhoseaccuracyversusspeedcharacteristicscanbecontrolled.Weanalyzetheperformanceofournumericalalgorithmonseveralexamples.KEYWORDS:pricingAsianoptions,jumpdiffusions,aniterativenumericalscheme,classicalsolu-tionsofintegropartialdifferentialequations.1.INTRODUCTIONWedevelopanefficientnumericalalgorithmtopriceAsianoptions,whicharederivativeswhosepay-offdependsontheaverageofthestockprice,forjumpdiffusions.Thejumpdiffusionmodelsareheavilyusedintheoptionpricingcontextsincethesemodelscancapturetheexcesskurtosisofthestockpricereturnsalongwiththeskewintheimpliedvolatilitysurface(seeContandTankov2003).Twowell-knownexamplesofthesemodelsare(i)themodelofMerton(1976),inwhichthejumpsizesarelog-normallydistributed,and(ii)themodelofKou(2002),inwhichthelogarithmofjumpsizeshavetheso-calleddoubleexponentialdistribution.Hereweconsideralargeclassofjumpdiffusionmodelsincludingthesetwo.ThepricingofAsianoptionsiscomplicatedbecauseitinvolvessolvingapartialdifferentialequation(PDE)withtwospacedimensions,onevariableaccountingfortheaveragestockprice,theotherforthestockpriceitself.However,Veˇceˇr(2001)andVeˇceˇrandXu(2004)wereabletoreducethedimensionoftheproblembyusingachangeofmeasureargument(alsoseeSection2.1).WhenthestockpriceisageometricBrownianmotion,Veˇceˇr(2001)showedthatthepriceoftheAsianoptionattimet=0,whichwewilldenotebyS0→V(S0),satisfiesV(S0)=S0v(z=z∗,t=0)forasuitableconstantz∗,inwhichthefunctionvsolvesaone-dimensionalparabolicPDE.Whenthestockpriceisajumpdiffusion,thenundertheassumptionsthatvt,vz,andvzzarecontinuousVeˇceˇrWearegratefultotheanonymousassociateeditorandtworefereesfordetailedcommentsthathelpedusimproveourpaper.E.BayraktarissupportedinpartbytheNationalScienceFoundationunderanappliedmathematicsresearchgrantandaCareergrant,DMS-0906257andDMS-0955463,respectively,andinpartbytheSusanM.SmithProfessorship.ManuscriptreceivedJuly2007;finalrevisionreceivedMarch2009.AddresscorrespondencetoErhanBayraktar,DepartmentofMathematics,UniversityofMichigan,530ChurchStreet,AnnArbor,MI48109;e-mail:[e-mail]:10.1111/j.1467-9965.2010.00426.xC2010WileyPeriodicals,Inc.117118E.BAYRAKTARANDH.XINGandXu(2004)(seetheirtheorem3.3andcorollary3.4)showedthatthefunctionvsolvesanintegroPDEusingItˆo’slemma.However,aprioriitisnotclearthattheseassumptionsaresatisfied.Inthispaper,weshowthatforthejumpdiffusionmodelstheseassumptionsareindeedsatisfied(seeTheorem2.1),i.e.,wedirectlyshowthatvistheuniqueclassicalsolutionofthepartialintegro-differentialequationinVeˇceˇrandXu(2004)(thisintegro-PDEisgivenin(2.17)inourpaper).Wedothisbyfirstshowingthatvisthelimitofasequenceoffunctionsconstructedbyiteratingasuitablefunctionaloperator,whichwewilldenotebyJ.ThisfunctionaloperatorJtakesfunctionswithcertainregularitypropertiesintotheuniqueclassicalsolutionsofparabolicdifferentialequationsandgivesthemmoreregularity.Weshowthatvisthefixedpointofthefunctionaloperator.Finally,weshowthatvsatisfiesthecertainregularityproperties,whichensuresthatitistheclassicalsolutionofthepartialintegro-differentialequationinVeˇceˇrandXu(2004).ThisprooftechniqueissimilartothatofBayraktar(2009),inwhichtheregularityoftheAmericanputoptionpricesisanalyzed.Inthecurrentpaper,somemajortechnicaldifficultiesarisebecausethepay-offfunctionsweconsiderarenotboundedandalsobecausethesequenceoffunctionsconstructedisnotmonotonous.(Bayraktar2009wasabletoconstructamonotonoussequencebecauseoftheearlyexercisefeatureoftheAmericanoptions.)TheiterativeconstructionofthesequenceoffunctionswhichconvergetotheAsianoptionpricenaturallyleadstoanefficientnumericalmethodforcomputingthepriceofAsianoptions.Weprovethattheconstructedsequenceoffunctionsconvergestothefunctionvuniformly(oncompactsets)andexponentiallyfast.Therefore,afterafewiterationsonecanobtainanapproximationofvwithinthedesiredlevelofaccuracy,i.e.,theaccuracyversusspeedcharacteristicsofournumericalmethodcanbecontrolled.Ontheotherhand,sinceeachelementoftheapproximatingsequencesolvesaparabolicPDE(notanintegro-differentialequation),wecanuseoneoftheclassicalfinitedifferenceschemestodetermineit.WeproposeanumericalschemeinSection3andanalyzetheperformanceofitinthesamesection.NumericalmethodsforpricingAsianoptionsfordiffusionmodelswerestudiedex-tensivelyintheliterature:Veˇceˇr(2001),RogersandShi(1995),andZhang(2001,2003)proposedvariousPDEmethods,GemanandYor(1993)developedasingleLaplaceinversionmethod,Linetsky(2004)investigatedaspectralexpansionapproach,CaiandKou(2007)discoveredadoubleLaplaceinversionmethod.Meanwhile,RogersandShi(1995)andThompson(1998)obtainedtightb