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1、3.One-factorshort-ratemodels“Itwillbeshort,theinterimismine.Andaman’slifeisnomorethantosay‘one’”Hamlet,V.2“Pilderon!”[‘Piloter’on!]KojiKabuto,assumingcontrolofMazingerZ(1972)3.1IntroductionandGuidedTourThetheoryofinterest-ratemodelingwasoriginallybasedontheassumptionofspec
2、ificone-dimensionaldynamicsfortheinstantaneousspotrateprocessr.Modelingdirectlysuchdynamicsisveryconvenientsinceallfundamentalquantities(ratesandbonds)arereadilydefined,byno-arbitragearguments,astheexpectationofafunctionaloftheprocessr.Indeed,theexistenceofarisk-neutralmeasu
3、reimpliesthatthearbitrage-freepriceattimetofacontingentclaimwithpayoffHTattimeTisgivenby!T−r(s)dsHt=Et{D(t,T)HT}=EtetHT,(3.1)withEtdenotingthetimet-conditionalexpectationunderthatmeasure.Inparticular,thezero-coupon-bondpriceattimetforthematurityTischaracterizedbyaunitamount
4、ofcurrencyavailableattimeT,sothatHT=1andweobtain!T−r(s)dsP(t,T)=Etet.(3.2)FromthislastexpressionitisclearthatwheneverwecancharacterizetheT−r(s)dsdistributionofetintermsofachosendynamicsforr,conditionalontheinformationavailableattimet,weareabletocomputebondpricesP.Aswehaves
5、eenearlierinChapter1,frombondpricesallkindofratesareavailable,sothatindeedthewholezero-couponcurveischaracterizedintermsofdistributionalpropertiesofr.ThepioneeringapproachproposedbyVasicek(1977)wasbasedondefin-ingtheinstantaneous-spot-ratedynamicsunderthereal-worldmeasure.H
6、isderivationofanarbitrage-freepriceforanyinterest-ratederivativefollowedfromusingthebasicBlackandScholes(1973)arguments,whiletakingintoaccountthenon-tradablefeatureofinterestrates.523.One-factorshort-ratemodelsTheconstructionofasuitablelocally-risklessportfolio,asinBlackan
7、dScholes(1973),leadstotheexistenceofastochasticprocessthatonlydependsonthecurrenttimeandinstantaneousspotrateandnotonthematuritiesoftheclaimsconstitutingtheportfolio.Suchprocess,whichiscommonlyreferredtoasmarketpriceofrisk,definesaGirsanovchangeofmeasurefromthereal-worldmea
8、suretotherisk-neutralonealsoincaseofmoregeneraldynamicsthanVasicek’s.Precisely,letusassum
