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页数:10页
时间:2018-02-10
《asset allocation in continuous time》由会员上传分享,免费在线阅读,更多相关内容在工程资料-天天文库。
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’sfirstlookatinvestinginjusttwoassets,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?Canwefindastochasticdifferentialequationfortheweal
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+···.Thefinalpartofthesto
8、chasticdifferentialequationisduetothechangeintheriskyasset.Sincetheassetpriceevolvesaccordingto(66.1)andweholdaproportionwofourwealthinthisasset,thereisachangeinourwealthofw(µWdt+σWdX).Notethattheasset
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