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1、AnOptimalPairs-TradingRule∗QingshuoSong†QingZhang‡February26,2013AbstractThispaperisconcernedwithapairstradingrule.Theideaistomonitortwohistoricallycorrelatedsecurities.Whendivergenceisunderway,i.e.,onestockmovesupwhiletheothermovesdown,apairstradeisenteredwhichconsistsofapairtoshortthe
2、outperformingstockandtolongtheunderperformingone.Suchastrategybetsthe“spread”betweenthetwowouldeventuallyconverge.Inthispaper,adifferenceofthepairisgovernedbyamean-revertingmodel.Theobjectiveistotradethepairsoastomaximizeanoverallreturn.Afixedcommissioncostischargedwitheachtransaction.Ina
3、ddition,astop-losslimitisimposedasastateconstraint.TheassociatedHJBequations(quasi-variationalinequalities)areusedtocharacterizethevaluefunctions.Itisshownthatthesolutiontotheoptimalstoppingproblemcanbeobtainedbysolvinganumberofquasi-algebraicequations.Weprovideasetofsufficientconditionsi
4、ntermsofaverificationtheorem.Numericalexamplesarereportedtodemonstratetheresults.Keywords:pairstrading,optimalstopping,quasi-variationalinequalities,mean-revertingprocessarXiv:1302.6120v1[q-fin.PR]25Feb2013∗ThisresearchissupportedinpartbytheResearchGrantsCouncilofHongKongNo.CityU103310an
5、dinpartbytheSimonsFoundation(235179).†DepartmentofMathematics,CityUniversityofHongKong,83TatCheeAve,Kowloon,HongKong,song.qingshuo@cityu.edu.hk‡DepartmentofMathematics,UniversityofGeorgia,Athens,GA30602,qingz@math.uga.edu11IntroductionThispaperisconcernedwithpairstrading.Theideaistoiden
6、tifyandmonitorapairofhistoricallycorrelatedstocks.Whenthetwostockpricesdiverge(onestockmovesupwhiletheothermovesdown),thepairstradewouldbetriggered:toshortthestrongerstockandtolongtheweakeronebettingtheeventualconvergenceoftheprices.ThepairstradingwasfirstdevelopedbyBambergerandfollowedb
7、yTartaglia’squantitativegroupatMorganStanleyinthe1980s.Amajoradvantageofpairstradingisits‘marketneutral’natureinthesensethatitcanbeprofitableunderanymarketconditions.Therearemanygooddiscussionsinconnectionwiththecauseofthedivergenceandsubse-quentconvergence.Wereferthereadertothe
