POT模型中GPD_厚尾_性及金融风险测度

©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(11;21)(POT),Pareto,01,y2/(1-),,VaRF830159ATailThicknessofGPDinPOTModelandFinanceRiskMeasurementAbstract:Extremevaluemodelisapowerfultooltoaccuratelyestimatemar2ketriskoffattaildistributionofreturnonfinancialassets1Inthisarticle,wean2alyzetherelationbetweentheshapeparameterandthethicknessoftheGPDtailofthePOTmodelanditsapplicationinriskmeasurement1Theresultsshowthatatthetime01,y2/(1-)thethicknessofthedistributiontailisfattailandwiththeincreaseofshapeparametersthickening1Atthistimeitismostsuit2ableformodelingtofattaildistributionofthefinancialassetstimeseriesanditsparameterpsMLE1ManyempiricalstudiesimprovethetheoryanddrawsomenewconclusionsaboutriskfeatureinstockmarketofChina1Keywords:GPD;ShapeParameters;Fat2tail;Value2at2Risk,(BMM)(POT),,BMM,701POTGPD©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(2001)POT,,,Pareto,,BMM,,,POTBMM,,POTPareto(GPD)Pickands1975,Davison(1984),Smith(19841985),VanMontfortWriter(1985),:POT,Zajdenweber(1996),Mc2neil(1997)Resnick(1997)POT,,,(2005)POT,(2002)POT,(2004)POTParetoDupuis(1996),(2008),Pareto0,0y-/,,,ParetoGill(2000)Q-QParetoCoronel2BrizioHF(2005)Ander2son2DarlingPareto(2008)Pare2to,Pareto,0,ParetoPareto,,,0,Pareto1/,k1/,E(xk)(Dacorogna,2001)015,,Pa2retoParetoPOTPareto,POT(POT)X1,X2,,Xn(IID),,F(x),xfF(x),,xf=supx:F(x)1u,uNu,,X1,X2,,XNu,YYi=Xi-u,i=1,2,3,,Nu,80120101©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(CEDF)Fu(y)(,2002;,2008)Fu(y)=P(X-uyXu)=F(u+y)-F(u)1-F(u)=F(x)-F(u)1-F(u):F(x)=1-F(u)Fu(y)+F(u)xu,0yxf-u(1)Balkema,deHaan(1974)Pickands(1975),u,Fu(y)Pareto,:uxf,supFu(y)-G,(y):0yxf-u0Pareto:G,(y)=1-(1+y)-1/01-exp(-y)=0(2),0,0,0y-/;0,y0,,,,ParetoParetoBMM(GEV),,Pareto,11GPD(2)yPareto:f,(y)=1exp(-y)=01(1+y)-1/-100,yGPD:lnf,(y)=-ln-(1+1)ln(1+y)y:f,(y)=5f,(y)5y=f,(y)5lnf,(y)5y=-(1+1)+yf,(y)(3)0,y,GPD:lnfy,()=-ln-(1+1)ln(1+y)901POTGPD©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.()5fy,()5=25lnfy,()5=ln(1+y)-(1+)y/1+yz=(/)y,fy,()0,,H(z)=ln(1+z)-(1+yz)z1+z0H(z)z:5H(z)5z=11+z-(1+)1(1+z)2-1+z=z-z-2(1+z)2=y(1-)-2(1+z)2=y(1-)-2(1+z)2(4)21=0,Pareto=0,Pareto1,,0,Pareto,,015215015,1,,1=0Pareto310,Pareto0,(3)f,(y)0,,260,0,(1)01,Pareto,y2/(1-),(4)H(z)0z0,H(z)H(0)=0,5fy,()/50,Pareto,0y2/(1-),H(z)0z0,H(z)H(0)=0,5fy,()/50,Pareto01,y2/(1-),Pareto,y,GPD,=1,01,GPDfy,(1),=1=0,lim0fy,()=fy,(0)01,GPDfy,(0),=0,01,GPD=0=1,=0,GPD,,0101120101©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=215,0101013401671,MAT2LAB710Pareto22/(1-)510505,715758,1511515,+,,4,y=10,0101,0141,0181,1121,1161,1071694610-23,23,,Pareto(2)1,Pareto1,y2/(1-),y0,,y2/(1-),H(z)0,H(z)0H(z)0,H(z)0,1,zy/,H(z)H(y/)l=y/,H(z)ln(1+l)-2l/(1+l)=G(l);dG(l)dl=l-1(1+l)2,liml0G(l)=0,G(l)1H(z)l0l1,G(l)l,l1,G(l)lG(l)l=1,4MATLABG(l)l0l=319215,G(l)00l319215,0y319215,G(l)0H(z)0,5fy,()/50,Paretol319215,y319215,G(l)0H(z),ParetoH(z)0,H(z)ln(1+z)-(1+yz)=I(z)k=/y=01159,ln(1+z),1+(/y)zk=/y01159,z1z2,5z[z1,z2],I(z)0,H(z)I(z)0111POTGPD©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=1,y31500319215419215519215,MATLAB710Pareto,60y319215,;y319215,y018726810-3,011826410-3,0100219510-3,010000010-3,6410,Pareto0,0,0y-/,Pareto,78,,,21120101©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(1-),H(z)0z0,H(z)H(0)=0,Pareto,:0y-,-10y21-,-10y2/(1-),H(z)0z0,H(z)H(0)=0,Pareto:21-y-,-10,-1:-10,(3)f,(y)0,Pareto0y2/(1-),Pareto2/(1-)y-/,,=1,-0185-0115,0115,MAT2LAB710Pareto7(2/(1-),-/)(110811,111765),(111765,114286),(112903,118182),(114286,215),(116,4)-1,(3)f,(y)0,Pareto0y-/,Pareto,=1,-10-1,2,MATLAB710Pareto8-/011,01125,01167,0125,01551Pareto,Pareto11:=000,Pareto,-1,Pare2to,1,Pareto,POTPareto,,01,y2/(1-),,,0015,311POTGPD©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(0)y=0y010y319215010y2/(1-)-100y2/(1-)-10y-/Pareto1y612893z[z1,z2]01y2/(1-)-102/(1-)y-/ParetoPOTPOT,,e(u)E(X-u|Xu)Pareto,(+u)/(1-),,,(SMEF),^e(u)=mi=1(xi-u)ni/N,Pareto,,,,,,,,(KearnsPagan,1997)Pareto,Pareto:L(,|y)=-nln-(1+1)ni=1ln(1+yi)0-nln-1ni=1yi=0HoskingWallis(1987),-015,,,(1)(2)^G,(y),(,2002;,2007):^F(x)=1-Nun(1+^y^)-1/^01-Nunexp(-y^)=041120101©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(,2002;,2007):VaR=^F-1(1-p)=u+^^[(nNup)-^-1]0u-^ln(nNup)=0(5)()P,LL(Bernoulli)X,P,X1/PL1/Pk=1/PPOT,23610,,19961216199012192007626A()319711420035302GPD[14]1996-12-162004-12-311939[7]1996-12-162000-09-19911[10]1999-05-192007-06-261952[15]1990-12-192003-12-253205[7]1996-12-162000-09-19911[15]1991-04-032003-12-253135[16]1997-01-022002-12-311444[8]1971-01-042003-05-308124,rt=lnPt-lnPt-1=ln[(Pt/Pt-