American Options under Proportional Transaction Co

arXiv:0709.1589v1[math.PR]11Sep2007AmericanOptionsunderProportionalTransactionCosts:Pricing,HedgingandStoppingAlgorithmsforLongandShortPositionsAletRouxandTomaszZastawniakDepartmentofMathematics,UniversityofYorkHeslington,YorkYO105DD,[e-mail],[e-mail]sarestudiedinageneraldiscretemarketinthepresenceofproportionaltransactioncosts,modelledasbid-askspreads.Pricingal-gorithmsandconstructionsofhedgingstrategies,stoppingtimesandmar-tingalerepresentationsarepresentedforshort(seller’s)andlong(buyer’s)positionsinanAmericanoptionwithanarbitrarypayoff.ThisgeneralapproachextendsthespecialcasesconsideredintheliteratureconcernedprimarilywithcomputingthepricesofAmericanputsundertransactioncostsbyrelaxinganyrestrictionsontheformofthepayoff,themagnitudeofthetransactioncostsorthediscretemarketmodelitself.Thelargelyunexploredcaseofpricing,hedgingandstoppingfortheAmericanoptionbuyerundertransactioncostsisalsocovered.Thepricingalgorithmsarecomputationallyefficient,growingonlypolynomiallywiththenumberoftimestepsinarecombinanttreemodel.Thestoppingtimesrealisingtheask(seller’s)andbid(buyer’s)optionpricescandifferfromoneanother.Theformerisgenerallyaso-calledmixed(randomised)stoppingtime,whereasthelatterisalwaysapure(ordinary)stoppingtime.1IntroductionInthispaperwestudytheseller’sandbuyer’spositionsinAmericanoptionswhentradingintheunderlyingassetissubjecttoproportionaltransactioncosts.Theresultsapplytooptionswitharbitrarypayoffsinanydiscretemarketmodelandproportionaltransactioncostsofanymagnitude.Weareconcernedwithcomputingtheseller’spriceofanAmericanoption,alsoknownastheupperhedgingpriceortheaskprice,aswellasthebuyer’sprice,oftenreferredtoasthelowerhedgingpriceorthebidprice.Apartfrompricing,weconstructoptimal1strategiessuperhedgingthepositionsoftheoptionsellerandbuyer,togetherwiththerespectivestoppingtimesrealisingtheoptionprices,generallyamixed(randomised)stoppingtimeforthesellerandapure(ordinary)stoppingtimeforthebuyer.Wealsoconsidermartingalerepresentationsfortheaskandbidoptionprices.ThefirsttoexamineAmericanoptionsunderproportionaltransactioncostsinasimilarsettingandlevelofgeneralityasinthepresentpaperwereChalasaniandJha[CJ01].Theyestablishedmartingalerepresentationsforoptionswithcashsettlement,subjecttothesimplifyingassumptionthattransactioncostsapplyatanytime,exceptatanyparticularstoppingtimechosenbythebuyertoexercisetheoption.AnimportantfeaturethatemergedinChalasaniandJha’srepresentationfortheoptionseller’spricewastheroleplayedbymixedstoppingtimesinplaceofpurestoppingtimes.ChalasaniandJhapointedoutthenon-trivialnatureofcomputingtheoptionpricesintheirrepresentationsandtheneedtodevelopalgorithmstoevaluatetheseprices.Ourpricingalgorithmssolvethisproblem.Moreover,weputforwardalgorithmsforconstructingthecorrespondinghedgingstrategies,stoppingtimes,andapproximatemartingales.BouchardandTemam[BT05]establishedadualrepresentationforthesetofinitialendowmentsallowingtosuperhedgetheseller’spositioninanAmericanoptioninadiscretetimemarketmodelwithproportionaltransactioncostsinthesettingofKabanov,R´asonyiandStricker[KRS03],andSchachermayer[Sch04].Inparticular,theyreproducedChalasaniandJha’s[CJ01]martingalerepresen-tationoftheseller’sprice.However,notethatBouchardandTemam[BT05]followadifferentconventionthanChalasaniandJha[CJ01]inthattheyrebal-ancetheportfoliosinahedgingstrategybeforeratherthanafteritbecomesknownwhetherornottheAmericanoptionistobeexercised.PapersconcernedwithvariousspecialcasesinvolvingthehedgingpricesofAmericanoptionsunderproportionaltransactioncostsincludeKoci´nski[Koc99],[Koc01],whostudiedsufficientconditionsfortheexistenceofper-fectlyreplicatingstrategiesforAmericanoptions,PerrakisandLefoll[PL00],[PL04],whoinvestigatedAmericancallsandputsinthebinomialmodel,andTokarzandZastawniak[TZ06],whoworkedwithgeneralAmericanpayoffsinthebinomialmodelundersmallproportionaltransactioncosts.Anothergroupofpapers,usingpreference-basedorriskminimisationap-proachesratherthansuperhedgingforAmericanoptionsunderproportionaltransactioncosts,includesDavisandZariphopoulou[DZ95],MercurioandVorst[MV97],ConstantinidesandZariphopoulou[CZ01],andConstantinidesandPer-rakis[CP04].TheworkbyLeventalandSkorohod[LS97],andJakubenas,Lev-entalandRyznar[JLR03]showsthatsuperhedgingincontinuoustimeleadstounrealisticresultsforAmericanoptionsunderproportionaltransactioncosts,thusprovidingmotivationforexploringdiscretetimeapproaches.ThepresentpapercomplementsandextendstheresultsobtainedbyCha-lasaniandJha[CJ01]andBouchardandTemam[BT05]byprovidingpricing,hedging,stoppingandapproximatemartingalealgorithmsforarbitraryAmeri-canoptionsunderproportionaltransactioncostsinageneraldiscretesetting.Italsoextendstheworkonhedgingpricesbyseveraloftheauthorslistedabove,2removinganyrestrictionsimposedinthevariousspecialcasesthathavebeenconsideredintheliterature.Asaby-product,weestablishthesamemartingalerepresentationsforAmericanoptionpricesundertransactioncostsasin[CJ01]or[BT05]byaverydifferentmethodbasedonanexplicitconstructionofthestoppingtimesa