Optimal Execution with Nonlinear Impact Functions and Trading -enhanced risk

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1、OptimalExecutionwithNonlinearImpactFunctionsandTrading-EnhancedRiskRobertF.Almgren¤October2001AbstractWedetermineoptimaltradingstrategiesforliquidationofalargesingle-assetportfoliotominimizeacombinationofvolatilityriskandmarketimpactcosts.Wetakethemarketimpactcostpersharetobeapowerlawfunctio

2、nofthetradingrate,withanarbitrarypositiveexponent.Thisincludes,forexample,thesquare-rootlawthathasbeenproposedbasedonmarketmicrostructuretheory.Inanalogytothelinearmodel,wedefinea“characteristictime”foroptimaltrading,whichnowdependsontheinitialportfoliosizeanddecreasesasexe-cutionproceeds.Wea

3、lsoconsideramodelinwhichuncertaintyoftherealizedpriceisincreasedbydemandingrapidexecution;weshowthatoptimaltrajectoriesaredescribedbya“criticalportfoliosize”abovewhichthiseffectisdominantandbelowwhichitmaybeneglected.Keywords:marketimpact,tradingstrategy,liquiditymodeling.¤UniversityofToronto

4、,DepartmentsofMathematicsandComputerScience,almgren@math.toronto.edu.ResearchsupportedbytheNationalSciencesandEngineeringResearchCouncilofCanadaandbytheUniversityofToronto.1October2001RobertAlmgren:NonlinearOptimalExecution21IntroductionIntheexecutionoflargeportfoliotransactions,atradingstra

5、tegymustbedeterminedthatbalancestheriskofdelayedexecutionagainstthecostofrapidexecution;thechoiceisroughlybetweenan“active”anda“passive”tradingstrategy(HasbrouckandSchwartz1988;WagnerandBanks1992).Severalrecentarticles(AlmgrenandChriss1999;AlmgrenandChriss2000;GrinoldandKahn1999;KonishiandMa

6、kimoto2001)haveconstructedoptimalstrategiesforthisproblem,undertheassumptionthatliquiditycostpersharetradedisalinearfunctionoftradingrateorofblocksize,andthattheonlysourceofuncertaintyinexecutionisthevolatilityoftheunderlyingasset.1Inpractice,linearityoftradingcostsisanunrealisticassumption.

7、Per-oldandSalomon(1991)havearguedthattheliquiditypremiumpersharedemandedbythemarketwillbeeitheraconvexoraconcavefunctionofblocksize,ifthemarket’sperceptionisthatthetraderisinformation-drivenorliquidity-driven,respectively.IntheBarraMarketImpactMode