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1、Buy-lowandsell-highinvestmentstrategiesMihailZervos∗,TimothyC.Johnson†andFaresAlazemiMay28,2011AbstractBuy-lowandsell-highinvestmentstrategiesarearecurrentthemeintheconsidera-tionsofmanyinvestors.Inthispaper,weconsideraninvestorwhoaimsatmaximisingtheexpecteddiscounte
2、dcash-flowthatcanbegeneratedbysequentiallybuyingandsellingoneshareofagivenassetatfixedtransactioncosts.Wemodeltheunderlyingassetpricebymeansofageneralone-dimensionalItˆodiffusionX,wesolvetheresultingstochasticcontrolprobleminaclosedanalyticform,andwecompletelycharacteri
3、setheoptimalstrategy.Inparticular,weshowthat,if0isanaturalboundarypointofX,e.g.,ifXisageometricBrownianmotion,thenitisneveroptimaltosequentiallybuyandsell.Ontheotherhand,weprovethat,if0isanentrancepointofX,e.g.,ifXisamean-revertingconstantelasticityofvariance(CEV)pro
4、cess,thenitmaybeoptimaltosequentiallybuyandsell,dependingontheproblemdata.KeyWords:optimalinvestmentstrategies,optimalswitching,sequentialentryandexitdecisions,variationalinequalities.1IntroductionWeconsideranassetwithpriceprocessXthatismodelledbytheone-dimensionalIt
5、ˆodiffusiondXt=b(Xt)dt+σ(Xt)dWt,X0=x>0,(1)whereWisastandardone-dimensionalBrownianmotion.Aninvestorfollowsastrategythatconsistsofsequentiallybuyingandsellingoneshareoftheasset.Weuseacontrolledfinitevariationc`agl`adprocessYthattakesvaluesin{0,1}tomodeltheinvestor’sposi
6、tioninthemarket.Inparticular,Yt=1(resp.,Yt=0)representsthestatewheretheinvestorholds(resp.,doesnothold)theasset,while,thejumpsofYoccuratthesequenceoftimes(τn,n≥1)∗DepartmentofMathematics,LondonSchoolofEconomics,HoughtonStreet,LondonWC2A2AE,UK,m.zervos@lse.ac.uk†Depar
7、tmentofActuarialMathematicsandStatisticsandtheMaxwellInstituteforMathematicalSci-ences,SchoolofMathematicalandComputerSciences,Heriot-WattUniversity,EdinburghEH144AS,UK,t.c.johnson@hw.ac.uk1atwhichtheinvestorbuysorsells.Givenaninitialcondition(Y0,X0)=(y,x)∈{0,1}×]0,∞
8、[,theinvestor’sobjectiveistoselectastrategyYthatmaximisestheperformancecriterionXnhhi−ΛτJy,x(Y)=limExejHs(Xτ)1{∆Yτ=−1}−Hb(Xτ)1{∆Yτ
