PortfolioSelectionUnderUncertainty(衍生证券的

Chp.4LifetimePortfolioSelectionUnderUncertaintyHaiLinDepartmentofFinance,XiamenUniversity,3610051.Introduction•Examinethecombinedproblemofoptimalportfolioselectionandconsumptionrulesforindividualinacontinuoustimemodel.•TheratesofreturnaregeneratedbyWienerBrownian-motionprocess.•Particularcase:–Twoassetmodelwithconstantrelativeriskaversionorisoelasticmarginalutility.–Constantabsoluteriskaversion.2.DynamicsoftheModel:TheBudgetEquation•W(t):thetotalwealthattimet;•Xi(t):thepriceofithassetattimet,i=1,2,…,m;•C(t):theconsumptionperunittimeattimet;•wi(t):theproportionoftotalwealthinvestedintheithassetattimet,i=1,2,…,m.miitw11)(Thebudgetequation•Attimet0,theinvestmentbetweent0andt(t0+h)is:•Thevalueofthisinvestmentattimetis:htCtW)()(00miiiimiiiihtCtWtXtXtwtXtXhtCtWtw1000010000))()(()()()()()())()()((miiimiiiihtChtCtWhtgtwhtChtCtWtXtXtwtWtW100000001000)())()(}](1])(){exp[([)())()()](1)()()(([)()(Theprocessofg(t)•Supposeg(t)isthegeometricBrownianmotion.Indiscretetime,•:theexpectedreturnofasseti;•:thevolatilityofasseti;;)2()(2iiiiYhhtgii),0(2hNYiimiiiiihtChtCtWYhtwtWtW1000200)())()(}(1])2){exp[(()()(Momentum)()()()(()()()()()(()())()()()(()())()()(1))(exp(()}()(){(0011200001100001000000hOhtCtWtwhtCtwhtCtWtwhtChtCtWhtwhtChtCtWhtwtWtWtEimiimiiiimiimiiimiii)()(}){()()(})]()(){[(0211000200hOtWYYtEtwtwtWtWtEmimjjijiContinuoustime.)()()()]()()([,)(110miiiimiiiiiidttZtWtwdttCtWtwdWdttZdYmiiihmiiitCtWtwhtWtWtEtWOtChtCtWtwhtWtWtE100000001000000)()()())()()((lim)()1()(])()([)())()()((3.Thetwoassetmodel•:theproportioninvestedintheriskyasset;•:theproportioninvestedinthesureasset.•:thereturnonriskyasset.)(1)(2twtw)()(1twtw)()(1tgtgrtg)(2Yhhtg)2/()(2Twoassetmodel(2))()()))((()(,)()()())()()))(((()()()())(()()(}))()(){(();()]()()))(([()()]()())](1[)([())()()((;)())()(])(1)[exp()](1[}1])2/){exp[((()()(202022002022000000000000000200tCtWrrtwtWdttWtZtwdttCtWrrtwdWhtWtwhOYtEtWtwtWtWtEhOhtCtWrrtwhOhtCtWrtwtwtWtWtEhtChtCtWrhtwYhtwtWtWTheobjectiveproblem0)('';0)(';0)0(;0)(,0)(..)},),(())(()exp({max00CUCUWWtWtCtsTTWBdttCUtETThedynamicprogrammingform•Define•Thentheobjectivefunctioncanbewritten:);),(()),((};),(())(()exp(){(max)),(()(),(TTWBTTWITTWIdssCUstEttWITtsWsC})),(())(()exp({max)0,(00tttWIdssCUsEWIThedynamicprogramming(2)•If,thenbytheMeanValueTheoremandTaylorRule,htt0],[)},()]()([)),((21)]()([)),(()),(()),(())(()){exp(()),((020200200000000},{00maxttthOtWtWWttWItWtWWttWIhtttWIttWIhtCUttEttWIwcThedynamicprogramming(3)•Taketheconditionalexpectationonbothsidesandusethepreviousresults,dividetheequationbyhandtakethelimitas))()(21)}()(]))(({[)]([)(exp(max022222))(),((tWtwWItCtWrrtwWItItCUttttwtc0hThesolution•Define0),;,(max),()(21)}()(]))(({[)()exp(),;,(},{22222tWCwtWtwWItCtWrrtwWItICUttWCwwctttFirstordercondition22*22*****)(0);;,()(')exp(0);;,(Secondordercondition•IfisconcaveinW,•)),((ttWI.0det,0,0.0)(,0)('')exp(,02222WItWCUttwwCCCwwCSummary•Themaximumproblemcanberewrittenas:)),(()),((;0;0;0),;,(**TTWBTTWItWCwwC4.Aspecialcase:constantrelativeriskaversion•Theabovementionednonlinearpartialequationcoupledwithtwoalgebraicequationsisdifficulttosolveingeneral.•Butfortheutilityfunctionwithconstantrelativeriskaversion,theequationscanbesolvedexplicitly.1)('/)(''.0,1,/)1()(CUCCUCCUOptimalityconditions)4......(....................//)()3.......(..........)(,)(0)2........(..........,.........])[exp()()1.....(..............................)exp(,)(')exp(0222*22*2222*22)1/(1*1**WIWIWrtOptimalityconditions(2)22222)1/(22222/)/(2)()1exp()(10),()(21)}()(]))(({[)()exp(0WIWIrrWWItItWItWtwWItCtWrrtwWItICUtttttttttBequestvaluefunction•Theboundaryconditioncancausemajorchangesinthesolution.•meansnobequest.•Aslightlymoregeneralformwhichcanbeusedaswithoutalteringtheresultingsolutionsubstantivelyis0/)]()[exp()),(()),((1TWTTTWBTTWI/)]()[(]),([TWTGTTWBThetrialsolution•Suppose)]()[exp(1)(/)/()]()[exp()()1()]()[exp()()]()[exp()()]()[exp()(,)]()[exp()()),((2222221tWttbWIWItWttbWItWttbtWttbtItWttbWItWttbttWItttttThetrialsolution(2)12*)1/(1*122)1/(}))(exp()1(1{)()1()()()]([)()(],)1(2/)[(,)]()[1()()(vTtvvtbrtwtWtbtCTbrrutbtubtbSufficientconditionforthesolution•bereal(feasibility);•Toensuretheaboveconditions,0)(,0*22tCWIt]),([ttWItTtvTtvv0,0)](exp[)1(1Theoptimalconsumptionandportfolioselectionrules*2**)1()(.0),(1;0),()](exp[)1(1)()(/1)(wrtwvtWtTvtWTtvvvtWtbtCTheBequestvaluationfunction•Theeconomicmotiveisthatthetruefunctionfornobequest•Thenwhen•Thisdoesnotmeantheinfinite