Optimal Control of Trading AlgorithmsA General Impulse Control Approach

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1、SIAMJ.FINANCIALMATH.c2011SocietyforIndustrialandAppliedMathematicsVol.2,pp.404–438∗OptimalControlofTradingAlgorithms:AGeneralImpulseControlApproachBrunoBouchard†,Ngoc-MinhDang‡,andCharles-AlbertLehalle§Abstract.Weproposeageneralframeworkforintradaytradingbasedonthecontroloftradingalgor

2、ithms.Givenasetofgenericparameterizedalgorithms(whichhavetobespecifiedbythecontrollerex-ante),ouraimistooptimizethedates(τi)iatwhichtheyarelaunched,thelength(δi)iofthetradingperiod,andthevalueoftheparameters(Ei)ikeptduringthetimeinterval[τi,τi+δi).Thisprovidesthefinancialagentadecisiontoo

3、lforselectingwhichalgorithm(andforwhichsetofparametersandtimelength)shouldbeusedinthedifferentphasesofthetradingperiod.Fromthemathematicalpointofview,thisgivesrisetoanonclassicalimpulsecontrolproblemwherenotonlytheregimeEibutalsotheperiod[τi,τi+δi)havetobedeterminedbythecontrollerattheim

4、pulsetimeτi.WeadapttheweakdynamicprogrammingprincipleofBouchardandTouzi[SIAMJ.ControlOptim.,49(2011),pp.948–962]toourcontexttoprovideacharacterizationoftheassociatedvaluefunctionasadiscontinuousviscositysolutionofasystemofpartialdifferentialequationswithappropriateboundaryconditions,forw

5、hichweproveacomparisonprinciple.Wealsoproposeanumericalschemefortheresolutionoftheabovesystemandshowthatitisconvergent.Wefinallyprovideasimpleexampleofapplicationtoaproblemofoptimalstocktradingwithanonlinearmarketimpactfunction.Thisshowshowparametersadapttothemarket.Keywords.optimalimpul

6、secontrol,discontinuousviscositysolutions,intradaytradingAMSsubjectclassifications.93E20,49L25,91B28DOI.10.1137/0907772931.Introduction.Tradingalgorithmsarenowadayswidelyspreadamongfinancialagents.Theyaretypicallyusedforhighfrequencyintradaytradingpurposes,e.g.,for“statisticalarbitrage”or

7、fortheexecutionoflargeordersbybrokers.Inbothcases,theuseofrobotsisjustifiedbythefactthatordershavetobeexecutedveryquickly,inordertomakeprofitof“goodprices,”and,typicallyforbrokers,bythelargesizeoftheportfoliostobehandledbyalimitednumberoftraders.Manyeffortshavebeendevotedinrecenty