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1、AHamiltonJacobiBellmanApproachtoOptimalTradeExecution1PeterA.Forsythy23August11,20104Abstract5Theoptimaltradeexecutionproblemisformulatedintermsofamean-variancetradeo ,asseen6attheinitialtime.Themean-varianceproblemcanbeembeddedinaLinear-Quadratic(L
2、Q)optimal7stochasticcontrolproblem,Asemi-Lagrangianschemeisusedtosolvetheresultingnon-linearHamilton8JacobiBellman(HJB)PDE.Thismethodisessentiallyindependentoftheformforthepriceimpact9functions.Providedastrongcomparisonpropertyholds,weprovethatthenum
3、ericalschemeconverges10totheviscositysolutionoftheHJBPDE.Numericalexamplesarepresentedintermsoftheecient11tradingfrontierandthetradingstrategy.Thenumericalresultsindicatethatinsomecasesthereare12manydi erenttradingstrategieswhichgeneratealmostidenti
4、calecientfrontiers.1314Keywords:Optimalexecution,mean-variancetradeo ,HJBequation,semi-Lagrangiandiscretization,15viscositysolution1617AMSClassi cation65N06,93C2018RunningTitle:AnHJBApproachtoOptimalTrading191Introduction20Alargeinstitutionalinvesto
5、r,whensellingalargeblockofshares,isfacedwiththefollowingdilemma.Ifthe21investortradesrapidly,thentheactualcashreceivedfromthesalewillbelessthananticipated,duetothe22marketimpactofthetrades.Marketimpactcanbeminimizedbybreakingupalargetradeintoanumber2
6、3ofsmallerblocks.However,inthiscase,theinvestorisexposedtotheriskofpricedepreciationduringthe24tradinghorizon.25Recently,therehasbeenconsiderableinterestinalgorithmictradingstrategies.Theseareautomated26strategiesforexecutionoftradeswiththeobjectiveo
7、fmeetingpre-determinedoptimalitycriteria[14,15].27Inthiswork,weconsideranidealizedmodelforpriceimpact.Inthecaseofsellingshares,themarket28pricewilldecreaseasafunctionofthetradingrate,whileatthesametimefollowingastochasticprocess.The29optimalcontrolpr
8、oblemisthentoliquidatetheportfoliooversome xedtime,andmaximizetheexpected30cashreceiptswhileminimizingthevarianceoftheoutcome[9,1,2,26,16,28].31Analternativeapproachistoposethisproblemintermsofmaximizingapower-laworexponentialutility32function[21,32,
