A two-factor lognormal model of the term structure

ATwo-factorLognormalModeloftheTermStructureandtheValuationofAmerican-StyleOptionsonBondsSandraPeterson1RichardC.Stapleton2MartiG.Subrahmanyam3February25,19991DepartmentofAccountingandFinance,TheManagementSchool,LancasterUniversity,Lan-casterLA14YX,UK.Tel:(44)524{593637,Fax:(44)524{847321.2DepartmentofAccountingandFinance,StrathclydeUniversity,Glasgow,UK.Tel:(44)524{381172,Fax:(44)524{846874,e{mail:[e-mail],NewYorkUniversity,ManagementEducationCenter,44West4thStreet,Suite9{190,NewYork,NY10012{1126,USA.Tel:(212)998{0348,Fax:(212)995{4233,e{mail:[e-mail]-factorLognormalModeloftheTermStructureandtheValuationofAmerican-StyleOptionsonBondsWebuildano-arbitragemodelofthetermstructure,usingtwostochasticfactors,theshort-terminterestrateandthepremiumoftheforwardrateovertheshort-terminterestrate.ThemodelextendsthelognormalinterestratemodelofBlackandKarasinski(1991)totwofactors.Itallowsformeanreversionintheshortrateandintheforwardpremium.Themethodiscomputationallyecientforseveralreasons.First,weuseLiborfuturesprices,enablingustosatisfytheno-arbitrageconditionwithoutresortingtoiterativemeth-ods.Second,themultivariate-binomialmethodologyofHo,StapletonandSubrahmanyam(1995)isextendedsothatamultiperiodtreeofrateswiththeno-arbitragepropertycanbeconstructedusinganalyticalmethods.Themethodusesarecombiningtwo-dimensionalbinomiallatticeofinterestratesthatminimizesthenumberofstatesandtermstructuresovertime.Third,theproblemofcomputingalargenumberoftermstructuresissimpliedbyusingalimitednumberof’bucketrates’ineachtermstructurescenario.Inadditiontothesecomputationaladvantages,akeyfeatureofthemodelisthatitisconsistentwiththeobservedtermstructureofvolatilitiesimpliedbythepricesofinterestratecapsandoors.WeillustratetheuseofthemodelbypricingAmerican-styleandBermudan-styleoptionsonbonds.Optionpricesforrealisticexamplesusingfortytimeperiodsareshowntobecomputableinseconds.Atwo-factorlognormalmodelofthetermstructure11IntroductionOneofthemostimportantanddicultproblemsfacingpractitioners,intheeldofin-terestratederivativesinrecentyears,hasbeentobuildinter-temporalmodelsofthetermstructureofinterestratesthatarebothanalyticallysoundandcomputationallyecient.Thesemodelsarerequiredbothtohelpinthepricingandintheoverallriskmanagementofabookofinterestratederivatives.Althoughmanyalternativemodelshavebeensuggestedintheliteratureandimplementedinpractice,thereareseriousdisadvantageswithmostofthem.Forexample,Gaussianmodelsofinterestrates,whichhavetheadvantageofanalyt-icaltractability,havethedrawbackofpermittingnegativeinterestrates,aswellasfailingtotakeintoaccountthepossibilityofskewnessinthedistributionofinterestrates.Also,manyoftheterm-structuremodelsusedinpracticearerestrictedtoonestochasticfactor.SincetheworkofHoandLee(1986),ithasbeenwidelyrecognizedthatterm-structuremodelsmustpossesstheno-arbitrageproperty.Inthiscontext,ano-arbitragemodelisoneinwhichtheforwardpriceofabondistheexpectedvalueoftheone-period-aheadspotbondprice,undertherisk-neutralmeasure.Buildingmodelsthatpossessthispropertyhasbeenamajorpre-occupationofbothacademicsandpractitionersinrecentyears.Onemodelthatachievesthisobjectiveinaone-factorcontextisthemodelproposedbyBlack,DermanandToy(1990)(BDT),andextendedbyBlackandKarasinski(1991)(BK).Inessence,themodelwhichwebuildinthispaperisatwo-factorextensionofthistypeofmodel.Inourmodel,interestratesarelognormalandaregeneratedbytwostochasticfactors.ThegeneralapproachwetakeissimilartothatofHullandWhite(1994)(HW),wheretheconditionalmeanoftheshortratedependsontheshortrateandanadditionalstochasticfactor,whichcanbeinterpretedastheforwardpremium.IncontrasttoHW,andinlinewithBK,webuildamodelwheretheconditionalvarianceoftheshortrateisafunctionoftime.Itfollowsthatthemodelcanbecalibratedtotheobservedtermstructureofinterestratevolatilitiesimpliedbyinterestratecaps/oors.Essentially,theaimhereistobuildatermstructuremodelwhichcanbeappliedtovalueAmerican-stylecontingentclaimsoninterestrates,whichisconsistentwiththeobservedmarketpricesofEuropean-stylecontingentclaims.Ourapproachtobuildingano-arbitragetermstructureforpricinginterestratederivativesisconsistentwiththegeneralframeworkproposedbyHeath,JarrowandMorton(1992)(HJM).IntheHJMformulation,assumptionsaremadeaboutthevolatilityoftheforwardinterestrates.Sincetheforwardratesarerelatedinano-arbitragemodeltothefuturespotinterestrates,thereisacloserelationshipbetweenthisapproachandtheonewearetaking.Infact,theHJMforward-ratevolatilitiescanbethoughtofastheoutputsofourmodel.IftheparametersoftheHJMmodelareknown,thisrepresentsasatisfactoryalternativeAtwo-factorlognormalmodelofthetermstructure2approach.However,theBDT-BK-HWapproachhastheadvantageofrequiringasinputsthevolatilitiesoftheshortrateandoflongerbondyieldswhicharemoredirectlyobservablefrommarketdataonthepricingofcaps,oorsandswaptions.Ourmodel,whichisanextensionofBK,hassimilaradvantages.Wewouldlikeanymodelofthestochastictermstructuretohaveanumberofdesirableproperties.Apartfromsatisfyingtheno-a