Perturbation Expansion for Option Pricing with Sto

arXiv:0708.3012v1[physics.soc-ph]22Aug2007PerturbationExpansionforOptionPricingwithStochasticVolatilityPetrJizba∗andHagenKleinert†ITP,FreieUniversit¨atBerlin,Arnimallee14D-14195Berlin,GermanyPatrickHaener‡NomuraInternational,NomuraHouse,1StMartin’s-le-Grand,London,EC1ANP,UK(Dated:February1,2008)AbstractWefitthevolatilityfluctuationsoftheS&P500indexwellbyaChidistribution,andthedistributionoflog-returnsbyacorrespondingsuperpositionofGaussiandistributions.TheFouriertransformofthisis,remarkably,oftheTsallistype.AnoptionpricingformulaisderivedfromthesamesuperpositionofBlack-Scholesexpressions.AnexplicitanalyticformulaisdeducedfromaperturbationexpansionaroundaBlack-Scholesformulawiththemeanvolatility.Theexpansionhastwoparts.Thefirsttakesintoaccountthenon-Gaussiancharacterofthestock-fluctuationsandisorganizedbypowersoftheexcesskurtosis,thesecondiscontractbased,andisorganizedbythemomentsofmoneynessoftheoption.WiththisexpansionweshowthatfortheDowJonesEuroStoxx50optiondata,aΔ-hedgingstrategyisclosetobeingoptimal.PACSnumbers:65.40.Gr,47.53.+n,05.90.+mKeywords:Black-Scholesformula;Volatility;Gammadistribution;Mellintransform∗Electronicaddress:[e-mail];OnleavefromFNSPE,CzechTechnicalUniversity,Bˇrehov´a7,11519Praha1,CzechRepublic†Electronicaddress:[e-mail]‡Electronicaddress:[e-mail]thispaperistodevelopanalyticexpressionsforoption-pricingofmarketswithfluctuatingvolatilitiesofagivendistribution.Thereareseveralgoodreasonsforcon-sideringthestatisticalpropertiesofvolatilities.Forinstance,anumberofimportantmodelsofpricechangesincludeexplicitlytheirtime-dependence,forexampletheHestonmodel[1],orthefamousARCH[2],GARCH[3],andmultiscaleGARCH[4]models.Changesinthedailyvolatilityarequalitativelywellexplainedbymodelsrelatingvolatilitytotheamountofinformationarrivinginthemarketatagiventime[5].Thereisalsoconsiderablepracticalinterestinvolatilitydistributionssincetheyprovidetraderswithanessentialquantitativeinformationontheriskinessofanasset[6,7].Assuchitprovidesakeyinputinportfolioconstruction.Unlikereturnswhicharecorrelatedonlyonveryshorttimescales[8]ofafewminutesandcanroughlybeapproximatedbyMarkovianprocess,thevolatilitychangesexhibitmemorywithtimecorrelationsuptomanyyears[9,10,11,12].InRef.[11]itwasshownthattheStandard&Poor500(hereafterS&P500)volatilitydatacanbefittedquitewellbyalog-normaldistribution.Thisresultappearstobeatoddswiththefactthatthelog-normalshapeofthedistributiontypicallysignalizesamultiplicativenature[13]ofanunderlyingstochasticprocess.Thisisrathersurprisinginviewofefficientmarkettheories[8]whichassumethatthepricechangesarecausedbyincomingnewinformationaboutanasset.Suchinformation-inducedpricechangesareadditiveandshouldnotgiverisetomultiplicativeprocess.WecurethiscontradictionbyobservingthatthesamevolatilitydatacanbefittedequallywellbyaChidistribution[14,15].InAppendixAweshowthatcorrespondingsamplepathsfollowtheadditiveratherthanmultiplicativeIt¯ostochasticprocess.WiththehelpofIt¯o’slemmaonemayshow(cf.AppendixA)thatthevariancev(t)=σ2(t)followstheIt¯o’sstochasticequationdv(t)=γ(t)[ν(t)−μ(t)v(t)−a(v(t),μ(t),ν(t))]dt+p2γ(t)v(t)dW(t).(1)whereW(t)isaWienerprocess.Hereγ(t),μ(t)andν(t)arearbitrarynon-singularpositiverealfunctionsonR+.Functiona(...)isnon-singularinallitsargumentsandittendstozeroatlarget’s.FromthecorrespondingFokker-Planckequationonemaydeterminethetime-dependentdistributionofvwhichreadsfμ(t),ν(t)(v)=1Γ(ν(t))[μ(t)]ν(t)vν(t)−1e−μ(t)v,withZ∞0dvfμ,ν(v)=1.(2)Thedistributionfμ,ν(v)isthenormalizedGammaprobabilitydensity[14],whoseprofileisshowninFig.6.Ithasanaverage¯v=ν/μ,avariance(v−¯v)2=ν/μ2,askewness(v−¯v)3=2/√ν,andanexcesskurtosis(v−¯v)4/(v−¯v)22−3=6/ν(see,e.g.,Refs.[14,16]).TheGammadistribution(2)willplayakeyroleinthefollowingreasonings.TheChidistributionρ(σ,t)isrelatedwiththeGammadistributionthroughtherelationρ(σ,t)=2σfμ(t),ν(t)(σ2).(3)2Oftenisthefunctionalform(3)itselfcalledaGammadistribution.Toavoidpotentialambiguities,weshallconfineinthefollowingtothenameChidistribution.Thederivationofρ(σ,t)fromtheunderlyingadditiveprocess,ratherthanamultiplicativeprocess,showsthattheChidistributioniswellcompatiblewithefficientmarkets.TheGammadistributionsforfluctuationsofvallowustogenerateanentirelynewclassofoptionpricingformulas.Forthisweusethewell-knownfactofnon-equilibriumstatisticalphysics[17],thatthedensitymatrixofasystemwithfluctuatingtemperaturecanbewrittenasadensitymatrixforthesystemwithfixedtemperatureaveragedwithrespecttoatemperaturedistributionfunction.Pathintegralsconvenientlyfacilitatethistask[16].Withthehelpoftheso-calledSchwingertrickweshowthatifthedistributionoftheinverse-temperaturesofthelog-returnsisoftheGammatype,thedistributioninmomentumspaceisoftheTsallistype.Toputthisobservationintoarelevantcontext,werecallthatTsallisdistributionsinmomentumspaceenjoyakeyroleinstatisticalphysicsasbeingoptimalinaninformationtheoreticalsense:givenpriorinformationonlyonthecovariancematrixandaso-calledescortparameter,theycontaintheleastpossibleassumptions,i