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1、CHAPTER24stronglypath-dependentderivativesInthisChapter...•strongpathdependence•pricingmanystronglypath-dependentcontractsintheBlack–Scholespartialdifferentialequationframe-work•howtohandlebothcontinuouslysampledanddis-cretelysampledpaths•jumpconditionsfordifferentialequations
2、24.1INTRODUCTIONTobeabletoturnthevaluationofaderivativecontractintothesolutionofapartialdifferentialequationisabigstepforward.Thepartialdifferentialequationapproachisoneofthebestwaystopriceacontractbecauseofitsflexibilityandbecauseofthelargebodyofknowledgethathasgrownuparoundth
3、efastandaccuratenumericalsolutionoftheseproblems.Thisbodyofknowledgewas,inthemain,basedaroundthesolutionofdifferentialequationsarisinginphysicalappliedmathematicsbutisstartingtobeusedinthefinancialworld.InthischapterIshowhowtogeneralizetheBlack–Scholesanalysis,deltahedgingandno
4、arbitrage,tothepricingofmanymorederivativecontracts,specificallycontractsthatarestronglypath-dependent.Iwilldescribethetheoryintheabstract,givingbriefexamplesocca-sionally,butsavingthedetailedapplicationtospecificcontractsuntillaterchapters.418PartTwoexoticcontractsandpathdepend
5、ency24.2PATH-DEPENDENTQUANTITIESREPRESENTEDBYANINTEGRALImagineacontractthatpaysoffatexpiry,T,anamountthatisafunctionofthepathtakenbytheassetbetweentimezeroandexpiry.Letussupposethatthispath-dependentquantitycanberepresentedbyanintegralofsomefunctionoftheassetovertheperiodzerot
6、oT:TI(T)=f(S,τ)dτ.0Thisisnotsuchastrongassumption;inparticularmostofthepath-dependentquantitiesinexoticderivativecontracts,suchasaverages,canbewritteninthisformwithasuitablechoiceoff(S,t).WearethusassumingthatthepayoffisgivenbyP(S,I)attimet=T.Priortoexpirywehaveinformationabo
7、utthepossiblefinalvalueofS(attimeT)inthepresentvalueofS(attimet).Forexample,thehigherSistoday,thehigheritwillprobablyendupatexpiry.Similarly,wehaveinformationaboutthepossiblefinalvalueofIinthevalueoftheintegraltodate:tI(t)=f(S,τ)dτ.(24.1)0Aswegetclosertoexpiry,sowebecomemorecon
8、fidentaboutthefinalvalueofI.Onecanimaginethatthevalueoftheoptio
