گزینه توقف تئوری تحت هزینه های معاملاتی

The problem of option hedging in the presence of proportional transaction costs can be formulated as a singular stochastic control problem. Hodges and Neuberger [1989. Optimal replication of contingent claims under transactions costs. Review of Futures Markets 8, 222–239] introduced an approach that is based on maximization of the expected utility of terminal wealth. We develop a new algorithm to solve the corresponding singular stochastic control problem and introduce a new approach to option hedging which is closer in spirit to the pathwise replication of Black and Scholes [1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–654]. This new approach is based on minimization of a Black–Scholes-type measure of pathwise risk, defined in terms of a market delta, subject to an upper bound on the hedging cost. We provide an efficient backward induction algorithm for the problem of cost-constrained risk minimization, whose associated singular stochastic control problem is shown to be equivalent to an optimal stopping problem. This algorithm is then modified to solve the singular stochastic control problem associated with utility maximization, which cannot be reduced to an optimal stopping problem. We propose to choose an optimal parameter (risk-aversion coefficient or Lagrange multiplier) in either approach by minimizing the mean squared hedging error and demonstrate that with this “best” choice of the parameter, both approaches have similar performance. We also discuss the different notions of risk in both approaches and propose a volatility adjustment for the risk-minimization approach, which is analogous to that introduced by Zakamouline [2006. European option pricing and hedging with both fixed and proportional transaction costs. Journal of Economic Dynamics and Control 30, 1–25] for the utility maximization approach, thereby providing a unified treatment of both approaches.

The problemofoptionpricingandhedgingwasinitiallystudiedinanidealizedsettingwhereaninvestorincursno transactioncostsfromtradinginamarketconsistingofarisk-freeasset(‘‘bond’’)withconstantrateofreturnandarisky asset (‘‘stock’’)whosepriceisageometricBrownianmotionwithconstantrateofreturnandvolatility.Forthissetting, Black andScholes(1973) demonstratedthatintheabsenceofarbitragethevalueofanoptionisanexpectationofthe discountedpayoffatexpirationunderthe‘‘risk-neutral’’measure,forwhichthestock’srateofreturnequalstherisk-free rate.Moreover,perfectreplicationoftheoptionispossibleandtheoptionisitself‘‘redundant’’insucha‘‘complete’’ market. However,theBlack–Scholes‘‘delta-hedging’’portfoliorequirescontinuoustrading.Inthepresenceoftransaction costs proportionaltotheamountoftrading,suchacontinuousstrategyisprohibitivelyexpensive.Henceitisimpossibleto perfectlyreplicatetheoptioninthissettingwhentherearetransactioncostsand,asaresult,tradinginanoptioninvolves an essentialelementofrisk. One approachtocharacterizethishedgingriskexaminesthedifferencebetweentherealizedcashflowfromahedging strategyandthedesiredpayoffatmaturity.Itembedsoptionhedgingwithintheframeworkofportfolioselection introducedby Magill andConstantinides(1976) and DavisandNorman(1990), andusesarisk-averseutilityfunctionto assess thisshortfall(‘‘replicationerror’’).Inthisway, HodgesandNeuberger(1989) formulatedtheproblemofoption hedging asthatofmaximizingtheinvestor’sexpectedutilityofterminalwealth.Makinguseofanindifferenceargument, the reservationsellingorbuyingpriceofanoptionisdefinedastheamountofmoneythatwouldmakeaninvestor indifferent, intermsofexpectedutility,betweentradinginthemarketwithandwithouta(shortorlong)positioninthe option.Thisinvolvesthevaluefunctionsoftwosingularstochasticcontrolproblemsandtheoptimalhedgeisgivenbythe differenceinthetradingstrategiescorrespondingtothesetwoproblems.Thenatureoftheoptimalhedgeisthatan investorwithanoptionpositionshouldrebalancehisportfolioonlywhenthenumberofsharesofstockfalls‘‘toofar’’out of line.Forthenegativeexponentialutilityfunction, Davisetal.(1993), ClewlowandHodges(1997) and Zakamouline (2006) havedevelopednumericalmethodstocomputetheoptimalhedgeandoptionpricebymakinguseofdiscrete-time dynamic programmingforanapproximatingbinomialtreeforthestockprice. WhalleyandWilmott(1997) and Barlesand Soner (1998) havedevelopedasymptoticapproximationsforthesehedgingstrategiesandoptionpricesasthetransaction costs approach0. ConstantinidesandZariphopoulou(1999,2001) haveprovidedoptionpriceboundsundergeneralutility functions (ratherthanthenegativeexponentialutilityfunctioncommonlyadoptedfornumericalstudies).Inthispaperwe make useofanewnumericalmethodforsolvingsingularstochasticcontrolproblems,recentlyintroducedby Lai etal. (2009), todevelopamuchsimpleralgorithmtocomputethebuy–sellboundariesandvaluefunctionsintheutility-based approach. In thepresenceoftransactioncosts,alternativestotheutility-basedapproachhavebeenbasedonsuper-replication(or replication)inadiscrete-timesettingandareconcernedwithfindingtradingstrategieswhichproducepayoffsat expirationthatareatleast(orexactly)asvaluableastheoptionpayoff.NotingthatusingtheBlack–Scholesdeltatoshort- sell deltasharesofstockatthebeginningofeachrevisionintervalintroducestoohightransactioncostsasthewidthofthe revision intervalshrinksto0, Leland (1985) proposedamodificationofthevarianceusedintheBlack–Scholesdeltasoasto yield thedesiredoptionpayoffatexpirationinclusiveoftransactioncosts.Thefactthatthismodifiedstrategyisnotself- financing hasprompted BoyleandVorst(1992) to workinadiscrete-state(binomialtree)frameworktoconstructa selffinancing discrete-timereplicatingstrategy,therebyextendingthetwo-periodmodelof Merton(1990,Chapter14). Explicit portfolio weightsateachnodeofthebinomialtreecanbecomputedbyusingabackwardinductionprocedure.However, these methodsrequiretheusertoexogenouslyspecifyarevisionintervalanditisunclearhowonecandosooptimally.In fact, asthewidthoftherevisionintervalapproaches0,thecostoftheoptionapproachesthepriceofasingleshareofstock, which turnsouttobetheleastexpensivewayofsuper-replicatingtheoptioninacontinuous-timemodel;see Soner etal. (1995). Forthebinomialtreemodel, Bensaid etal.(1992) derivedboundsontheoptionvalueatinceptionbyminimizing the initialcostoftheself-financingstrategyusedtoproduceasuper-replicatingportfolioofstockandbondatexpiration. As theyhaveshown,byrebalancingonlyintheearlierperiods,itispossibletohaveasuper-replicatingportfoliothatis less expensive than thecorrespondingreplicatingportfolio.Ingeneral,theoptimaldiscrete-timesuper-replicatingstrategyis such thattheinvestorwithanoptionpositiondoesnottransactatatradingdateiftheinheritedamountofstockisina certain range(whichdependsonthepasthistoryofthestockprice);otherwiseheadjustshisportfoliobacktothisrange. Notingthatthiscostminimizationproblemassociatedwithsuper-replicationispathdependentandthatthedynamic programmingalgorithmiscomputationallyexpensiveifthenumberofperiodsisnotsufficientlysmall, Edirisinghe etal. (1993) developedalinearprogrammingalgorithmandatwo-stagedynamicprogrammingmethodtoapproximatethe optimalsolution.Morerecently, Primbs (2009) providedanalternativeformulationofsuper-replicationintermsofthefirst two momentsofthereplicationerror. In thispaperweproposeanewapproachwhichformulatestheoptionhedgingprobleminthepresenceoftransaction costs asaconstrainedriskminimizationproblemthatminimizesameasureofpathwiseriskundertheconstraintthatthe hedging cost(centraltothereplication/super-replicationapproach)doesnotexceedaprescribedlevel.Thismeasureof pathwise riskisimplicitintheBlack–Scholestheorythatcontinuouslyrebalancestheportfoliotomakesuchriskzero when therearenotransactioncosts,andleadsinourapproachtoanaturalmodificationoftheBlack–Scholesdelta-hedging scheme. Inthepresenceoftransactioncosts,thismodificationconsistsofbuyingorsellingtheunderlyingstockwhenever the holdingofsharesfallsoutsideano-transactionbandcontainingtheoption’sdelta.Thecorrespondingsingular stochasticcontrolproblem,whoseno-actionregionistheno-transactionband,isequivalenttoanoptimalstopping problem.Thisequivalenceisusedtocomputethebuy-andsell-boundariesefficiently,therebyreducingsubstantiallythe computational complexityoftheoriginalsingularstochasticcontrolproblem,whichrequiresdeterminationofboth when to applythecontrol(intheformofbuyingorselling)and how much controltoapply. This paperisorganizedasfollows.Section2definesthehedgingcostofaself-financingstrategyandusesittoformulate the singularstochasticcontrolproblemsassociatedwiththeutilitymaximizationandthecost-constrainedrisk minimization approachestooptionhedginginthepresenceoftransactioncosts.Itwillbeshownthatbothformulationsof the optionhedgingproblembelongtothegeneralframeworkofminimizingexpectedhedgingcostunderariskconstraint. Section 3presentsnewalgorithms,whichareconsiderablysimplerthanthosecurrentlyavailable,tocomputetheoptimal buy-andsell-boundariesforbothsingularstochasticcontrolproblems.Section4givessomenumericalresultstoillustrate the computationalschemesandcomparetheirhedgingerrorswiththoseof Leland (1985) and Black andScholes(1973) with prescribedrevisionintervals.SomeconcludingremarksaregiveninSection5.

The precedingsectionshavefocusedonEuropeancalloptionsonstocksthatdonotpaydividends.Since Ds ðSÞ ¼ IfSoKg for ashortputand Db ðSÞ ¼ IfSoKg for alongput,Algorithms1and2canbeappliedtoEuropeanputoptionsbyredefining DsðzÞ ¼ Ifzo0g for ashortputand DbðzÞ ¼ Ifzo0g for alongput.Recallalsothatfortheput, pBSðt; S; sÞ ¼ KerðTtÞFðd2ðt; S; sÞÞ  SFðd1ðt; S; sÞÞ and DBSðt; S; sÞ ¼ Fðd1ðt; S; sÞÞ. Whenthestockpaysdividendsattherate q, we cansimplyreplace S in theprecedingby SeqðTtÞ. Moregenerally,thealgorithmscanbemodifiedinastraightforward manner toaccommodatecombinationsofcallsandputs(e.g.,bullspreads,butterflyspreads,etc.)bychangingthe definitions ofterminalwealthin(4)and(5). In thepresenceoftransactioncosts,thereisatradeoffbetweenminimizingtheriskassociatedwithwritinganoption (since thewritercannothedgeawaytheriskentirelybycontinuoustradingoftheunderlyingstock)andkeepingthe hedging costataminimum.Theutility-maximizationapproachinitiatedby HodgesandNeuberger(1989) introducesa concaveutilityfunction(usuallychosentobeoftheCARAtypefortractabilityandforitsnaturalquantificationofrisk aversion)ofterminalwealthandtheformulationleadstothesingularstochasticcontrolprobleminSection2.2.We introduceanewapproachtooptionhedgingwhichiscloserinspirittothepathwisereplicationof Black andScholes(1973) and thehedging-costminimizationof Bensaid etal.(1992). Ouralternativecost-constrainedpathwise-riskminimization formulation leadstothesingularstochasticcontrolprobleminSection2.3whichissimplerandforwhichanefficient backwardinductionalgorithm(Algorithm1)canbeusedtosolvefortheoptimalbuyandsellboundaries.Wehavealso modified thealgorithmtoobtainarelativelysimplealgorithm(Algorithm2)forutility-basedoptionpricingandhedging. Even thoughtheutilitymaximizationandthecost-constrainedpathwise-riskminimizationapproachesusedifferent notionsofrisk,wehaveshowninSection4.3thatbychoosingtheirassociatedparameters‘‘optimally,’’bothapproaches havesimilarperformanceintermsoftherootmeansquaredhedgingerror.Moreover,inSection4.4wehavedevelopeda volatilityadjustmentforthecost-constrainedpathwise-riskminimizationapproach,whichisanalogoustothatintroduced by Zakamouline (2006) for theutilitymaximizationapproach.Inthiswayandalsointhecomputationalalgorithmsof Section 3,aunifiedtreatmentofbothapproachesisprovided.