The Bismut-Elworthy-Li formula for jump-diffusions

arXiv:math/0604311v3[math.PR]8May2007TheBismut-Elworthy-Liformulaforjump-diffusionsandapplicationstoMonteCarlomethodsinfinanceThomasR.CassandPeterK.FrizStatisticalLaboratory,UniversityofCambridge,WilberforceRoad,Cambridge,CB30WB,UK.March22,2007AbstractWeextendtheBismut-Elworthy-Liformulatonon-degeneratejumpdiffusionsand”payoff”functionsdependingontheprocessatmul-tiplefuturetimes.InthespiritofFourni´eetal[14]andDavisandJohansson[10]thiscanimproveMonteCarlonumericsforstochasticvolatilitymodelswithjumps.Tothisendoneneedsso-calledMalli-avinweightsandwegiveexplicitformulaevalidinpresenceofjumps:(a)Inanon-degeneratesituation,theextendedBELformularepresentspossibleMalliavinweightsasItointegralswithexplicitintegrands;(b)inahypoellipticsettingwereviewworkofArnaudonandThalmaier[1]andalsofindexplicitweights,nowinvolvingtheMalliavincovari-ancematrix,butstillstraight-forwardtoimplement.(Thisisincon-trasttorecentworkbyForster,L¨utkebohmertandTeichmannwhereweightsareconstructedasanticipatingSkorohodintegrals.)Wegivesomefinancialexamplescoveredby(b)butnotethatmostpracticalcasesofpoorMonteCarloperformance,DigitalCliquetcontractsforinstance,canbedealtwithbytheextendedBELformulaandhencewithoutanyrelianceonMalliavincalculusatall.Wethendiscusssomeoftheapproximations,oftenignoredintheliterature,neededtojustifytheuseoftheMalliavinweightsinthecontextofstandardjumpdiffusionmodels.Finally,asallthisismeanttoimprovenumer-ics,wegivesomenumericalresultswithfocusonCliquetsundertheHestonmodelwithjumps.11IntroductionModernarbitragetheoryreducesthepricingof(non-American)optionstothecomputationofanexpectationunderariskneutralmeasure.Itiscommonpracticetoassumethattheriskneutralmeasureisinducedbyaparametricfamilyofjumpdiffusionswhichcanthenbecalibratedtoliq-uidoptionprices.Wecanthereforeassumethatallexpectationsarewithrespecttoafixedpricingmeasure.Atypicaloptiononsomeunderlying(St)thenhas(undiscounted)priceE[f(ST1,ST2,...,STn)]≡E[f(S)].Forhedgingandrisk-managementpurposesitiscrucialtounderstandthedependenceonS0andothermodelparameters.ComputingΔ=∂∂S0E[f(S)]=E∇f(S)∂S∂S0viafinitedifferencescanpresentcomputationalchallengesinMonteCarlo;justthinkofanat-the-moneydigitaloptionnearexpiration.BroadieandGlasserman[7]showedthatthisproblemisovercomeby∂∂S0E[f(S)]=E[f(S)π](1)whereπisthelogarithmicderivativeofthejointdensityoftherandomvectorS.Ontheotherhand,therandomweightπaddsnoiseitselfanditisimportanttolocalise:forinstancebyusing(1)foranirregular,butcom-pactlysupportandbounded,˜fandtheusualfinitedifferencetechniqueforf−˜f,assumedtobenice(C1willusuallysuffice).Intwoseminalpapers,Fourni´eetal[14]and[15]useMalliavincalcu-lustocomputeπwhennoexplicittransitiondensityisknown.Theyworkwithnon-degenerate(or:elliptic)continuousdiffusionsbutalsocoversomehypoellipticsituations.Asiswellknown,ellipticresultscanbeobtainedbytheBismut-Elworthy-Liformula(ElworthyandLi[11],Bismut[6])andthereare,infact,otherwaystoobtainsuchresultswithoutMalliavincal-culus:wementioninparticulartheideaofThalmaier[27]ofdifferentia-tionattheleveloflocalmartingaleswhichwasemployedbyGobetandMunos[18]inthepresentcontext.Thepointwasthatinmanycasesofpracticalinterest,atleastinabsenceofjumps,onedoesnotneedMalliavincalculus.(SpecialistswillnotethatMalliavintechniquesaremoreflexibleinthesensethatdifferentperturbationsofBrownianmotionyielddifferentweightsandthereisanaprioriinteresttopickweightswithsmallvariance.Inreality,itishardtojustifymucheffortinthisdirectionasthepotentialgainsarenegligibletotheimprovementsobtainedbylocalisation.)Overthelastdecadeithasbecomeclearthatpurediffusionmodelsareunabletofittheshort-datedsmileandjumpshavebeenincludedtomod-elstorectifythissituation;ContandTankov[9]andGatheral[16]provide2twoexcellentaccounts.ThequestionhasarisenastohowtheaboveideascanbeadaptedtomodelsbasedonjumpdiffusionprocessesandweshallproposeaquitesimplesolutiontothisalongtheideasofElworthy-Liby-passingbothclassicalMalliavintechniquesanditsextensionstoL´evypro-cessesthathavebeenusedinthisfinancialMonteCarlocontext.WenotethatasimilarextensionoftheBELformula,slightlylessgeneralthanours,wasusedrecentlybyPriolaandZabczyk[22]toestablishLiovilletheoremsfornon-localoperators.Letusbrieflymentionthatinsomecasesarandomweightπcanbeconstructedbyconditioningarguments.ConsiderforinstancethetrivialexampleXt=z+Bt+Nt,whereBisastandardBrownianmotionandNaPoissonprocess.ConditionalonNt,anyfunctionofXtisa(different)functionofz+Bt,apurediffusionwithnojumps,andsincetheassociatedrandomweightπisuniversal(i.e.donotdependontheparticularpay-offfunction)thisalsosolvestheproblemforthejumpdiffusionX.Thiskindofreasoningleadsimmediatelytotheclassof”separable”jumpdif-fusions,consideredinDavisandJohansson[10]viaMalliavincalculusforsimpleL´evyprocesses.Weshallomitadetaileddiscussionsincearefined,iteratedconditioningargumentcanbeusedassumingonlyfiniteactivityofthejumps(andwithoutassumingseparabilityinthesenseof[10]).Tothisend,wequicklyrecalltheBELforcontinuousdiffusions(seeSection3fornotationandassumptions)∂∂zjE[f(xzT)]=Ef(xzT)ZT0a(t)R(t,xzt)∂xzt∂zjTdW