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1、CHAPTER76overviewofnumericalmethodsInthisChapter...•finite-differencemethods•MonteCarlosimulation•numericalintegration•programsofstudy76.1INTRODUCTIONWeareinthehomestraightnow,onlynumericalmethodsseparateusfromthebibliography.Theaimofthischapter,beforewegettoth
2、emeatofthenumericalmethods,istoputthesetechniquesintocontext,toexplainwhattheyareusedfor,why,andgiveyousomeinitialthoughtsonhowtheyareimplemented.I’llalsoexplainhowefficientthesetechniquesareintermsofcomputationaltime,andsuggestaprogramofselfstudytohelpyoubuild
3、upyourexperienceofimplementation.76.2FINITE-DIFFERENCEMETHODSFinite-differencemethodsaredesignedforfindingnumericalsolutionsofdifferentialequations.Sinceweworkwithamesh,notunlikethebinomialmethod,wewillfindthecontractvalueatallpointsisstockprice-timespace.Inquan
4、titativefinancethatdifferentialequationisalmostalwaysofdiffusionorparabolictype,asexplained.Theonlyrealdifferencebetweenthepartialdifferentialequationsarethefollowing:•Numberofdimensions•Functionalformofcoefficients•Boundary/finalconditions1192PartSixnumericalmet
5、hodsandprograms•Decisionfeatures•LinearornonlinearNumberofdimensionsIsthecontractanoptiononasingleunderlyingormany?Isthereanystrongpathdepen-denceinthepayoff?Answerstothesequestionswilldeterminethenumberofdimensionsintheproblem,aswesawinChapter24.Attheveryleas
6、twewillhavetwodimen-sions:Sorr,andt.Finite-differencemethodscopeextremelywellwithsmallernumberofdimensions,uptofour,say.Abovethattheygetrathertimeconsuming.We’lllookatfinite-differencemethodsinuptothreedimensions,meaningtimeplustwoothers,althoughtheexplicitfinit
7、e-differencemethodcanbeeasilyextendedtoasmanydimensionsasyouwant.FunctionalformofcoefficientsThemaindifferencebetweenanequityoptionproblemandasingle-factorinterestrateoptionproblemisinthefunctionalformofthedriftrateandthevolatility.Theseappearinthegoverningpart
8、ialdifferentialequationsascoefficients.Thestandardmodelforequitiesisthelognormalmodel,buttherearemanymore‘standard’modelsinfixedincome.Doesthismatter?No,notifyouaresolvingtheequation
