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首页 shreve(金融随机分析习题解答)

shreve(金融随机分析习题解答).pdf

shreve(金融随机分析习题解答)

永贞互联网分析
2017-06-08 0人阅读 举报 0 0 暂无简介

简介:本文档为《shreve(金融随机分析习题解答)pdf》,可适用于经济金融领域

StochasticCalculusforFinance,VolumeIandIISolutionofExerciseProblemsYanZengAugust,ContentsStochasticCalculusforFinanceI:TheBinomialAssetPricingModelTheBinomialNoArbitragePricingModelProbabilityTheoryonCoinTossSpaceStatePricesAmericanDerivativeSecuritiesRandomWalkInterestRateDependentAssetsStochasticCalculusforFinanceII:ContinuousTimeModelsGeneralProbabilityTheoryInformationandConditioningBrownianMotionStochasticCalculusRiskNeutralPricingConnectionswithPartialDifferentialEquationsExoticOptionsAmericanDerivativeSecuritiesChangeofNumeacuteraireTermStructureModelsIntroductiontoJumpProcessesThisisasolutionmanualforthetwovolumetextbookStochasticcalculusforfinance,byStevenShreveIfyouhaveanycommentsorfindanytyposerrors,pleaseemailmeatyzcornelleduThecurrentversionomitsthefollowingproblemsVolumeI:,,,VolumeII:,,,ndash,,,AcknowledgmentIthankHuaLi(agraduatestudentatBrownUniversity)forreadingthroughthissolutionmanualandcommunicatingtomeseveralmistakestyposIalsothankHidekiMurakamiforpointingoutatypoinExercise,VolumeIIChapterStochasticCalculusforFinanceI:TheBinomialAssetPricingModelTheBinomialNoArbitragePricingModelProofIfwegettheupsate,thenX=X(H)=∆uS(r)(Xminus∆S)ifwegetthedownstate,thenX=X(T)=∆dS(r)(Xminus∆S)IfXhasapositiveprobabilityofbeingstrictlypositive,thenwemusteitherhaveX(H)orX(T)(i)IfX(H),then∆uS(r)(Xminus∆S)PluginX=,wegetu∆(r)∆Byconditiondru,weconclude∆Inthiscase,X(T)=∆dS(r)(Xminus∆S)=∆Sdminus(r)(ii)IfX(T),thenwecansimilarlydeduce∆andhenceX(H)SowecannothaveXstrictlypositivewithpositiveprobabilityunlessXisstrictlynegativewithpositiveprobabilityaswell,regardlessthechoiceofthenumber∆Remark:HeretheconditionX=isnotessential,asfarasapropertydefinitionofarbitrageforarbitraryXcanbegivenIndeed,fortheoneperiodbinomialmodel,wecandefinearbitrageasatradingstrategysuchthatP(XgeX(r))=andP(XX(r))First,thisisageneralizationofthecaseX=second,itisldquoproperrdquobecauseitiscomparingtheresultofanarbitraryinvestmentinvolvingmoneyandstockmarketswiththatofasafeinvestmentinvolvingonlymoneymarketThiscanalsobeseenbyregardingXasborrowedfrommoneymarketaccountThenattime,wehavetopaybackX(r)tothemoneymarketaccountInsummary,arbitrageisatradingstrategythatbeatsldquosaferdquoinvestmentAccordingly,werevisetheproofofExerciseasfollowsIfXhasapositiveprobabilityofbeingstrictlylargerthanX(r),theeitherX(H)X(r)orX(T)X(r)Thefirstcaseyields∆S(uminusminusr),ie∆SoX(T)=(r)X∆S(dminusminusr)(r)XThesecondcasecanbesimilarlyanalyzedHencewecannothaveXstrictlygreaterthanX(r)withpositiveprobabilityunlessXisstrictlysmallerthanX(r)withpositiveprobabilityaswellFinally,wecommentthattheaboveformulationofarbitrageisequivalenttotheoneinthetextbookFordetails,seeShreve,ExerciseProofX(u)=∆timesGammatimesminus(∆Gamma)=∆Gamma,andX(d)=∆timesminus(∆Gamma)=minus∆minusGammaThatis,X(u)=minusX(d)SoifthereisapositiveprobabilitythatXispositive,thenthereisapositiveprobabilitythatXisnegativeRemark:NotetheaboverelationX(u)=minusX(d)isnotacoincidenceIngeneral,letVdenotethepayoffofthederivativesecurityattimeSupposeXmacrand∆macrarechoseninsuchawaythatVcanbereplicated:(r)(Xmacrminus∆macrS)∆macrS=VUsingthenotationoftheproblem,supposeanagentbeginswithwealthandattimezerobuys∆sharesofstockandGammaoptionsHethenputshiscashpositionminus∆SminusGammaXmacrinamoneymarketaccountAttimeone,thevalueoftheagentrsquosportfolioofstock,optionandmoneymarketassetsisX=∆SGammaVminus(r)(∆SGammaXmacr)PlugintheexpressionofVandsortoutterms,wehaveX=S(∆∆macrGamma)(SSminus(r))Sinced(r)u,X(u)andX(d)haveoppositesignsSoifthepriceoftheoptionattimezeroisXmacr,thentherewillnoarbitrageProofV=rrminusduminusdS(H)uminusminusruminusdS(T)=Srrminusduminusduuminusminusruminusdd=SThisisnotsurprising,sincethisisexactlythecostofreplicatingSRemark:ThisillustratesanimportantpointTheldquofairpricerdquoofastockcannotbedeterminedbytheriskneutralpricing,asseenbelowSupposeS(H)andS(T)aregiven,wecouldhavetwocurrentprices,SandSprimeCorrespondingly,wecangetu,danduprime,dprimeBecausetheyaredeterminedbySandSprime,respectively,itrsquosnotsurprisingthatriskneutralpricingformulaalwaysholds,inbothcasesThatis,S=rminusduminusdS(H)uminusminusruminusdS(T)r,Sprime=rminusdprimeuprimeminusdprimeS(H)uprimeminusminusruprimeminusdprimeS(T)rEssentially,thisisbecauseriskneutralpricingreliesonfairprice=replicationcostStockasareplicatingcomponentcannotdetermineitsownldquofairrdquopriceviatheriskneutralpricingformulaProofXn(T)=∆ndSn(r)(Xnminus∆nSn)=∆nSn(dminusminusr)(r)Vn=Vn(H)minusVn(T)uminusd(dminusminusr)(r)ptildeVn(H)qtildeVn(T)r=ptilde(Vn(T)minusVn(H))ptildeVn(H)qtildeVn(T)=ptildeVn(T)qtildeVn(T)=Vn(T)ProofThebankrsquostradershouldsetupareplicatingportfoliowhosepayoffistheoppositeoftheoptionrsquospayoffMoreprecisely,wesolvetheequation(r)(Xminus∆S)∆S=minus(SminusK)ThenX=minusand∆=minusThismeansthetradershouldsellshortshareofstock,puttheincomeintoamoneymarketaccount,andthentransferintoaseparatemoneymarketaccountAttimeone,theportfolioconsistingofashortpositioninstockand(r)inmoneymarketaccountwillcanceloutwiththeoptionrsquospayoffThereforeweendupwith(r)intheseparatemoneymarketaccountRemark:ThisproblemillustrateswhyweareinterestedinhedgingalongpositionIncasethestockpricegoesdownattimeone,theoptionwillexpirewithoutanypayoffTheinitialmoneywepaidattimezerowillbewastedByhedging,weconverttheoptionbackintoliquidassets(cashandstock)whichguaranteesasurepayoffattimeoneAlso,cfpage,paragraphAstowhywehedgeashortposition(asawriter),seeWilmott,pageProofTheideaisthesameasProblemThebankrsquostraderonlyneedstosetupthereverseofthereplicatingtradingstrategydescribedinExampleMoreprecisely,heshouldshortsellshareofstock,investtheincomeintomoneymarketaccount,andtransferintoaseparatemoneymarketaccountTheportfolioconsistingashortpositioninstockandinmoneymarketaccountwillreplicatetheoppositeoftheoptionrsquospayoffAftertheycancelout,weendupwith(r)intheseparatemoneymarketaccount(i)Proofvn(s,y)=(vn(s,ys)vn(s,ys))(ii)Proof(iii)Proofdeltan(s,y)=vn(us,yus)minusvn(ds,yds)(uminusd)s(i)ProofSimilartoTheorem,butreplacer,uanddeverywherewithrn,unanddnMoreprecisely,setptilden=rnminusdnunminusdnandqtilden=minusptildenThenVn=ptildenVn(H)qtildenVn(T)rn(ii)Proof∆n=Vn(H)minusVn(T)Sn(H)minusSn(T)=Vn(H)minusVn(T)(unminusdn)Sn(iii)Proofun=Sn(H)Sn=SnSn=Snanddn=Sn(T)Sn=SnminusSn=minusSnSotheriskneutralprobabilitiesattimenareptilden=minusdnunminusdn=andqtilden=RiskneutralpricingimpliesthepriceofthiscallattimezeroisProbabilityTheoryonCoinTossSpace(i)ProofP(Ac)P(A)=AcP()AP()=ΩP()=(ii)ProofByinduction,itsufficestoworkonthecaseN=WhenAandAaredisjoint,P(AA)=AAP()=AP()AP()=P(A)P(A)WhenAandAarearbitrary,usingtheresultwhentheyaredisjoint,wehaveP(AA)=P((AminusA)A)=P(AminusA)P(A)leP(A)P(A)(i)ProofPtilde(S=)=ptilde=,Ptilde(S=)=ptildeqtilde=,Ptilde(S=)=ptildeqtilde=,andPtilde(S=)=qtilde=(ii)ProofEtildeS=Ptilde(S=)Ptilde(S=)=ptildeqtilde=,EtildeS=ptildemiddotptildeqtildemiddotqtilde=,andEtildeS=middotmiddotmiddotmiddot=SotheaverageratesofgrowthofthestockpriceunderPtildeare,respectively:rtilde=minus=,rtilde=minus=andrtilde=minus=(iii)ProofP(S=)=()=,P(S=)=middot()middot=,P(S=)=middot=,andP(S=)=Accordingly,ES=,ES=andES=SotheaverageratesofgrowthofthestockpriceunderPare,respectively:r=minus=,r=minus=,andr=minus=ProofApplyconditionalJensenrsquosinequality(i)ProofEnMn=MnEnXn=MnEXn=Mn(ii)ProofEnSnSn=EnesigmaXnesigmaeminussigma=esigmaeminussigmaEesigmaXn=(i)ProofIn=nminusj=Mj(MjminusMj)=nminusj=MjMjminusnminusj=Mjminusnminusj=Mj=nminusj=MjMjMnminusnminusj=Mjminusnminusj=Mj=Mnminusnminusj=(MjminusMj)=Mnminusnminusj=Xj=Mnminusn(ii)ProofEnf(In)=Enf(InMn(MnminusMn))=Enf(InMnXn)=f(InMn)f(InminusMn)=g(In),whereg(x)=f(xxn)f(xminusxn),sinceInn=|Mn|ProofEnInminusIn=En∆n(MnminusMn)=∆nEnMnminusMn=ProofWedenotebyXntheresultofnthcointoss,whereHeadisrepresentedbyX=andTailisrepresentedbyX=minusWealsosupposeP(X=)=P(X=minus)=DefineS=XandSn=Snbn(X,middotmiddotmiddot,Xn)Xn,wherebn(middot)isaboundedfunctionon{minus,}n,tobedeterminedlateronClearly(Sn)ngeisanadaptedstochasticprocess,andwecanshowitisamartingaleIndeed,EnSnminusSn=bn(X,middotmiddotmiddot,Xn)EnXn=Foranyarbitraryfunctionf,Enf(Sn)=f(Snbn(X,middotmiddotmiddot,Xn))f(Snminusbn(X,middotmiddotmiddot,Xn))Thenintuitively,Enf(SncannotbesolelydependentuponSnwhenbnrsquosareproperlychosenThereforeingeneral,(Sn)ngecannotbeaMarkovprocessRemarkIfXnisregardedasthegainlossofnthbetinagamblinggame,thenSnwouldbethewealthattimenbnisthereforethewagerforthe(n)thbetandisdevisedaccordingtopastgamblingresults(i)ProofNoteMn=EnMNandMprimen=EnMprimeN(ii)ProofIntheproofofTheorem,weprovedbyinductionthatXn=VnwhereXnisdefinedby()ofChapterInotherwords,thesequence(Vn)lenleNcanberealizedasthevalueprocessofaportfolio,whichconsistsofstockandmoneymarketaccountsSince(Xn(r)n)lenleNisamartingaleunderPtilde(Theorem),(Vn(r)n)lenleNisamartingaleunderPtilde(iii)ProofVprimen(r)n=EnVN(r)N,soVprime,Vprimer,middotmiddotmiddot,VprimeNminus(r)Nminus,VN(r)NisamartingaleunderPtilde(iv)ProofCombine(ii)and(iii),thenuse(i)(i)Proofu=S(H)S=,d=S(H)S=,u(H)=S(HH)S(H)=,d(H)=S(HT)S(H)=,u(T)=S(TH)S(T)=andd(T)=S(TT)S(T)=Soptilde=rminusduminusd=,qtilde=,ptilde(H)=r(H)minusd(H)u(H)minusd(H)=,qtilde(H)=,ptilde(T)=r(T)minusd(T)u(T)minusd(T)=,andqtilde(T)=ThereforePtilde(HH)=ptildeptilde(H)=,Ptilde(HT)=ptildeqtilde(H)=,Ptilde(TH)=qtildeptilde(T)=andPtilde(TT)=qtildeqtilde(T)=TheproofsofTheorem,TheoremandTheoremstillworkfortherandominterestratemodel,withpropermodifications(iePtildewouldbeconstructedaccordingtoconditionalprobabilitiesPtilde(n=H|,middotmiddotmiddot,n):=ptildenandPtilde(n=T|,middotmiddotmiddot,n):=qtildenCfnotesonpage)SothetimezerovalueofanoptionthatpaysoffVattimetwoisgivenbytheriskneutralpricingformulaV=EtildeV(r)(r)(ii)ProofV(HH)=,V(HT)=,V(TH)=andV(TT)=SoV(H)=ptilde(H)V(HH)qtilde(H)V(HT)r(H)=,V(T)=ptilde(T)V(TH)qtilde(T)V(TT)r(T)=,andV=ptildeV(H)qtildeV(T)r(iii)Proof∆=V(H)minusV(T)S(H)minusS(T)=minusminus=minus(iv)Proof∆(H)=V(HH)minusV(HT)S(HH)minusS(HT)=minusminus=(i)ProofEtildenXn(r)n=Etilden∆nYnSn(r)n(r)(Xnminus∆nSn)(r)n=∆nSn(r)nEtildenYnXnminus∆nSn(r)n=∆nSn(r)n(uptildedqtilde)Xnminus∆nSn(r)n=∆nSnXnminus∆nSn(r)n=Xn(r)n(ii)ProofFrom(),wehave{∆nuSn(r)(Xnminus∆nSn)=Xn(H)∆ndSn(r)(Xnminus∆nSn)=Xn(T)So∆n=Xn(H)minusXn(T)uSnminusdSnandXn=EtildenXnrTomaketheportfolioreplicatethepayoffattimeN,wemusthaveXN=VNSoXn=EtildenXN(r)Nminusn=EtildenVN(r)NminusnSince(Xn)lenleNisthevalueprocessoftheuniquereplicatingportfolio(uniquenessisguaranteedbytheuniquenessofthesolutiontotheabovelinearequations),thenoarbitragepriceofVNattimenisVn=Xn=EtildenVN(r)Nminusn(iii)ProofEtildenSn(r)n=(r)nEtilden(minusAn)YnSn=Sn(r)nptilde(minusAn(H))uqtilde(minusAn(T))dSn(r)nptildeuqtilded=Sn(r)nIfAnisaconstanta,thenEtildenSn(r)n=Sn(r)n(minusa)(ptildeuqtilded)=Sn(r)n(minusa)SoEtildenSn(r)n(minusa)n=Sn(r)n(minusa)n(i)ProofFNPN=SNminusK(KminusSN)=(SNminusK)=CN(ii)ProofCn=EtildenCN(r)Nminusn=EtildenFN(r)NminusnEtildenPN(r)Nminusn=FnPn(iii)ProofF=EtildeFN(r)N=(r)NEtildeSNminusK=SminusK(r)N(iv)ProofAttimezero,thetraderhasF=SinmoneymarketaccountandoneshareofstockAttimeN,thetraderhasawealthof(FminusS)(r)NSN=minusKSN=FN(v)ProofBy(ii),C=FPSinceF=Sminus(r)NS(r)N=,C=P(vi)ProofBy(ii),Cn=PnifandonlyifFn=NoteFn=EtildenSNminusK(r)Nminusn=Snminus(r)NS(r)Nminusn=SnminusS(r)nSoFnisnotnecessarilyzeroandCn=PnisnotnecessarilytrueforngeProofFirst,thenoarbitragepriceofthechooseroptionattimemmustbemax(C,P),whereC=Etilde(SNminusK)(r)Nminusm,andP=Etilde(KminusSN)(r)NminusmThatis,CisthenoarbitragepriceofacalloptionattimemandPisthenoarbitragepriceofaputoptionattimemBothofthemhavematuritydateNandstrikepriceKSupposethemarketisliquid,thenthechooseroptionisequivalenttoreceivingapayoffofmax(C,P)attimemTherefore,itscurrentnoarbitragepriceshouldbeEtildemax(C,P)(r)mBytheputcallparity,C=SmminusK(r)NminusmPSomax(C,P)=P(SmminusK(r)Nminusm)Therefore,thetimezeropriceofachooseroptionisEtildeP(r)mEtilde(SmminusK(r)Nminusm)(r)m=Etilde(KminusSN)(r)NEtilde(SmminusK(r)Nminusm)(r)mThefirsttermstandsforthetimezeropriceofaput,expiringattimeNandhavingstrikepriceK,andthesecondtermstandsforthetimezeropriceofacall,expiringattimemandhavingstrikepriceK(r)NminusmIfwefeelunconvincedbytheaboveargumentthatthechooseroptionrsquosnoarbitragepriceisEtildemax(C,P)(r)m,duetotheeconomicalargumentinvolved(likeldquothechooseroptionisequivalenttoreceivingapayoffofmax(C,P)attimemrdquo),thenwehavethefollowingmathematicallyrigorousargumentFirst,wecanconstructaportfolio∆,middotmiddotmiddot,∆mminus,whosepayoffattimemismax(C,P)Fix,ifC()P(),wecanconstructaportfolio∆primem,middotmiddotmiddot,∆primeNminuswhosepayoffattimeNis(SNminusK)ifC()P(),wecanconstructaportfolio∆primeprimem,middotmiddotmiddot,∆primeprimeNminuswhosepayoffattimeNis(KminusSN)Bydefining(mlekleNminus)∆k()={∆primek()ifC()P()∆primeprimek()ifC()P(),wegetaportfolio(∆n)lenleNminuswhosepayoffisthesameasthatofthechooseroptionSothenoarbitragepriceprocessofthechooseroptionmustbeequaltothevalueprocessofthereplicatingportfolioInparticular,V=X=EtildeXm(r)m=Etildemax(C,P)(r)m(i)ProofNoteunderbothactualprobabilityPandriskneutralprobabilityPtilde,cointossesnrsquosareiidSowithoutlossofgenerality,weworkonPForanyfunctiong,Eng(Sn,Yn)=Eng(SnSnSn,YnSnSnSn)=pg(uSn,YnuSn)qg(dSn,YndSn),whichisafunctionof(Sn,Yn)So(Sn,Yn)lenleNisMarkovunderP(ii)ProofSetvN(s,y)=f(yN)ThenvN(SN,YN)=f(Nn=SnN)=VNSupposevnisgiven,thenVn=EtildenVnr=Etildenvn(Sn,Yn)r=rptildevn(uSn,YnuSn)qtildevn(dSn,YndSn)=vn(Sn,Yn),wherevn(s,y)=vtilden(us,yus)vtilden(ds,yds)r(i)ProofFornleM,(Sn,Yn)=(Sn,)SincecointossesnrsquosareiidunderPtilde,(Sn,Yn)lenleMisMarkovunderPtildeMoreprecisely,foranyfunctionh,Etildenh(Sn)=ptildeh(uSn)htilde(dSn),forn=,,middotmiddotmiddot,MminusForanyfunctiongoftwovariables,wehaveEtildeMg(SM,YM)=EtildeMg(SM,SM)=ptildeg(uSM,uSM)qtildeg(dSM,dSM)AndforngeM,Etildeng(Sn,Yn)=Etildeng(SnSnSn,YnSnSnSn)=ptildeg(uSn,YnuSn)qtildeg(dSn,YndSn),so(Sn,Yn)lenleNisMarkovunderPtilde(ii)ProofSetvN(s,y)=f(yNminusM)ThenvN(SN,YN)=f(NK=MSkNminusM)=VNSupposevnisalreadygivena)IfnM,thenEtildenvn(Sn,Yn)=ptildevn(uSn,YnuSn)qtildevn(dSn,YndSn)Sovn(s,y)=ptildevn(us,yus)qtildevn(ds,yds)b)Ifn=M,thenEtildeMvM(SM,YM)=ptildevM(uSM,uSM)vtilden(dSM,dSM)SovM(s)=ptildevM(us,us)qtildevM(ds,ds)c)IfnM,thenEtildenvn(Sn)=ptildevn(uSn)qtildevn(dSn)Sovn(s)=ptildevn(us)qtildevn(ds)StatePricesProofNoteZtilde():=P()Ptilde()=Z()ApplyTheoremwithP,Ptilde,ZreplacedbyPtilde,P,Ztilde,wegettheanalogousofproperties(i)(iii)ofTheorem(i)ProofPtilde(Ω)=ΩPtilde()=ΩZ()P()=EZ=(ii)ProofEtildeY=ΩY()Ptilde()=ΩY()Z()P()=EYZ(iii)ProofPtilde(A)=AZ()P()SinceP(A)=,P()=foranyASoPtilde(A)=(iv)ProofIfPtilde(A)=AZ()P()=,byP(Z)=,weconcludeP()=foranyASoP(A)=AP()=(v)ProofP(A)=lArrrArrP(Ac)=lArrrArrPtilde(Ac)=lArrrArrPtilde(A)=(vi)ProofPicksuchthatP(),defineZ()={,if=P(),if=ThenP(Zge)=andEZ=P()middotP()=ClearlyPtilde(Ω{})=EZΩ{}==Z()P()=ButP(Ω{})=minusP()ifP()HenceinthecaseP(),PandPtildearenotequivalentIfP()=,thenEZ=ifandonlyifZ()=InthiscasePtilde()=Z()P()=AndPtildeandPhavetobeequivalentInsummary,ifwecanfindsuchthatP(),thenZasconstructedabovewouldinduceaprobabilityPtildethatisnotequivalenttoP(i)ProofZ(HH)=,Z(HT)=,Z(TH)=andZ(TT)=(ii)ProofZ(H)=EZ(H)=Z(HH)P(=H|=H)Z(HT)P(=T|=H)=Z(T)=EZ(T)=Z(TH)P(=H|=T)Z(TT)P(=T|=T)=(iii)ProofV(H)=Z(HH)V(HH)P(=H|=H)Z(HT)V(HT)P(=T|=T)Z(H)(r(H))=,V(T)=Z(TH)V(TH)P(=H|=T)Z(TT)V(TT)P(=T|=T)Z(T)(r(T))=,andV=Z(HH)V(HH)()()P(HH)Z(HT)V(HT)()()P(HT)Z(TH)V(TH)()()P(TH)ProofUprime(x)=x,soI(x)=x()givesEZ(r)N(r)NlambdaZ=XSolambda=XBy(),wehaveXN=(r)NlambdaZ=XZ(r)NHenceXn=EtildenXN(r)Nminusn=EtildenX(r)nZ=X(r)nEtildenZ=X(r)nZnEnZmiddotZ=Xn,wherethesecondtolastldquo=rdquocomesfromLemmaProofUprime(x)=xpminusandsoI(x)=xpminusBy(),wehaveEZ(r)N(lambdaZ(r)N)pminus=XSolveitforlambda,wegetlambda=XEZppminus(r)Nppminuspminus=Xpminus(r)Np(EZppminus)pminusSoby(),XN=(lambdaZ(r)N)pminus=lambdapminusZpminus(r)Npminus=X(r)NppminusEZppminusZpminus(r)Npminus=(r)NXZpminusEZppminus(i)Proofddx(U(x)minusyx)=Uprime(x)minusySox=I(y)isanextremepointofU(x)minusyxBecauseddx(U(x)minusyx)=Uprimeprime(x)le(Uisconcave),x=I(y)isamaximumpointThereforeU(x)minusy(x)leU(I(y))minusyI(y)foreveryx(ii)ProofFollowingthehintoftheproblem,wehaveEU(XN)minusEXNlambdaZ(r)NleEU(I(lambdaZ(r)N))minusElambdaZ(r)NI(lambdaZ(r)N),ieEU(XN)minuslambdaXleEU(XlowastN)minusEtildelambda(r)NXlowastN=EU(XlowastN)minuslambdaXSoEU(XN)leEU(XlowastN)(i)ProofXn=EtildenXN(r)NminusnSoifXNge,thenXngeforalln(ii)Proofa)Iflexga

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