基于GARCH模型的VaR方法

GARCHVaR*GARCHVaRTGEDGARCHValueatRisk;GED;GARCHJ.P.MorgenVaR[1]JorionVaR[2]VaRVaR,HistoricalSimulationMonteCarloSimulationNormal,T-distribution,GEDGARCHVaRVaRVaRGARCH()VaRVaR,,E,a,VaR,)][(0arrEWVaR−=arrPa−=1)(aZ0War][rar),(~2tNrt∆σµt∆−µrZaa=σ−a,arttZraa∆+∆−=µσt∆σZWa0rrEWa=−=0)][(VaRVaRGARCHVaRthtathZpVaR1−t=t-1Za1−tpa*GARCH(GARCH-MEGARCH-MLGARCH)GARCH1∑=++=kitiitxbar1εttthv=ε2hvhvNelson1991Hamilton(1994)GeneralizederrordistributionGEDttGEDttttvtt2/)1(22/1))2/(1()2/(])2[()2/)1((),(+−−+Γ−+Γ=ddxddddxfπdGED∞≤Γ−=+ddvdvfdddtt0;)/1(2/21exp)(]/)1[(λλGammad•Γdzezxzx−∞−∫=Γ01)(λλ2/1)/2)/3()/1(2ΓΓ=−dddλtail-thicknessparameterd=2GEDd2GEDd2GEDthGARCHGARCH(p,q)∑∑=−=−++=qjjtjpiitithch121εβαthGARCH(p,q)Nelson(1991)EGARCHEGARCHp,q∑∑=−=−++=qiitipiititGhch11)ln(exp[βαttttthLhGεπε−−=/2/tttttthLddddhGεε−+Γ−ΓΓ−−=)2/)1(()1()2/1()2/(22/GEDttddttthLddddhεε−ΓΓ×ΓΓ−=−)/1()/2()/3()/1(22/)/2)/1(GLEGARCHdtGEDLeverageGARCH(LGARCH)LGARCHGlostenJagannathanRunkel1993th21121*−−−=−+++=∑∑ttqiitipiititILevhchεεβαLevIh≥=0,001tεεttIth1t∑=+⋅++=kittiithmxbar1ε3GARCH-MEGARCH-MLGARCH-MGARCH-MEGARCH-MLGARCH-M19961216200152319961216T+1GARCH(1,1)-MEGARCH(1,1)-MLGARCH(1,1)-MMLEtGEDtGEDEGARCH-MLLGARCH-MLevtGEDttthMArε+⋅+=1GARCH-MnormaldistributionA1CQ1P1M-2.2230e-3(-1.74)2.1930e-5(6.70)0.2627(10.58)0.6870(29.62)0.2049(2.11)-1.5496e-3(-1.18)1.5111e-5(4.95509)0.2049(7.73)0.7561(27.41)0.1501(1.56)GARCH-MtdistributionA1CQ1P1MD-0.0006(-0.62)1.0000e-5(3.20)0.1985(4.72)0.7717(20.53)0.1459(1.75)4.2958(7.823)-0.0010(-0.91)1.3537e-5(3.22)0.2225(5.09)0.7505(19.56)0.1554(1.90)5.4653(6.44)GARCH-MGEDdistributionA1CQ1P1MD-0.0013(-1.23)1.5000e-5(3.51)0.1988(4.84)0.7550(18.65)0.1951(2.34)1.1776(23.28)-0.0009(-0.86)1.4657e-5(3.37)0.2161(5.07)0.7469(18.31)0.1581(1.90)1.3112(21.04)EGARCH-MnormaldistributionA1CQ1P1ML1-0.0015(-1.30)-0.6136(-5.72)0.3776(10.18)0.9244(72.16)0.1437(1.60)6.5966e-5(0.001)-0.0015(-1.30)-0.6133(-5.72)0.3775(10.18)0.9245(72.19)0.1436(1.60)4.5824e-5(-0.01)EGARCH-MtdistributionA1CQ1P1ML1D0.0001(0.15)-0.3971(-2.46)0.3532(6.49)0.9356(51.54)0.0774(1.01)0.1695(1.78)4.3798(7.81)-0.0003(-0.36)-0.4128(-2.81)0.3752(6.86)0.9440(60.27)0.0959(1.33)0.1445(1.72)5.6006(6.33)EGARCH-MGEDdistributionA1CQ1P1ML1D-0.0007(-0.75)-0.5627(-3.53)0.3420(6.06)0.9341(50.00)0.1396(1.79)0.0866(0.98)1.1916(22.26)-0.0003(-0.33)-0.5021(-3.76)0.3683(6.65)0.9407(59.27)0.0970(1.31)0.1071(1.39)1.3311(20.35)LGARCH-MNormaldistributionA1CQ1P1MLEV-2.2232e-3(-1.74)2.1989e-5(6.48)0.2601(9.54)0.6868(27.88)0.2041(2.09)4.6079e-3(0.12)-1.4305e-3(-1.09)1.5266e-5(5.34)0.1705(5.92)0.7579(28.37)0.1302(1.35)0.0586(1.99)LGARCH-MtdistributionA1CQ1P1MLEVD-0.0005(-0.46)1.0425e-5(3.39)0.1590(3.29)0.7599(20.09)0.1252(1.50)0.0917(1.47)4.3032(7.89)-0.0008(-0.73)1.3958(3.43)0.1776(3.49)0.7460(20.07)0.1311(1.61)0.0878(1.45)5.4991(6.49)LGARCH-MGEDdistributionA1CQ1P1MLEVD-0.0012(-1.13)1.5518e-5(3.61)0.1744(3.65)0.7520(18.77)0.1817(2.17)0.4679(0.85)1.1767(22.54)-0.0008(-0.78)1.5067(3.59)0.1805(3.70)0.7442(18.81)(0.1433)(1.72)0.0660(1.23)1.3137(21.10)VaRtatthZp1−=VaRt-1Za12Noramal,t,GED95%99%395%VaR499%VaRVaRVaR95%tGEDVaRVaR95%tGEDVaR99%tGEDVaRVaR99%tGEDVaRVaRVaRVaR1−tpa1Noramal,t,GED95%99%GARCH-MEGARCH-MLGARCH-M4.29584.37984.30321.5231.5361.528Dt95%99%2.6472.6322.6431.17761.19161.17671.6451.6461.645DGED95%99%2.6492.6352.64995%99%1.6452.3272Noramal,t,GED95%99%GARCH-MEGARCH-MLGARCH-M5.46535.60065.49911.5731.5781.574Dt95%99%2.5862.5812.5851.31121.33111.31371.6501.6521.650DGED95%99%2.5742.5712.57395%99%1.6452.327395%VaRGarch-mGarch-mt.distGarch-mGed.distEgarch-mEgarch-mt.distEgarch-mGed.distLgarch-mLgarch-mt.distLgarch-mGed.distSH37.5034.6436.7036.7934.4236.1737.5034.8136.73SZ39.0537.5039.0138.4937.0838.4639.0737.5939.06(SHSZ)499%VaRGarch-mGarch-mt.distGarch-mGed.distEgarch-mEgarch-mt.distEgarch-mGed.distLgarch-mLgarch-mt.distLgarch-mGed.distSH53.0560.2059.1052.0458.9857.9153.0560.2259.14SZ55.2561.6560.8654.4560.6559.8955.2761.7460.91GARCHVaRVaRVaRtGEDGARCHGARCHGARCHVaR[1]Morgan,P.J..RiskMetricsTechnologyDocument:3rded.[M].NewYork:MorganTrustCompanyGlobalResearch,1995.[2]Jorison,P..ValueatRisk:Thenewbenchmarkforcontrollingmarketrisk[M].NewYork:TheMcGrawHillCompanies,1997.[3]Nelson,D.B..Conditionalheteroskedasticityinassetreturns:anewapproach[J].Econometrica,1991,(59):347-370.[4]Hamilton,D..TimeSeriesAnalysis[M].Princeton:PrincetonUniversityPress,1994.[5]Kendall,P..Techniquesforverifyingtheaccuracyofriskmeasurementmodels[J].JournalofDerivatives,1995,(3):73-84.[6]Engle,R.F.,Ng,V.K..Measuringandtestingtheimpactofnewsonvolatility[J].JournalofFinance,(48):1749-1778.[7]BasleCommitteeonBankingSupervision.AmendmenttotheCapitalAccordtoIncorporateMarketRisks[M].Basle:BankforInternationalSettlements,1996.[8]SmithsonC,MintonL.ValueatRisk[J].Risk,Jan,1996,9(1):25-27.[9]DuffieD,PanJ.AnoverviewofValueatRisk[J].JournalofDerivatives,1997,(4):7-49.[10]VaR[J].20003:26-32[11.VaR[J].1999,(6):15-18.