期权、期货及其他衍生产品 课件 hull-OFOD11-ppt-33

Options,Futures,andOtherDerivativesEleventhEditionChapter33ModelingForwardRatesCopyright©2022,2018,2012PearsonEducation,Inc.AllRightsReservedH

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MModel:Notation(1of2)priceattimetofadiscountbondwithprincipalof$1maturingatTvectorofpastandpresentvaluesofinterestratesandbondpricesattimetthatarerelevantfordeterminingbondpricevolatilitiesatthattimevolatilityofH

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MModel:Notation(2of2)forwardrateasseenattfortheperiodbetweeninstantaneousforwardrateasseenattforacontractmaturingatTshort-termrisk-freeinterestrateattWienerprocessdrivingtermstructuremovementsModelingBondPrices(Equation33.1)WecanchooseanyvfunctionprovidingforalltBecauseWecanuseIto’slemmatodeterminetheprocessforTheProcessforFoftandUpperT

(Equation33.4and33.5)ThisresultmeansthatifwewritewemusthaveSimilarresultsholdwhenthereismorethanonefactorTreeEvolutionofTermStructureisNonrecombiningTreefortheshortraterisnon-MarkovTheB

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MModelTheB

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Mmodelisamodelconstructedintermsoftheforwardratesapplicabletoperiodssuchas3monthsor6months(notinstantaneousforwardrates)TheforwardratevolatilitiesfortheperiodsconsideredcanbedeterminedfromcappricesNotationresetdateforwardratebetweentimesindexfornextresetdateattimetVolatilityStructureWeassumeastationaryvolatilitystructurewherethevolatilityofdependsonlyonthenumberofaccrualperiodsbetweenthenextresetdateandInTheorytheLambdaPrimesCanBeDeterminedFromCapPricesDefineasthevolatilityofisthevolatilityforthecaplet.IfthemodelprovidesaperfectfittocappriceswemusthaveThisallowsthetobedeterminedinductivelyExample33.1IfBlackvolatilitiesforthefirstthreecapletsare24%,22%,and20%,thenExample33.2n12345sigmasubnleftparenthesis%rightparenthesis15.5018.2517.9117.7417.27Lambdasubstartexpressionnminus1endexpressionleftparenthesis%rightparenthesis15.5020.6417.2117.2215.25n678910sigmasubnleftparenthesis%rightparenthesis16.7916.3016.0115.7615.54Lambdasubstartexpressionnminus1endexpressionleftparenthesis%rightparenthesis14.1512.9813.8113.6013.40TheProcessforFSubkinaOne-FactorB

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MMarketModelThedriftdependsontheworldchoseninaworldthatisforwardrisk-neutralwithrespecttothedriftiszero.RollingRisk-Neutrality(Equation33.12)Itisoftenconvenienttochooseaworlddefinedbyanumerairethatisalwaysthebondmaturingatthenextresetdate.Inthiscase,wecandiscountfromrateobservedattimeTheprocessforTheB

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MandH

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MmodelsInthelimitasthetimebetweenresetstendstozero,theB

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MmodelwithrollingriskneutralitybecomestheH

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Mmodelinthetraditionalrisk-neutralworld.MonteCarloImplementationofB

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MModel(Equation33.14)WeassumenochangetothedriftbetweenresetdatessothatMultifactorVersionsofB

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MB

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McanbeextendedsothatthereareseveralcomponentstothevolatilityAfactoranalysiscanbeusedtodeterminehowthevolatilityofissplitintocomponents.RatchetCaps,StickyCaps,andFlexiCapsAplainvanillacapdependsonlyononeforwardrate.Itspriceisnotdependentonthenumberoffactors.Ratchetcaps,stickycaps,andflexicapsdependonthejointdistributionoftwoormoreforwardrates.TheirpricestendtoincreasewiththenumberoffactorsValuingEuropeanOptionsintheB

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MModelThereisananalyticapproximationthatcanbeusedtovalueEuropeanswapoptionsintheB

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Mmodel.CalibratingtheB

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MModelIntheory,B

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Mcanbeexactlycalibratedtocappricesasdescribedearlier.Inpractice,weproceedasforshortratemodelstominimizeafunctionoftheformwhereisthemarketpriceoftheithcalibratinginstrument,isthemodelpriceoftheithcalibratinginstrumentandPisafunctionthatpenalizesbigchangesorcurvatureinTypesofAgencyMortgage-BackedSecurities(M

B

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s)Pass-ThroughCollateralizedMortgageObligation(C

M

O)InterestOnly(I

O)PrincipalOnly(P

O)Option-AdjustedSpread(O

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S)TocalculatetheO

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SforaninterestratederivativewevalueitassumingthattheinitialyieldcurveistheTreasurycurve+aspreadWeuseaniterativeproceduretocalculatethespreadthatmakesthederivative’smodelprice=marketprice.ThisspreadistheO

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