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1、CHAPTER27derivativesandstochasticcontrolInthisChapter...?anintrotothesubjectofstochasticcontrol?valuingarathercomplexexotic27.1INTRODUCTIONSomeoptionsgivetheholderanelementofcontrolduringthelifeofthecontract.ThemostcommonoftheseistheAmericanoptionwhichtheholdercanexercisewheneverhewantsa
2、nddoesn’thavetowaituntilexpiry.Somecontractsgivetheholderevengreater?exibilityandrequirehimtomakemanyandcomplexdecisions.InthischapterIwanttodescribeanoptionthatsitsoutsidetheframeworkwehavedevelopedsofar.Itisnottoodif?culttoanalyzebutitintroducessomenewideas,andinparticularitisagentlein
3、troductiontothesubjectofstochasticcontrol.27.2PERFECTTRADERANDPASSPORTOPTIONSSupposethatyouinvestinaparticularstock,keepingtrackofitsmovementsandbuyingorsellingaccordingtoyourviewofitsfuturedirection.Theamountofmoneythatyouaccumulateduetotradinginthisstockiscalledthetradingaccount.Ifyoua
4、reagoodtraderorluckythentheamountintheaccountwillgrow;ifyouareabadtraderorunluckythentheamountwillbenegative.Howmuchwouldyoupaytobeinsuredagainstlosingmoneyoveragiventimehorizon?Aperfecttraderorpassportoptionisacalloptiononthetradingaccount,givingtheholdertheamountinhisaccountattheendoft
5、hehorizonifitispositive,orzeroifitisnegative.Tovaluethiscontractwemustintroduceanewvariableπwhichisthevalueofthetradingaccount,meaningthevalueofthestocksheldtogetherwithanycashaccumulated.Wealsointroduceqastheamountofstockheldattimet.Thisisthecontrolvariable.Wewillbedeciding,aspartofthev
6、aluationprocess,whatthisqshouldbeateachtime.SinceweholdqofthestockourtotalstockpositionisqS,andsinceourtotalworthisπwemustbeholdingπ?qSincash.OurstockpositionwillchangebyqdSwhileourcash454PartTwoexoticcontractsandpathdependencyClassificationTimedependenceCashflowDecisionsPathdependenceDi
7、mensionOrderClassi?cationoptiontableforPassportOption.positionearnsinterest,anamountr(π?qS)dt.1Thereforethestochasticdifferentialequationforπisdπ=r(π?qS)dt+qdS.(27.1)Thequantityqisofourchoosing,willvaryastimeandthestockpricechange,andiscalledthestrategy;itwillbeafunctiono
