Abstract Pricing Interest Rate Derivatives A Gener

PricingInterestRateDerivatives:AGeneralApproach1GeorgeChackoHarvardUniversitySanjivDasHarvardUniversityAugust19991Incomplete,commentswelcome.Thisisasubstantiallyrevisedversionofthepaper\PricingAverageInterestRateOptions:AGeneralApproach(1998)basedontheoriginalpaper\AverageInterest(NBERWorkingPaperNo.6045[1997]).Thecommentsoftheeditorandananonymousrefereearegratefullyacknowledged,andhavehelpedtremendouslyinimprovingthecontentandexpositionofthepaper.ThanksalsotoMarcoAvellaneda,EricReiner,VladimirFinklestein,AlexLevinandseminarparticipantsattheCourantInstituteofMathematicalSciences,NewYorkUniversity,theComputationalFinanceGroupatPurdueUniversity,andtheRisk99conferencefortheircomments.PleaseaddressallcorrespondencetotheauthorsatHarvardUniversity,GraduateSchoolofBusinessAdministration,MorganHall,SoldiersField,Boston,MA02163.AbstractPricingInterestRateDerivatives:AGeneralApproachTherelationshipbetweenanestochasticprocessesandbondpricingequationsinex-ponentialtermstructuremodelshasbeenwell-established(seeDueandKan[42]).Weextendthislinkagetothepricingofinterestratederivatives.Thispapershowsthat,ifthetermstructuremodelisexponential-ane,thenthereisasimplelinkagebetweenthebondpricingsolutionandthepricesofmanywidelytradedinterestratederivativesecurities.Ourresultsaregeneral,andapplytom-factorprocesseswithndiusionsandljumpprocesses.Regardlessofthenumberofshocks,thepricingsolutionsrequireatmostasinglenumericalintegral,makingthemodeleasytoimplement.Weprovidemanyexamplesofoptionsthatyieldsolutionsusingthemethodsofthepaper.Fastestimationofthesemodelsispossiblebyvectorizingtheequationsforthepricingsolutions.Arangeofnumericalsolutionsillustratestheuseofthemodels.InterestRateDerivatives:AGeneralApproach.........................................11IntroductionTheliteratureontermstructuremodellinghasevolvedfromone-factordiusionmodelssuchasCox,Ingersoll,&Ross[32]andVasicek[83]tomultifactormodelssuchasBrennan&Schwartz[17],Longsta&Schwartz[67],Balduzzi,Das,&Foresi[7],andDue&Singleton[43]aswellasjump-diusionmodelssuchasAhn&Thompson[1],Das&Foresi[36],andDas[35].ThemotivationforthisevolutionintermstructuremodelshascomefromempiricalpaperssuchasAit-Sahalia[2],Chan,Karolyi,Longsta,&Sanders[26].1.However,asworkproceedsonbettermatchingthedynamicsoftheshortratetotheobservedtermstructure,theareaofxedincomederivativepricing,themainapplicationformodellingtheshortrateprocess,haslaggedbehind.Inthispaper,weattempttobridgethegapbetweenthemulti-factor,jump-diusionmodelsoftheshortratethataresocommonlyusednowandthepricingofxedincomederivatives.Specically,weshowthatanyinterestrateprocess(withanynumberoffactorsincludingstochasticvolatility,stochasticcentraltendency,etc.,orutilizingdiusionorjump-diusionprocesses)thatleadstoanexponentialtermstructuremodel,alsolendsitselftoanalyticsolutionsforthreelargeclassesofxedincomesecurities.Thesemethodssupportnumericaltechniqueswhichallowforeasyimplementationinthecontextofano-arbitrageapproach.Itisourhopethattheresultsofthispaperwillallowbothresearchersandpractitionerstofocusontheappropriatestochasticprocessfortheshortrateanditsfactors,andobviateconcernsastowhetherspecicformsoftheshortrateleadtotractablesolutionsforpopularxedincomesecurities.ThebenchmarkpaperofDueandKan[42]establishedthelinkbetweenanestochas-ticprocessesandexponential-anetermstructuremodels.2TheyshowedthatthefactorcoecientsofthesetermstructuremodelsaresolutionstoasystemofsimultaneousRiccatiequationsandthatthesecoecientsarefunctionsofthetimetomaturity.Thekernelofourtechniqueresidesinthefactthatthesolutionfordierenttypesofinterestrateoptionssolvesanalmostidenticalsystemsofequations.Theonlydierencebetweenthetwosetsofequationsisintheconstanttermsunderlyingtheequations.BymanipulatingtheRiccatiequationsandvaryingtheconstantterms,wedevelopaproceduretopriceoptionsusingtheknowncomponentsoftheoriginaltermstructuremodel.Thus,weessentiallyshowthatoncetheexponential-anetermstructuremodelisderived,thepricingformulasforawiderangeofpopularxed-incomederivativescanbewrittenbyinspectionfromthecomponentsofthetermstructuremodel.3Specically,weshowthatthisapproachisfeasibleforthreelargeclassesofxedincomederivatives:thosewith(1)payosthatarelinearintheshortrateandfactors;(2)payosthatareexponential-aneintheshortrateandfactors;and(3)payosthatareanintegralovertimeofalinearcombinationoftheshortrateandfactors.Thesethreepayostructuresencompassmostofthepopularxed-incomederivatives.1ManyotherpapersincludingthosebyBrown&Dybvig(1989),Litterman&Scheinkman(1991),andStambaugh(1988)havetosimilarconclusions.2Dai&Singleton(1997)providesacharacterizationoftheexponential-aneclassoftermstructuremodelsastheyunifyandgeneralizethisclass.3Subsequenttotheoriginalversionofthispaper,BakshiandMadan[9],andDue,Pan,andSingleton[44]haveindenpendentlydevelopedresultsthatparallelsomeofthosederivedinthispaper.InterestRateDerivatives:AGeneralApproach.........................................2Ourtechniqueisgeneralinthatitappliestoanymulti-factor,exponential-anetermstructure