基于日内价格幅度与回报的随机波动率模型

1[1][2][1],200433;[2]200093,F830.91CAPMAPT[1][2][3][4]123RealizedVolatility:(2005BJB002)[e-mail]2GARCHAnderson2001Chou2001[4]ConditionalAutoRegressiveRangeCARRAlizadeh,Brandt,andDiebold(2001)[3]Brandt2002EGARCH[5][1][3][6][7]11.1tPt1tttdWσµdtPdP+=t2ttβdWdtσσασd+−=)ln(lnln1t1Wt2WdtσPθdWdWttt2t1),(=3ttPdP/tσdlnσln0α0σPθtt=),([0T]NH=T/NtσiHt(i+1)Hi=1,2,,NiHtσσ=itiHttdWσµdtPdP+=HitiH)1(+≤2)ln(ttPs=ItotstiHiHtdWσdtσµds+−=)21(2HitiH)1(+≤3HH1iσ)(ln+]),ln(ln[ln~ln|ln)1(HβσσρσNσσ2iHHiHHi−++4αHρH−=112))1((HitiHst+≤)))1(((HitiHsft+≤iHσiHσiHσ4)))1((()))1(((HitiHsfσHitiHsftiHt+≤=+≤∗γ5|)))1(((|lnln|)))1(((|lnHitiHsfσHitiHsftiHt+≤+γ=+≤∗6))1((HitiHst+≤∗))1((HitiHst+≤1=∗iHσ6|)))1(((|lnHitiHsft+≤iHσlniHσlniHσln45HiiHHHivHβσσρσσ)1()1()ln(lnlnln+++−+=7aHitiHtεHitiHsfEσHitiHsf)1(|])))1(((|[lnln|)))1(((|ln+∗++≤+γ=+≤7b)1,0(N~)1(Hiv+Hiε)1(+0(7a)7b|)(|lnfiHσlni(i+1)HiH||lnln||ln|)))1(((|ln*)1()1(iHHiiHiHHitssσssHitiHsf−γ+γ=−γ=+≤∗++8γ=12γγ=15)infsupln()ln()infsupln(|)))1(((|ln*1)(*1)(1)(1)(tHitiHtHitiHiHtHitiHtHitiHtssσssHitiHsf+≤+≤+≤+≤−+=−=+≤9Alizadeh,Brandt,andDiebold(2001)[3]17b11/41MeanSt.DevSkewnessKurtosis||ln)1(iHHiss−+)ln(21ln0.64HσiH++−1.11-1.536.93)infsupln(1)(1)(tHitiHtHitiHss+≤+≤−)ln(21ln0.43HσiH++0.290.172.8022116231Alizadeh,Brandt,andDiebold(2001)12[3]31=H11iiissr−=++)1(1titititiissd1)(1)(1infsup+≤+≤+−=SV-Return)1()1()1(1+++=iiiεσr)1()1()ln(lnlnln+++−+=iiHiβvσσρσσ10N(0,1)~)1()1(+iε)1,0(N~)1(+iv10||ln)1(iiss−+=)1(ln0.64+++−iiεσ(SV-Range))ln(1+id=1ln0.43++iσ+σ(2)1)(+iε)1()1()ln(lnlnln+++−+=iiiβvσσρσσ11)0,1(~(2)1)(Nεi+)1,0(N~)1(+iv-(SV-Return+Range))1()1()1()1(+++=iiiεσr)ln(1+id=1ln0.43++iσ+σ(2)1)(+iε)1()1()ln(lnlnln+++−+=iiiβvσσρσσ12(1)1)(+iε1(2)1)(+iε2)1,0(N~)1(+iv22MCMC[8]SV-Return-Range7NiiNr1}{==RNiiNd1}{==D()',,lnβρσ=θN0iiNσ==}{Σ),|,(θΣRNNNDf)(θ|ΣNf)(θf),|,(),()(),(θΣDRθΣθDR|Σθ,TNNNNNNfffKf×=13Jacquier[10][11]1δ2δ−=**δ).,(~*5120Betaδδ0.86ω)100,0(~Nω)ln(v)73.0,49.2(~)ln(−Nv0.20.125SVMarkovChainMonteCarloMCMC[12]MCMC(Metropolis1953)Hastings(1970)GelfandSmith(1990)MCMCTannerwong(1987)GibbsHastings-metropolisMCMCGibbsSVGibbsψΘ)(TR|ψfNkk1)(}{=ψNmkk=}{)(ψ))((ψgE8∑=+−≈NmkTkgmNgE)|(11))(()(RψR|ψT))((ψgEgTRGibbsψB:))()1(B,...,(ψψψ=)()()1()1(1)()(TBbbbR;ψ,...,ψ,ψ,...,ψ|ψ+−fGibbs1)()0()()0()1()0(Bψ,...,ψψ=0=i2)(~)()()()2()1(1)()1(TiBiiR;ψ,...,ψ|ψψf+)|(~)()()(1)(1)()1()1()2()1()1()(1)()(TiBibibiibibR;ψ,...,ψ,ψ,...,ψ,ψψψ++−+++f)|(~1)(1)()1((2))1()1()()(TiiiB(i)BR;ψ,...,ψ,ψψψ+−++Bf)(iψ1)(+iψ31+=ii,2,...)(1)(0)(iψ,...,ψ,ψ,)(iψ)1(+iψ∏=++−++++=BbGfK1)()()()1(1)()1(1)()2(1)(1)(1)()(1)()()|()(iBibibiiibiiψ,...,ψ,,...ψψ,ψψψ,ψMCMCWINBUGSGilks,Spiegelhalter1996[12]33.1199710-2004591600221Std.DevSkewnessKurtosis2MeanStd.DevSkewnessKurtosis1251020ir0.0002260.01430.39858.97440.017-0.045-0.017-0.0120.048||lnir-5.14211.2028-1.18466.24380.0580.1080.0790.0690.055)ln(id-4.30820.53750.38103.22310.5100.4450.3570.2890.2253220000MCMC.40005001200001500044,sdMC,MCMCHρ0.92SV-Return+RangeHρSV-RangeSV-ReturnSV-RangeβSV-Return+RangeSV-ReturnHρβAICSV-Return+RangeSV-ReturnSV-Range5)1()1(Hiε+(2)1)(Hiε+110J-B2SV-Return3SV-Range4SV-Return+Range4σlnρβσAICSV-Return-4.6336(0.2201)[0.0110]0.9291(0.0177)[0.0012]0.3284(0.0416)[0.0034]-6.0123SV-Range-4.7374(0.1843)[0.0171]0.9309(0.0178)[0.0014]0.1422(0.0121)[6.136E-4]0.36950.0093[3.933E-4]-6.2341SV-Return+Range-4.7341(0.1655)[0.0132]0.9558(0.0244)[0.0015]0.2435(0.0121)[4.919E-4]0.32730.0087[3.672E-4]-6.5632:[]MC5SVSV-Return+RangeSV-ReturnSV-RangeReturnRange0.92390.91531.10731.14250.04150.06430.07340.04832.87183.18423.21493.2754J-B1.5586(0.000)2.1494(0.002)2.7831(0.004)1.8531(0.000)Q(1)1.5012(0.220)18.425(0.000)2.5666(0.109)3.2763(0.049)Q(2)1.6698(0.434)41.466(0.000)2.6482(0.266)3.8354(0.104)Q(5)6.7606(0.239)67.915(0.000)7.7101(0.173)9.9544(0.069)ResidualsQ(20)23.0460(0.287)78.092(0.000)26.087(0.163)27.2549(0.157)Q(1)16.6310(0.000)3.4717(0.062)2.2358(0.135)3.2857(0.096)Q(2)23.5960(0.000)3.4997(0.174)4.7587(0.093)4.2475(0.129)Q(5)40.152(0.000)9.8915(0.078)10.424(0.064)11.4735(0.059)Box-PierceQ:SquaredresidualsQ(20)69.913(0.000)25.092(0.198)30.147(0.068)27.2984(0.075):.3311(1)∑=−−=N1i22i2iσrNMSE)ˆ(1∑=−−=T1tiiσrNMAE|ˆ|221∑=−−=NiiiσrNLE122212)}ˆln(){ln(][∑=−−=NiiiσrNLE1221|)ˆln()ln(|||∑=−−−=NiiiσdNMSE121))ˆln(0.43)(ln(∑=−−−=NiiiσdNMAE11|)ˆln(0.43)ln(|6Vol-Return-RangeVol-RangeVol-Return6MSEMAE[LE]2|LE|Vol-Return2.50E-070.000196.73441.7787Vol-Range2.51E-070.000165.20051.5371Vol-Return-Range2.20E-070.000144.80141.4214Vol-Return0.22790.3903Vol-Range0.05890.1938Vol-Return-Range0.03070.1405AIC(2)VaR,ttUσ=α−1VaRα−1Uα−1VaR95-VaR129555VaR5795%99%SV-ReturnVaRSV-RangeVaRVol-Return-RangeVaR7SV-ReturnSV-RangeVol-Return-Range95%4.674%5.271%5.115%99%0.836%1.195%1.056%4[1]EngleR,GalloG.Amultipleindicatorsmodelforvolatilityusingintra-dailydata[R].NBERWorkingPaper10117,2003[2]AndersenTG,TBollerslev,FXDiebold.Thedistributionofrealizedstockreturnvolatility[J].JournalofFinancialEconomics,2001,61:43–76[3]AlizadehS,MWBrandt,FXDiebold.RangeBasedEstimationofStochasticVolatilityModels[J].Journalof