Hedging with risk for game options in discrete tim

HEDGINGWITHRISKFORGAMEOPTIONSINDISCRETETIME.YANDOLINSKYANDYURIKIFERINSTITUTEOFMATHEMATICSHEBREWUNIVERSITYJERUSALEM,ISRAELAbstract.Westudytheproblemsofefficienthedgingofgame(Israeli)op-tionswhentheinitialcapitalintheportfolioislessthanthefairoptionprice.Inthiscaseaperfecthedgingisimpossibleandonecanonlytrytominimisetherisk(whichcanbedefinedindifferentways)ofhavingnotenoughfundsintheportfoliotopaytherequiredamountattheexcercisetime.Wesolvetheminimizationproblemsandfindviadynamicalprogrammingappropriateefficienthedgingstrategiesfordiscretetimegameoptionsinmultinomialmar-kets.TheapproachandsomeoftheresultsarenewalsoforstandardAmericanoptions.1.IntroductionThegame(Israeli)optionswereintroducedin[6]inastandardfinancialmarketwhichconsistsofanonrandomcomponentBtdescribedasasavingsaccountattimetwithaninterestrandofarandomcomponentStdescribedasthepriceofastockattimet.Theformalsetupconsistsofaprobabilityspace(Ω,F,P)togetherwithastochasticprocessSt≥0,t∈Z+(discretetime),ort∈R+(continuoustime)describingthepriceofaunitofthestock,ofarightcontinuousfiltration{Ft},andoftworightcontinuouswithleftlimitsstochasticpayoffprocessesXt≥Yt≥0adaptedtothegivenfiltration.Thegamecontingentclaim(GCC)isdefinedasacontractbetweenthesellerandthebuyeroftheoption,suchthatbothhavetherighttoexerciseatanytimeuptoamaturitydate(horizon).IfthebuyerexercisesthecontractattimetthenhegetsthepaymentYt,butifthesellercancelsbeforethebuyerthenthelatterreceivesthepaymentXt.Thedifferenceδt=Xt−Ytisthepenaltywhichthesellerpaystothebuyerforthecontractcancellation.AhedgeagainstGCCisdefinedasapair(π,σ)whichconsistsofaselffinancingstrategyπ(i.e.atradingstrategywithnoconsumptionandnoinfusionofcapitalsothatonlytransferbetweenstockandbondaccountsareallowed)andacancellationtimeofthecontractσ,whichisastoppingtimewithrespecttothegivenfiltration.Iftheseller’scancellationtimeisσandthebuyer’sexercisetimeisτthenthebuyergetsthepaymentH(σ,τ)=Xσχ{στ}+Yτχ{τ≤σ},whereχQ=1ifaneventDate:November1,2006.2000MathematicsSubjectClassification.Primary91B28:Secondary:60G40,91A05,62L15,91B30.Keywordsandphrases.gameoptions,risk,hedging,optimalstopping.PartiallysupportedbytheISFgrantno.130/06.12Y.DolinskyandY.KiferQoccursand=0ifnot.Ahedgeiscalledperfectifnomatterwhatexercisetimethebuyerchooses,thesellercancoverhisliabilitytothebuyer(withprobabilityone).TheoptionpriceV∗isdefinedastheminimalinitialcapitalthatrequiredforaperfecthedge,i.e.foranyxV∗thereisaperfecthedgewithaninitialcapitalx.Itwasshownin[6]thatpricingtheoptioninacompletemarketsleadstoazerosumoptimalstoppinggame(Dynkin’sgame)withthediscountedpayoffs˜Xt=B0XtBtand˜Yt=B0YtBt,undertheuniquemartingalemeasureP∗∼PwhichmeansthattheoptionpriceistheDynkingamevaluewhenthepayoffsare˜Xt,˜Yt,andtheprobabilitymeasureisP∗.Foranyselffinancingstrategyπthewealthprocesscorrespondingtoπisastochasticprocess{Vπt}suchthatVπtistheportfoliovalueofπatthemomentt.Inthispaperwedefinehedgeasapair(π,σ)withtherestrictionthatthewealthprocesscorrespondingtoπshouldbepositiveandsuchselffinancingstrategyπwillbecalledadmissible.Itiseasytocheckthattheresultsfrom[6]describedaboveremainvalidunderthisrestriction.Observethatgameoptionsinincompletemarketswerestudiedin[5]usingtheoptionaldecompositiontheorem(see[8]).Foradditionalinformationaboutpricingofgameoptionswereferthereaderto[6],[9]and[10].Inarealmarketconditionsaninvestor(seller)maynotbewillingforvariousreasonstotieinahedgingportfoliothefullinitialcapitalrequiredforaperfecthedge.Inthiscasethesellerisreadytoacceptariskthatatanexcercisetimehisportfoliovaluewillbelessthanhisobligationtopayandhewillneedadditionalfundstofulfilthecontract.Thisleadstothenaturalquestionofminimizationofriskforagivensmalleramountofinitialcapital.Inordertomakethisquestionpreciseweneedtodefineexplicitlytheriskmeasure.MostofourstudywillberelatedtotheminimizationoftheshortfallriskforgameoptionswhichisdefinedasR(π,σ)=supτEl˜H(σ,τ)−B0Vπσ∧τBσ∧τ)+
,wherelisanondecreasingcontinuouslossfunction,Vπtisthewealthprocessofanadmissibleselffinancingstrategyπ,˜H(σ,τ)isthediscountedpayment,andEdenotestheexpectationwithrespecttotheobjectiveprobabilityP.WeadmitonlystrategiesπforwhichVπt≥0forallt,i.e.borrowingisprohibited,butsimilarresultscanbeobtainedwithoutimposinganylowerboundonVt.Thereasonthatwetakeasupremumoverallstoppingtimesisthatthesellerdoesnotknowwhenthebuyerwillexerciseandsoheshouldconsidertheworstcase,i.e.wewanttominimizethelargestpossibleexpectationoftheshortfallwhichisweightedbyagivenlossfunction.TheproblemofminimizationofshortfallriskshasbeenmostlystudiedintheEuropeanoptionscase(see,forinstance,CvitaniˆcandKaratzas[1],F¨ollmerandLeukert[3],ScagnellatoandVargiolu[16]).Forcontinuoustimemodels(see[1],[3])theauthorsusedapproachesconnectedwiththeNeyman-PearsonLemmaandconvexdualitymethods,whichdonotworkforAmericanandIsraelioptionscase.RecentlyMulinacci[12]studiedtheshortfallriskfortheAmericanoptionscaseprovingthatifthelossfunctionisconvexthereisahedgethatminimizestheshortfallriskunderaconstraintontheinitialcapital.Theproofreliedheavilyonthefactthattheshortfallriskinthiscaseisaconvexfun