A BOUND ON LIBOR FUTURES PRICES FOR HJM YIELD CURV

ABOUNDONLIBORFUTURESPRICESFORHJMYIELDCURVEMODELSVLADIMIRPOZDNYAKOVANDJ.MICHAELSTEELEAbstract.Weprovethatforalargeclassofwidelyusedtermstructuremod-elsthereisasimpletheoreticalupperboundforvalueofLIBORfuturesprices.Whenthisboundiscomparedtoobservedfuturesprices,oneneverthelessßndsthatthetheoreticalboundissometimesviolatedbymarketprices.Themainconsequenceofthisobservationisthatvirtuallyalloftheimportantßxedin-comemodelshavetheoreticalimplicationsthataresometimesatoddswithmarketrealities,atleastwhentheyareappliedtofuturesmarkets.1.IntroductionThemainpurposeofthisarticleistoprovethatforalargeclassofstochasticmodelsforthetermstructureofinterestrates,onehasthefuturespriceinequality:(1)Fµ(t;T)´100²1µ+1³ ²100µ³1+(T t+µ)LT t+µ(t)1+(T t)LT t(t);whereFµ(t;T)denotestheLIBORfuturespriceatdatetforsettlementatdateTandwhereLµ(t)denotestheLondoninterbankoÞeredrate(LIBOR)attimetforatermdepositwithatermofµyears.Anintriguingfeatureofthisinequalityisthatitholdsforessentiallyallstochasticyieldcurvemodelsthatarecurrentlyused,and,evenso,oneßndsthattheinequalityissometimesviolatedbymarketprices.Thisarticlefocusesalmostentirelyonthemathematicalconsequencesoftheyieldcurvemodels,andtheresolutionoftheresultingempiricalparadoxisleftasanopenproblem.2.BackgroundontheHJMModelSinceitsintroductionalmostadecadeago,thetermstructuremodelofHeath,Jarrow,andMorton(1992)hasbecomeanalmostuniversalstandardforthetheo-reticalanalysisofßxedincomesecuritiesandtheirassociatedderivatives(seee.g.,BaxterandRennie(1996),Duáe(1996),orMusielaandRutkowski(1997)).TheHJMmodelalsohasthehonorofincludingessentiallyallearlierbondmodelsasspecialcases,and,inparticular,theHJMmodelcontainsboththewidelyusedmodelofHoandLee(1986)andtheseminalmodelofVasicek(1977).Still,themostfundamentalbeneßtoftheHJMmodelisthatitprovidesabroadclassofnaturalconditionsunderwhichonecanruleoutthepossibilityofrisk-freearbitragebetweenbondsofdiÞerentmaturities,andfreedomfromarbitrageDate:January15,2001.1991MathematicsSubjectClassißcation.Primary:91B28;Secondary:60H05,60G44.Keywordsandphrases.Heath-Jarrow-Mortonmodel,HJMmodel,interestrates,LIBOR,futuresprices,arbitragepricing,equivalentmartingalemeasures.12VLADIMIRPOZDNYAKOVANDJ.MICHAELSTEELEisarguablythemostfundamentalconditionofeconomicrealism.Forallthesereasons,theHJMmodelsdeservethecarefulattentionofanyoneinterestedtheanalyticunderstandingofàuctuatinginterestrates.OneofthebasicinsightsofHeath,Jarrow,andMorton(1992)isthatthepriceP(t;T)attimetofabondthatpaysonedollaratthematuritydateT´¼isbestviewedintermsofanintegralrepresentation:(2)P(t;T)=expÀ ZTtf(t;u)du!0´t´T´¼:Asimple(butuseful)consequenceofsucharepresentationisthatitimmediatelyguaranteesP(T;T)=1andP(t;T)0,twoessentialpropertiesofanybondpricemodel.Nevertheless,thisßrststeponlydefersthemodelingoftheensembleofprocessesfP(t;T):0´t´T´¼gtothemodelingofthecorrespondingensembleofkernelprocessesff(t;T):0´t´T´¼g.FortheintegralrepresentationofP(t;T)tobegenuinelyuseful,onemustbeabletospecifymodelsforf(t;T)thatarebothßnanciallyfeasibleandmathematicallytractable.TheHJMmodeladdressestheseaimsbyfocusingonrandomkernelsf(t;T)thathavearepresentationasanItÝointegraloftheform(3)f(t;T)=f(0;T)+Zt0«(u;T)du+Zt0»(u;T)?dBu;whereBtdenotesann-dimensionalBrownianmotionandwheref«(u;T):0´u´T´¼gandf»(u;T):0´u´T´¼garerespectivelyRandRnvaluedprocessesthatareadaptedtothestandardßl-trationFtoffBtg.Thetransposesymbol?inthesecondintegralof(3)servestoremindusthatBtand»(u;T)arebothviewedascolumnvectors.Theintuitiveideabehindthekernelprocessf(t;T)isthatitshouldrepresentaninterestrateforwhichonecancontractattimetforarisklessloanthatbeginsatdateTandwhichispaidbackÕaninstantÔlater.Traditionally,suchquantitiesarecalledforwardrates,andHeath,Jarrow,andMorton(1992)showedthatmanyimportantpropertiesofthebondpricesP(t;T)canbeexpressedmosteasilyintermsofamodelforf(t;T).Inparticular,theyshowedthatbyimposinganaturalrestrictionontheSDEmodelforf(t;T)onecanguaranteethatthepriceprocessesfP(t;T)gwillnotoÞeranyopportunityfortheconstructionofarisklessarbitragebetweenbondsofdiÞeringmaturities.3.TheForwardRateDriftRestrictionIfthecoeácientprocesses«(t;T)and»(t;T)oftheSDEforf(t;T)havethepropertythat«(t;T)maybewrittenas(4)«(t;T)=»(t;T)?¢­(t)+ZTt»(t;u)du£where­(t)isanadaptedn-dimensionalprocesssuchthat(5)Eexp² Z¼0­(u)?dBu 12Z¼0j­(u)j2du³=1;LIBORFUTURESPRICEBOUND3thenwesaythatf(t;T)satisßestheforwardratedriftrestriction,andthepro-cess­(t)thatappearsinthisconditioniscalledthemarketpriceforrisk.TheimportanceofthesenotionsforthetheoryofbondpriceprocessesP(t;T)isthatwhentheforwardratedriftrestrictionapplies,onecanshowthattheprobabilitymeasureePdeßnedonF¼by(6)eP(A)=E´1Aexp² Z¼0­(u)?dBu 12Z¼0j­(u)j2du³µhasseveralusefulproperties.Inparticular,ifwelet(7)r(t)=f(t;t)and¬(t)=exp Zt0r(u)du¡;thenforeachTthediscountedprocessfP(t;T)=¬(t):0´t´TgisaeP-martingalewithrespecttotheßltrationFt.Theprocessr(t)=f(t;t)iscalledthespotrater(t)and¬(t)iscalledtheaccumulationfactor(ordiscountfactor).Theintuitionbehind¬(t)isthatitshouldrepresenttheamountofmoneythatonehasinamoneymarketacc