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1、CHAPTER66assetallocationincontinuoustimeInthisChapter...?thewealthprocess?optimizingoverassetholdingsandconsumption?theBellmanequation?theeffectoftransactioncosts66.1INTRODUCTIONInChapter18welookedatwaysinwhichtoinvestoptimallyassumingthatwecannotchangeourm
2、indslater;thisistheone-periodinvestment.Inpractice,wecanchangeourmindsasoftenaswewantaccordingtothewaythatthefutureunveilsitself.Inthischapterweexaminemodelsforcontinuous-timeinvestmentandportfoliorebalancing.MostofthisistakenfromMerton’swork.Oneoftherestri
3、ctionsonrebalancingourportfolioisthecostoftransacting.Attheendofthechapterwetakeabriefpeekattransactioncosts.66.2ONERISK-FREEANDONERISKYASSETTogettheballrolling,let’s?rstlookatinvestinginjusttwoassets,onerisk-freeearningasurerateofrandtheotherriskyfollowing
4、therandomwalkdS=μSdt+σSdX.(66.1)Wearegoingtoinvestafractionw(t)ofourwealthintheriskyassetandtheremainingfraction1?wintherisk-free.Wewillalsochoosetoconsumeorspend,spend,spend,atarateofC(t).Ouraimistodecidehowtosplitourmoneybetweenthetwoassetsandhowmuchtocon
5、sume;thatiswemustdecideonthebestw(t)andC(t).Obviously,wewillhavetooptimizesomeproblemaswe’llshortlysee.1052PartFiveadvancedtopics66.2.1TheWealthProcessHowdoesourwealthevolveastheassetpriceschangeandweconsume?Canwe?ndastochasticdifferentialequationfortheweal
6、thW?Fromonetimesteptothenextourwealthchangesforthreereasons:TheassetSchangesrandomly;Weconsume;Weearninterestonourcash.Thefractionwonlychangesattheendofthetimestep,notduring(it’sratherliketheintheBlack–Scholesanalysis).Let’sbuildupthestochasticdifferential
7、equationforWinstages.Theeasiestbitistheconsumption.Wealthchangesbytheamountweconsume:dW=?C(t)dt+···.The···representallthetermstocome.Wegetinterestonthecash.Thecashweholdis(1?w)Wandsotheinterestreceivedisr(1?w)Wdt:dW=?C(t)dt+r(1?w)Wdt+···.The?nalpartofthesto
8、chasticdifferentialequationisduetothechangeintheriskyasset.Sincetheassetpriceevolvesaccordingto(66.1)andweholdaproportionwofourwealthinthisasset,thereisachangeinourwealthofw(μWdt+σWdX).Notethattheasset
