derivatives and stochastic control

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1、CHAPTER27derivativesandstochasticcontrolInthisChapter...•anintrotothesubjectofstochasticcontrol•valuingarathercomplexexotic27.1INTRODUCTIONSomeoptionsgivetheholderanelementofcontrolduringthelifeofthecontract.ThemostcommonoftheseistheAmericanoptionwhichtheholdercanexercisewheneverhewantsa

2、nddoesn’thavetowaituntilexpiry.Somecontractsgivetheholderevengreaterflexibilityandrequirehimtomakemanyandcomplexdecisions.InthischapterIwanttodescribeanoptionthatsitsoutsidetheframeworkwehavedevelopedsofar.Itisnottoodifficulttoanalyzebutitintroducessomenewideas,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、mensionOrderClassificationoptiontableforPassportOption.positionearnsinterest,anamountr(π−qS)dt.1Thereforethestochasticdifferentialequationforπisdπ=r(π−qS)dt+qdS.(27.1)Thequantityqisofourchoosing,willvaryastimeandthestockpricechange,andiscalledthestrategy;itwillbeafunctiono