金融研究中的数学方法

12006.9.282311990H.MarkowitzW.SharpeM.Miller——H.MarkowitzW.SharpeM.Miller721997R.MertonM.Schole,F.BlackBlack-SholesBlack-Sholes1973R.MertonBlack-SholesBlack-SholesMertonBlack-Sholes1032003RobertF.EngleCliveGrangerEngle1982ARCHGranger,,,,,Engle1982ARCH,2070,,,,,,13„„Ross„„„Black-Scholes„ARCH„„„141.µMarkowitz()µσ===Σ=wXEXEwtswwTpTTwp)()(11..minmin2w*()∆+−=cbapµµσ2222111),()(),(1,11bacXEXEcXEbaTTT−=∆Σ=Σ=Σ=−−−16-)(pXE2pσ17-)(pXEpσ18wpdgpwAAww)1(*−+=∆−=)(abacAµ11111−−ΣΣ=Tgw)(1)(11XEXEwTd−−ΣΣ=19-()rwrXEtswwTTwp−=−Σ=µσ)1)((..minmin2()()12222−+−−=arrbcrpµσ-0r()2122arrbc+−±σ)(pXEarrbcr22+−+=σµ()∆+−=cbapµµσ222tw“”212.CAPM,(Sharp-Lintner-Monssin)))(()(fMifirXErXE−=−β)(),(MMiiXVarXXCov=βXbeta,XXM——npii=0,1,…,n,KkiNik∑∑∑===⎟⎟⎠⎞⎜⎜⎝⎛=niiKkkiKkikimipNpNw011()mnmmm=wt23CAPMXi))(,(iiXEβ)(MXE))(()(fMifirXErXE−+=β)(XEβfrAB124CAPMCAPMjPeP0()00PPPXej−=),(1)(0MjeXXCovrPEPλ++=)()(MMXVarrXE−=λ),(1),()(000,MeMeMjXPCovPXPPPCovXXCov=−=rXPCovPEPMee+−=1),()(0λ26CAPM———I1100I2VI2VI2$1000$800P0.50.52.0)(,09.0)(,1.0,045.0),(====MMfMIXEXVarrXXCov6.782045.009.01.02.01.015.08005.01000)1(=×−++×+×=IV7.83$273.RossAPTCAPM“”GAPTRossAPT“”“”Ross1976JournalofEconomicTheoryArbitrageTheoryofCapitalAssertPricing“”“”iXifk,k=1,…,K,εi,iKkkikiifbaXε++=∑=1()(3.1-a)()()()()0=====mkkijikiffEfEEfEEεεεε(3.1-b)()()1,2222==kiifESSEεi(3.1-c)εikKkkibrXE∑==−10(λλkfk(3.2-c)APTCAPMAPTCAPM12345314.212————YtY0=0,⎩⎨⎧≥0}0{0TTYPYurd+1001)(uSHS=HT0S01)(dSTS=K01V0=?+−=)(11KSVV0,V00∆01))(1(000101SVrSV∆−++∆=1))(1()()(000101SVrHSHV∆−++∆=))(1()()(000101SVrTSTV∆−++∆=⎥⎦⎤⎢⎣⎡−+−+−−++=−−=∆)()1()(111,)()()()(11011110TVduruHVdudrrVTSHSTVHV*0∆pduruqdudrp~1)1(~,1~−=−+−=−−+=*][~11)](~)(~[111110VErTVqHVprV+=++=)~,~(qp“”35SuSdSSu2udSSd2Su3Sd3Sud2dSu2Su4Sd4dSu3Sdu22Sud3Stockprice…,…+−=)(KSVnn365.Black-ScholesBlack-Scholes⎪⎩⎪⎨⎧==∈+=∈=pPpPTttdBtPdttbPtdPTtdttrPtdP)0()0(],0[)()()()(],0[)()(,0000σTV(T)=g(P(T));0V(0);0X(0)=V(0)t∆(t),dttPttXrtdPttdX)]()()([)()()(∆−+∆=)()()()]())(()([tdBttPdttPrbttrX∆+−∆+=σ(5.1)V(t,x)tx,dBPVdtVPbPVVtdPVdxVdtVxtdVxxxxtxxxtσσ+++=++=]21[))((21),(222(5.2)38V(0,P(0))=X(0),V(t,P(t))=X(t),g(P(T))=X(T)(4.1),(4.2)dt,dB))(,(tPtVxt=∆),(),(21),(),(22xtrVxtVxxtrxVxtVxxxt=++σ)(),(xgxTV=(5.3)——Black-Scholes39tQ~)(tPert−]|))(([~),(txTPgExtV=(5.4)(5.3)(5.4))()(),(2)(1dKedxxtVtTrΦ−Φ=−−;))(21()log(21tTtTrKxd−−++=σσtTdd−−=σ12Black-ScholesBlack-Scholes40„„„„41(0PT)5436ARCH,,,,,,,ARMAARIMAMARKOV,Engle1982ARCHARCH20,ARCH45EngleARCH(1){Yt}1,,1∀++=−ϕεϕµtYYttttεtEtt∀=−0)|(1εε(6.1)tttuac++=−212εε(6.2))(tuu=(6.1)-(6.2)hthhhttYYYE−−+−−=ϕϕϕµ11)|((6.3)(6.4)211)|(−−+=tttacYYVarε(6.3)(6.4)47GARCHARCH(1),Engle,,(),,,EngleARCH,ARCH-M,48ARCH(1)•ARCH(q)Engle1982)0)|(212=++=−−=∑itttitqiitEuacεεεε•GARCH(p,q)(Bollerslev1986)⎪⎩⎪⎨⎧++===∑∑=−=−−−pjjtjqiititiiiihchVarE121211)|(0)|(βεαεεεε•GARCH-M(Engle,Lilien,Robbins1987)ttttthbXYεεδ++=GARCH[1]EngleRobertF.AutoregressionconditionalheteroskedasticitywithestimatesofthevarianceofU.K.inflation,Econometrica,1982,50(4):987_1008[2]ChristainGARCHModelsandFinancialApplicationsSpringer,1997[3]T.Bollerslev.Generalizedautoregressiveconditionalheteroskedasticity,JournalofEconometrics31,307-327,(1986).507Cox-Ingersoll-RossHull-White-Vasicek“”“”528121.2.3.4.5.MillerMillerMiller1990559