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1、CHAPTER79finite-differencemethodsfortwo-factormodelsInthisChapter...•theexplicitfinite-differencemethodfortwo-factormodels•theADIandHopscotchmethods79.1INTRODUCTIONManycurrentlypopularfinancialmodelshavetworandomfactors.Convertiblebondsareusuallypricedwithbothrandomu
2、nderlyingandrandomriskofdefaultorinterestrate.Exoticequityderivativesareoftenpricedwithstochasticvolatility,thisisespeciallytrueofbarrieroptions.Finite-differencemethodsarequitesuitedtosuchproblems.Onceyougettothreefactors,orthreesourcesofrandomness,thenfinite-diff
3、erencemethodsstarttobecomeslowandcumbersome.Whentherearefourrandomfactorsthen,ifpossible,youshoulduseaMonteCarloapproachforpricing.ThesearediscussedinChapter80.MonteCarloworkswellinhighdimensions.Finite-differencemethodsworkwellinlowdimensions,andhandleearlyexerci
4、severyeffi-ciently.Somepathdependencyisalsoeasytocopewith.InthischapterIwilldescribethesimplestformsoffinite-differencemethodsusedwithtwoormorefactors.79.2TWO-FACTORMODELSIamgoingtoreferthroughoutthischaptertothetwo-factorequation∂V∂2V∂V+a(S,r,t)+b(S,r,t)+c(S,r,t)V∂
5、t∂S2∂S∂2V∂2V∂V+d(S,r,t)+e(S,r,t)+f(S,r,t)=0.(79.1)∂r2∂S∂r∂rAsIwroteinthepreviouschapter,everythingwillbeperfectlyapplicabletoanyproblem,whatevertheunderlyingvariables.Iwritetheproblemtobesolvedas(79.1)assumingthat1254PartSixnumericalmethodsandprogramsmyreaderspref
6、ertheconcrete‘asset’Sand‘interestrate’rtothemoreabstractxandy.Actually,itwillbequitehelpfulifwethinkofsolvingthetwo-factorconvertiblebondproblemofChapter33.Thisisbecausethatproblemcontainsmanyimportantfeaturessuchaschoiceofinterestratemodelandhowitisdiscretized,an
7、dearlyexercise.Forthegeneraltwo-factorproblem(79.1)tobeparabolicweneede(S,r,t)2<4a(S,r,t)d(S,r,t).Asintheone-factorworld,thevariablesmustbediscretized.Thatis,wesolveonathree-dimensionalgridwithS=iδS,r=jδr,andt=T−kδt.Expiryist=Tork=0.TheindicesrangefromzerotoIandJf
8、oriandjrespectively.Ihaveassumedthattheinterestratemodelisonlyspecifiedonr≥0.Thismaynotbethecase,assomesimpleinterestratemodelssuchasVasicekaredefinedover
