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1、CHAPTER37theHeath,Jarrow&MortonandBrace,Gatarek&MusielamodelsInthisChapter...•theHeath,Jarrow&Morton(HJM)forwardratemodel•therelationshipbetweenHJMandspotratemodels•theadvantagesanddisadvantagesoftheHJMapproach•howtodecomposetherandommovementsoftheforwardratecurv
2、eintoitsprincipalcomponents•theBrace,Gatarek&MusielaorLIBORMarketModel37.1INTRODUCTIONTheHeath,Jarrow&Mortonapproachtothemodelingofthewholeforwardratecurvewasamajorbreakthroughinthepricingoffixed-incomeproducts.Theybuiltupaframeworkthatencompassedallofthemodelsweh
3、aveseensofar(andmanythatwehaven’t).Insteadofmodelingashort-terminterestrateandderivingtheforwardrates(or,equivalently,theyieldcurve)fromthatmodel,theyboldlystartwithamodelforthewholeforwardratecurve.Sincetheforwardratesareknowntoday,thematterofyield-curvefittingis
4、containednaturallywithintheirmodel;itdoesnotappearasanafterthought.Moreover,itispossibletotakerealdatafortherandommovementoftheforwardratesandincorporatethemintothederivative-pricingmethodology.37.2THEFORWARDRATEEQUATIONThekeyconceptintheHJMmodelisthatwemodelthee
5、volutionofthewholeforwardratecurve,notjusttheshortend.WriteF(t;T)fortheforwardratecurveattimet.Thustheprice610PartThreefixed-incomemodelingandderivativesofazero-couponbondattimetandmaturingattimeT,whenitpays$1,isT−F(t;s)dsZ(t;T)=et.(37.1)Letusassumethatallzero-co
6、uponbondsevolveaccordingtodZ(t;T)=µ(t,T)Z(t;T)dt+σ(t,T)Z(t;T)dX.(37.2)Thisisnotmuchofanassumption,otherthantosaythatitisaone-factormodel,andIwillgeneralizethatlater.Inthisd·meansthattimetevolvesbutthematuritydateTisfixed.NotethatsinceZ(t;t)=1wemusthaveσ(t,t)=0.Fro
7、m(37.1)wehave∂F(t;T)=−logZ(t;T).∂TDifferentiatingthiswithrespecttotandsubstitutingfrom(37.2)resultsinanequationfortheevolutionoftheforwardcurve:∂∂dF(t;T)=1σ2(t,T)−µ(t,T)dt−σ(t,T)dX.(37.3)∂T2∂TInFigure37.1isshowntheforwardratecurvetoday,timet∗,andafewdayslater.T
8、hewholecurvehasmovedaccordingto(37.3).Wherehasthisgotus?Wehaveanexpressionforthedriftoftheforwardratesintermsofthevolatilityoftheforwardrates.Thereisalsoaµterm
