Numerical Solution of Jump-Diffusion LIBOR Market

NumericalSolutionofJump-DiffusionLIBORMarketModelsPaulGlasserman∗andNicolasMerener†ColumbiaUniversityFebruary2001AbstractThispaperdevelops,analyzes,andtestscomputationalproceduresforthenumericalsolu-tionofLIBORmarketmodelswithjumps.Weconsider,inparticular,aclassofmodelsinwhichjumpsaredrivenbymarkedpointprocesseswithintensitiesthatdependontheLIBORratesthemselves.Whilethisformulationofferssomeattractivemodelingfeatures,itpresentsachallengeforcomputationalwork.Asafirststep,wethereforeshowhowtoreformulateatermstructuremodeldrivenbymarkedpointprocesseswithsuitablyboundedstate-dependentintensitiesintoonedrivenbyaPoissonrandommeasure.ThisfacilitatesthedevelopmentofdiscretizationschemesbecausethePoissonrandommeasurecanbesimulatedwithoutdis-cretizationerror.JumpsinLIBORratesarethenthinnedfromthePoissonrandommeasureusingstate-dependentthinningprobabilities.Becauseofdiscontinuitiesinherenttothethin-ningprocess,thisprocedurefallsoutsidethescopeofexistingconvergenceresults;weprovidesometheoreticalsupportforourmethodthrougharesultestablishingfirstandsecondorderconvergenceofschemesthataccommodatesthinningbutimposesstrongerconditionsonotherproblemdata.Thebiasandcomputationalefficiencyofvariousschemesarecomparedthroughnumericalexperiments.∗403UrisHall,GraduateSchoolofBusiness,ColumbiaUniversity,NewYork,NY10027,[e-mail].†DepartmentofAppliedPhysicsandAppliedMathematics,ColumbiaUniversity,NewYork,NY10027,[e-mail]1IntroductionTheinterestratemodelingapproachadvancedinMiltersen,Sandmann,andSondermann[24],Brace,Gatarek,andMusiela[4],MusielaandRutkowski[25],Jamshidian[12]andalargesubse-quentliteraturehasgainedwidespreadacceptanceinthederivativesindustryandhasstimulatedagrowingbodyofresearch.Thismarketmodelapproachemphasizestheuseofmarketobserv-ablesasmodelprimitivesandeaseofcalibrationtomarketdata.Thisentailsmodelingthetermstructureofinterestratethrough,e.g.,simplycompoundedforwardLIBORratesorforwardswapratesratherthanthecontinuouslycompoundedinstantaneousforwardratesattheheartoftheHeath,Jarrow,andMorton[11](HJM)frameworkorthroughtheinstantaneousshortrateofmoretraditionalmodels.Easeofcalibrationtomarketpricesofderivativesrequirestractableformulasforliquidinstrumentslikecapsorswaptions.Thecenterpieceofthemarketmodelframeworkisaclassofmodels([24,4,12])inwhichthepricesofcapsorswaptionscoincidewiththe“Black[3]formulas”traditionallyusedinindustrysomewhatheuristically—thatis,withoutasupportingarbitrage-freemodelofthetermstructure.Withinthisclassofmodels,calibrationtovolatilitiesimpliedbytheBlackformulasisessentiallyautomatic.However,preciselybecausethesemodelsreproduceBlack-formulapricesexactly,theycannotgenerateaskeworsmileinimpliedvolatilities;forexample,allcapletsofagivenmaturitymustshareacommonimpliedvolatilityregardlessofstrike.Thiscontradictsempiricalevidencethatimpliedvolatilitiesinmarketpricesdovarywithstrike,sometimesquitedramaticallyasintheYenmarket.AndersenandAndreasen[1]andZ¨uhlsdorff[31]havedevelopedextensionsofthebasicLIBORmarketmodelcombiningmoregeneralvolatilityspecificationswithcomputationaltractability;theseextensionsproducenon-constantimpliedvolatilities.GlassermanandKou[8]extendthemarketmodeltoincludejumpsininterestratesgovernedbymarkedpointprocessesandillustratethevarietyofimpliedvolatilitypatternssuchamodelcanproduce.ThispaperaddressescomputationalissuesintheextensionoftheLIBORmarketmodeltoincludejumpsandalsosomemodelformulationissuesarisingfromcomputationalconsiderations.Thepotentialimportanceofjumpsinfinancialmarketshasbeenwidelydocumented.Theirimpactisperhapsmostpronouncedinequitymarkets,buthasbeendocumentedinforeignexchangeandinterestratemarketsaswell.Jumpsplaytworelatedbutsomewhatdistinctrolesinmodeling:oneisprovidingabetterfittotimeseriesdataandtheotherisprovidinggreaterflexibilityinmatchingderivativeprices—i.e.,inmodelingdynamicsunderanequivalentmartingalemeasure.Equivalenceofphysicalandmartingalemeasuresrequiresthatbothadmitjumpsifeitherdoes,buttheirfrequencyandmagnitudescanbequitedifferentunderthetwomeasures.Numerousreferences1totheliteratureonmodelingwithjumpsarediscussedin[8]soherewementionjustafew.Inempiricalwork,Das[6]andJohannes[14]arguethatthekurtosisinshort-terminterestratesisincompatiblewithapure-diffusionmodel.ModelsaddingjumpstotheHJMframework(andthusfocusedonderivatives)includeBjork,Kabanov,andRunggaldier[2](onwhichGlassermanandKou[8]build)andShirakawa[29].Jamshidian[13]hasdevelopedaverygeneralextensionofthemarketmodelframeworkinwhichinterestratedynamicsaredrivenbydiscontinuoussemimartingales.Thispaperaddressesthenumericalsolution,throughdiscretizationandsimulation,ofthemarketmodelswithjumpsdevelopedin[8].GlassermanandKoushowhowmarkedpointprocessintensitiescanbechosentoproduceclosed-formexpressionsforcapletsorswaptions,butforpricinggeneralpath-dependentinterestratederivativessimulationisnecessary.Thenumericalsolutionofcontinuousprocessesmodeledthroughstochasticdifferentialequationshasbeenstudiedindepth(see,inparticular,KloedenandPlaten[15]anditsmanyreferences),buttherehasbeenfarlessworkonsim