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1、Chapter32HJM,LMM,andMultipleZeroCurvesOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20141HJMModel:NotationOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20142P(t,T):priceattimetofadiscountbondwithprincipalof$1maturingatTWt:vectoro
2、fpastandpresentvaluesofinterestratesandbondpricesattimetthatarerelevantfordeterminingbondpricevolatilitiesatthattimev(t,T,Wt):volatilityofP(t,T)NotationcontinuedOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20143ƒ(t,T1,T2):forwardrateasseenattforthepe
3、riodbetweenT1andT2F(t,T):instantaneousforwardrateasseenattforacontractmaturingatTr(t):short-termrisk-freeinterestrateattdz(t):WienerprocessdrivingtermstructuremovementsModelingBondPrices(Equation32.1,page741)Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.H
4、ull20144TheprocessforF(t,T)Equation32.4and32.5,page742)Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20145TreeEvolutionofTermStructureisNon-RecombiningOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20146Treefortheshortraterisnon-Ma
5、rkovTheLIBORMarketModelTheLIBORmarketmodelisamodelconstructedintermsoftheforwardratesunderlyingcapletpricesOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20147NotationOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20148VolatilityStr
6、uctureOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20149InTheorytheL’scanbeDeterminedfromCapPricesOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201410Example32.1(Page745)IfBlackvolatilitiesforthefirstthreecapletsare24%,22%,and20%
7、,thenL0=24.00%L1=19.80%L2=15.23%Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201411Example32.2(Page745-746)Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201412TheProcessforFkinaOne-FactorLIBORMarketModelOptions,Futures,andOtherDe
8、rivatives,9thEdition,Copyright©JohnC.Hull201413RollingForwardRisk-Neutrality(Equation32.12,page7