믹폚죕쓚볛룱럹뛈폫믘놨뗄쯦믺늨뚯싊쒣탍 붯쿩쇖 [ 1]ꎬ컢쿾쇘[ 2] [1] 뢴떩듳톧 뷰죚퇐뺿풺, 짏몣200433; [2]짏몣샭릤듳톧 맜샭톧풺ꎬ짏몣 200093 햪튪ꎺ뷰죚닺늨뚯싊닢솿폫붨쒣쫇뷰죚샭싛폫쪵볹훐뗄튻룶훘튪뿎쳢ꎬ틑뺭폐쇋탭뛠닢솿폫붨쒣랽램ꆣ놾컄틽죫쇋믹폚죕쓚볛룱럹뛈폫믘놨솽룶닢뛈횸뇪뗄쯦믺늨뚯싊쒣탍ꆣ샻폃훐맺막쫐쫽뻝뷸탐뗄쪵횤뷡맻뇭쏷ꎬ폫떥닢뛈횸뇪뗄쯦믺늨뚯싊쒣탍쿠뇈ꎬ믹폚솽룶닢뛈횸뇪뗄쯦믺늨뚯싊쒣탍쓜룼뫃뗘쏨쫶막욱쫐뎡늨뚯싊뫍쫐뎡늨뚯럧쿕ꆣ 맘볼듊ꎺ늨뚯싊붨쒣ꎬ죕쓚볛룱럹뛈ꎬ죕볤믘놨ꎬ쯦믺늨뚯싊쒣탍, 퓚쿕볛횵 훐춼럖샠뫅ꎺF 틽퇔 뷼쓪살ꎬ뷰죚닺볛룱늨뚯싊붨쒣쫇뷰죚뺭볃톧폫뷰죚볆솿톧뗄튻룶훘튪뗄뿎쳢ꆣ쫗쿈ꎬퟷ캪뷰죚닺볛룱럧쿕뗄뛈솿횸뇪ꎬ늨뚯싊뛔폚샭뷢닺볛룱뗄뚯첬쳘헷쫇벫캪훘튪뗄ꆣ쫂쪵짏ꎬ늨뚯싊쫇횤좯ퟩ뫏샭싛ꆢ놾닺뚨볛쒣탍ꎨCAPMꎩꆢ쳗샻뚨볛쒣탍ꎨAPTꎩ벰웚좨뚨볛릫쪽뗄뫋탄뇤솿ꆣ웤듎ꎬ늨뚯싊뛔웳튵뗄춶폫닆컱룜룋뻶닟ꆢ쿻럑헟뗄쿻럑탐캪뫍쒣쪽ꆢ뺭볃훜웚벰쿠맘뫪맛뺭볃뇤솿뗈뚼뻟폐훘튪펰쿬ꆣퟮ뫳ꎬ평폚늨뚯싊폫뷰죚쫐뎡뗄릦쓜폫컈뚨탔쏜쟐쿠맘ꎬ쫇쳥쿖뷰죚쫐뎡훊솿뫍킧싊뗄ퟮ볲뷠뫍ퟮ폐킧뗄횸뇪횮튻ꎬ틲듋늨뚯싊튲쫇뻶닟헟폫볠맜떱뻖쏜쟐맘힢뗄뛔쿳ꆣ 떫쫇ꎬ늨뚯싊놾짭쫇튻룶쒣뫽뗄룅쓮ꎬ폐ퟅ탭뛠뗄닢솿랽램뫍쒣탍ꆣ룹뻝쿖[1][2][3][4]폐뗄컄쿗ꎬ듓닢솿쫽뻝살풴짏뾴ꎬ늨뚯싊폐죽훖샠탍뗄닢솿횸뇪ꎺꎨ1ꎩ믹폚믘놨믲뻸뛔횵믘놨뗄늨뚯싊닢솿ꎬ벴쪹폃쫕엌볛룱뗄뛔쫽닮럖ퟷ캪늨뚯싊뗄닢솿믹뒡ꎻꎨ2ꎩ믹폚죕쓚볛룱뇤뮯럹뛈뗄늨뚯싊닢솿ꎬ죕쓚볛룱뇤뮯럹뛈횸뗄쫇죕쓚뗄ퟮ룟볛폫ퟮ뗍볛횮닮ꎻꎨ3ꎩ믹폚죕쓚룟욵쫽뻝뗄쿖쪵늨뚯싊ꎨRealized Volatilityꎩ닢솿ꎬ쿖쪵늨뚯싊횸뗄쫇죕쓚뗈쪱볤볤룴믘놨뗄뇪ힼ닮ꆣ헢죽룶횸 믹뷰쿮쒿:놾퇐뺿뗃떽짏몣쫐헜톧짧믡뿆톧맦뮮쿮쒿훺(2005BJB002). ퟷ헟볲뷩ꎺ붯쿩쇖ꎨ1974ꆪꎩꎬ쓐ꎬ맣컷죋ꎬ늩쪿ꆢ뢴떩듳톧뷰죚퇐뺿풺붲쪦ꎬ퇐뺿랽쿲캪뷰죚릤돌ꆢ놾쫐뎡뗄샭싛폫쪵횤퇐뺿ꎬ emailꎺjiangxianglin@ꆣ 1
뇪횮볤듦퓚ퟅ튻킩늻춬뗄쳘뗣ꎺ죕믘놨탨튪쪹폃잰튻붻틗죕뗄쫕엌볛룱ꎬ죕쓚볛룱럹뛈폫쿖쪵늨뚯싊쪹폃뗄쫇죕쓚맛닢떽뗄쫽뻝ꎻ뛸죕쓚볛룱럹뛈탨튪쎿튻룶듎붻틗뗄볛룱탅쾢ꎬ떫쿖쪵늨뚯싊쫇믹폚샫즢뗄죕쓚볛룱쫽뻝뗃떽뗄ꆣ 뒫춳뗄늨뚯싊쒣탍ꎬ죧GARCH쒣탍ꆢ쯦믺늨뚯싊쒣탍뚼쫇믹폚믘놨헢튻횸뇪붨솢뗄ꎻAndersonꎨ2001ꎩ뗈붨솢쇋믹폚쿖쪵늨뚯싊쒣탍ꎻChou뗈ꎨ2001ꎩ쳡[4]돶뗄믹폚죕쓚볛룱뇤뮯럹뛈뗄쳵볾ퟔ믘맩럹뛈쒣탍ꎨConditional AutoRegressive RangeꎬCARRꎩꎬAlizadeh, Brandt, and Diebold (2001)붨솢[3]쇋믹폚죕쓚볛룱뇤뮯럹뛈뗄쯦믺늨뚯싊쒣탍ꎬBrandtꎨ2002ꎩ붨솢쇋믹폚죕쓚[5][1][3][6][7]볛룱뇤뮯럹뛈뗄EGARCH쒣탍ꆣ듓쿖폐뗄퇐뺿뷡싛뾴ꎬ믹폚쿖쪵늨뚯싊닢솿뗄킧맻뗄킧맻ퟮ뫃ꎬ떫웤좱쿝쫇탨튪쪹폃죕쓚룟욵쫽뻝ꎬ쟒쫜떽탭뛠캢맛뷡릹틲쯘뗄펰쿬ꎨ죧싲싴볛닮ꆢ럇춬늽붻틗뗈뗈ꎩ뛸틽웰닢솿컳닮ꎻ쿠뇈뷏뛸퇔ꎬ믹폚죕쓚볛룱뇤뮯럹뛈뗄늨뚯싊닢솿쓜릻뗃떽폫죕쓚뗄쿖쪵늨뚯싊뗄닢솿쿠닮늻뛠뗄킧맻ꆣퟛ뫏살뾴ꎬ룷룶횸뇪뿉쓜냼삨쇋웤쯼횸뇪쯹쎻폐뗄탅쾢ꎬ죧믘놨뻍늻쓜돤럖냼삨붻틗죕쓚볛룱랢짺뚯첬뇤뮯뗄탅쾢ꎬ뛸뛔폚쿠춬듳킡뗄쿖쪵늨뚯싊뛔펦쇋늻춬듳킡뗄죕쓚볛룱뇤뮯럹뛈ꆣ틲듋ꎬ쓜릻ퟛ뫏쪹폃헢킩닢솿횸뇪뿉쓜뿉틔룄짆늨뚯싊쒣탍뗄탔쓜ꆣ 놾컄뗄쒿뗄쫇틽죫믹폚죕쓚볛룱럹뛈폫믘놨뗄솽룶닢솿횸뇪뗄쯦믺늨뚯싊쒣탍쏨쫶컒맺막욱쫐뎡뗄늨뚯싊ꎬ늢폫떥닢솿횸뇪뗄쯦믺늨뚯싊쒣탍뷸탐뛔뇈럖컶ꆣ 1 쯦믺늨뚯싊쒣탍 솬탸쪱볤늨뚯싊맽돌뗄샫즢뮯 볙뚨막욱볛룱Pퟱ톭죧쿂뗄솬탸쪱볤쯦믺늨뚯맽돌ꎺ tdPt=µdt+σdWt1tPt dlnσ=α(lnσ−lnσ)dt+βdW tt2tꎨ1ꎩ 헢샯ꎬW뫍W뚼캬쓉맽돌ꎬ쯼쏇뻟폐쿠맘탔dWdW=θ(P,σ)dtꆣ짏쫶1t2t1t2ttt 2
뗄탎쪽뇭쏷ꎬ믘놨dP/P뗄뛔쫽늨뚯싊dlnσ쫇튻룶잱퓚뗄뇤솿ꎬퟱ톭튻룶뻹횵ttt캪lnσꆢ뻹횵믘뢴닎쫽캪α>0뗄뻹횵믘뢴맽돌ꆣ놾컄캪쇋뒦샭짏뗄랽뇣ꎬ볙뚨θ(P,σ)=0ꎬ룃볙짨쳵볾틢캶ퟅ업돽ꆰ룜룋킧펦ꆱꆣ tt퓚쪵횤퇐뺿훐ꎬ횻쓜틀삵폚샫즢뗄퇹놾뛔솬탸쪱볤쯦믺맽돌뷸탐췆뛏ꆣ폫샫즢뗄퇹놾쿠뛔펦ꎬ컒쏇붫쪱볤뎤뛈[0ꎬT]뻹럖캪N룶볤룴ꎬ쎿튻룶볤룴캪튻룶붻틗죕뗄뎤뛈ꎺH=T/Nꆣ볙뚨늨뚯싊σ퓚붻틗죕쓚캪튻뎣쫽ꎬ떫죝탭웤퓚죕볤t뇤뚯ꆣ룃볙뚨틢캶ퟅ늨뚯싊쫇룟뛈돖탸뗄ꎬ틲듋늨뚯싊퓚죕쓚쎻폐뻧쇒뗄뇤뚯ꎬ뿉틔볙짨캪뎣쫽ꆣ퓚룃볙뚨쿂ꎬ뿉틔붫짏쫶뗄솬탸쪱볤늨뚯싊뚯첬맽돌뮯캪럖뛎뗄뎣쫽늨뚯싊맽돌ꎬ벴떱iH<t<(i+1)Hꎨi=1,2,ꆭ,Nꎩ쪱ꎬσ=σꆣ tiH퓚짏쫶뷼쯆쳵볾쿂ꎬ틢캶ퟅ막욱볛룱퓚뗚i쳬쓚ퟱ톭죧쿂뗄벸뫎늼샊퓋뚯ꎺ dPt=µdt+σdW iH<t≤(i+1)H ꎨ2ꎩ iHtPt쇮s=ln(P)ꎬ룹뻝Ito뚨샭ꎬsퟱ톭틔쿂탎쪽뗄늼샊퓋뚯ꎺ ttt12ds=(µ−σ)dt+σdW iH<t≤(i+1)H ꎨ3ꎩ tiHiHt2뛔폚뷏뛌뗄쟸볤뎤뛈Hꎬ뛔쫽늨뚯싊lnσ뗄쳵볾럖늼뗄뷼쯆캪ꎺ (i+1)H2lnσ|lnσ~N[lnσ+ρ(lnσ−lnσ),βH] ꎨ4ꎩ (i+1)HiHHiH웤훐ꎬρ=1−αHꆣ H1ꎮ2 믹폚늻춬듺샭횸뇪뗄늨뚯싊닢뛈 틲캪늻뿉쓜췪좫맛닢떽쎿튻룶쪱볤쟸볤쓚뗄솬탸뗄볛룱뇤뮯슷뺶ꎬ샫즢뫳뗄쯦믺늨뚯싊쒣탍죔좻늻쫇뫜죝틗뒦샭뫍맀볆ꆣ퓚쪵볊펦폃훐ꎬ컒쏇뺭뎣쪹폃뿉맛닢떽뗄춳볆솿뛔샫즢뮯뫳뗄늨뚯싊벰웤뚯첬맽돌뷸탐췆뛏ꎬ죧쎿튻쪱볤쟸볤짏뗄믘놨뻸뛔횵ꆢ믘놨욽랽뗈ꆣ 튻냣뗘ꎬ냑솬탸쪱볤뇤솿s(iH<t≤(i+1)H)뗄춳볆솿tf(s(iH<t≤(i+1)H))ퟷ캪쟸볤쓚뗄늨뚯싊σ뗄듺샭횸뇪ꎬ폃헢룶듺샭횸뇪뛔tiH늨뚯싊σ뷸탐췆뛏ꆣ볙뚨헢룶듺샭횸뇪쫇σ뗄웫듎쏝ꎬ뿉틔뗃떽죧쿂랽돌ꎺ iHiH 3
γ∗f(s(iH<t≤(i+1)H))=σf(s(iH<t≤(i+1)H)) ꎨ5ꎩ tiHt솽뇟좡뛔쫽ꎬ뿉틔뗃돶늨뚯싊닢뛈랽돌ꎺ ∗ln|f(s(iH<t≤(i+1)H))|=γlnσ+ln|f(s(iH<t≤(i+1)H))| ꎨ6ꎩ tiHt∗헢샯ꎬ s(iH<t≤(i+1)H)캪폫s(iH<t≤(i+1)H)뻟폐쿠춬뗄쯦믺탐캪뗄tt∗뇪ힼ뮯뗄솬탸쪱볤퇹놾ꎬ떫σ=1ꆣ iH듓ꎨ6ꎩ뿉틔뾴돶ꎬ춳볆솿ln|f(s(iH<t≤(i+1)H))|쫇뛔쫽늨뚯싊lnσ뗄tiH뻟폐퓫짹뗄듺샭횸뇪ꎬ웤훐랽돌폒뇟뗄뗚튻쿮폫뛔쫽늨뚯싊lnσ돉헽뇈ꎬ뛸뗚iH뛾쿮닢뛈컳닮ꆣ틲듋ꎬ닢솿컳닮뗄랽닮풽듳ꎬ늨뚯싊듺샭횸뇪뻟폐뗄탅쾢몬솿뻍풽킡ꎬ뛔뛔쫽늨뚯싊lnσ뗄췆뛏뻍풽늻뺫좷ꆣ iH붫ꎨ4ꎩ폫ꎨ5ꎩ듓탂낲업ꎬ뿉틔뗃떽죧쿂뗄쿟탔ힴ첬뿕볤쾵춳ꎺ lnσ=lnσ+ρ(lnσ−lnσ)+βHv ꎨ7aꎩ (i+1)HHiH(i+1)H∗ln|f(s(iH<t≤(i+1)H))|=γlnσ+E[ln|f(s(iH<t≤(i+1)H))|]+ε tiHt(i+1)Hꎨ7bꎩ 웤훐ꎬv~N(0,1)ꎻε캪뻹횵캪0뗄럖늼ꎬ떫쎻폐헽첬탔뗄쿞훆ꆣ랽(i+1)H(i+1)H돌(7a)쏨쫶쇋뛔쫽늨뚯싊뗄뚯첬맽돌ꎻꎨ7bꎩ캪뛔쫽늨뚯싊듺샭횸뇪ln|f()|폫뛔쫽늨뚯싊lnσ횮볤뗄솪쾵랽쪽ꆣ iH퓚듳솿뗄쯦믺늨뚯싊쒣탍뗄퇐뺿훐ꎬ춨뎣쪹폃뻸뛔횵믘놨믲욽랽믘놨ퟶ캪늨뚯싊듺샭횸뇪ꆣ뗚i룶쟸볤솬탸쪱볤믘놨헽뫃뗈폚 (i+1)H쪱뿌폫iH쪱뿌뗄막욱볛룱뛔쫽뗄닮횵ꎬ틲듋웤뛔쫽늨뚯싊듺샭횸뇪싺ퟣꎺ ∗*ln|f(s(iH<t≤(i+1)H))|=γln|s−s|=γlnσ+γln|s−s| t(i+1)HiHiH(i+1)HiHꎨ8ꎩ γ=1ꆢ2럖뇰폫뻸뛔횵믘놨뫍욽랽믘놨쿠뛔펦ꆣ틲캪γ뷶뷶쫇ퟷ폃폚뛔쫽늨뚯싊듺샭횸뇪뗄튻룶뇈샽돟뛈ꎬ쯹틔늻믡펰쿬떽늨뚯싊닢뛈랽돌컳닮쿮뗄럖늼ꆣ늻쪧튻냣탔ꎬ놾컄ퟖ퇐뺿쇋γ=1뗄쟩뿶ꎬ벴붫뻸뛔횵믘놨ퟷ캪늨뚯싊뗄듺샭횸 4
뇪ꆣ 뷼쓪살ꎬ듳솿뗄샭싛폫쪵횤퇐뺿뇭쏷ꎬ뛔쫽볛룱뇤뮯럹뛈쿠뛔폚뻸뛔횵믘놨뫍욽랽믘놨쫇룼캪폐킧뗄늨뚯싊듺샭횸뇪ꆣ웤뛔쫽늨뚯싊듺샭횸뇪쫇ꎺ ln|f(s(iH<t≤(i+1)H))|=ln(sups−infs)tttiH<t≤(i+1)HiH<t≤(i+1)H **=ln(σ)+ln(sups−infs)iHttiH<t≤(i+1)HiH<t≤(i+1)Hꎨ9ꎩ Alizadeh, Brandt, and Diebold (2001)룸돶쇋럖뇰틔뻸뛔횵믘놨뫍뛔쫽[3]볛룱뇤뮯럹뛈ퟷ캪늨뚯싊뗄듺샭횸뇪쪱ꎬ듺샭횸뇪샭싛짏뗄룷뷗뻘뗄쟩뿶ꎨ볻뇭1ꎩꎬ튲쫇퓚늻춬늨뚯싊듺샭횸뇪쿂랽돌ꎨ7bꎩ뗄쳘탔ꆣ듓뇭1뗄욫뛈ꆢ럥뛈춳볆솿뿉틔뾴돶ꎬ뛔쫽뻸뛔횵믘놨뗄럖늼폫헽첬럖늼쿠닮짵풶ꎻ쿠뇈횮쿂ꎬ뛔쫽볛룱뇤뮯럹뛈뗄럖늼폫헽첬럖늼뷌캪뷓뷼ꆣ룼캪훘튪뗄쫇ꎬ뛔쫽볛룱뇤뮯럹뛈뗄뇪ힼ닮듳풼쫇뛔쫽뻸뛔횵믘놨뗄뇪ힼ닮뗄1/4ꎬ틲듋떱믹폚뛔쫽볛룱뇤뮯럹뛈뛔늨뚯싊뷸탐닢뛈쪱뻟폐룼룟뗄뺫좷돌뛈ꆣ 뇭1 늨뚯싊듺샭횸뇪뗄뻘춳볆솿뗄샭싛횵 춳볆솿 Mean St. Skewness Kurtosis늨뚯싊듺샭횸뇪 Dev ln|s−s| −+lnσ+ln(H)(i+1) ln(sups−infs) +lnσ+ln(H) iH<t≤(i+1)HiHiH<t≤(i+1)H2 2 쪵횤쒣탍폫맀볆 2ꎮ1 쪵횤쒣탍짨볆 듓짏쫶뗄럖컶뿉틔뾴돶ꎬ잱퓚늨뚯싊맽돌쫇뿍맛듦퓚뗄맽돌ꎬ컒쏇뿉틔쪹폃늻춬듺샭횸뇪뗄탅쾢뛔잱퓚뗄늨뚯싊맽돌뷸탐췆뛏ꆣ평폚듺샭횸뇪쫇뻟폐퓫짹뗄ꎬ쯹틔룹뻝늻춬뗄듺샭횸뇪뗄탅쾢뛔늨뚯싊맽돌췆뛏쪱ꎬ맀볆돶뗄늨뚯싊맽돌늻뿉쓜랴펳돶좫늿헦쪵뗄늨뚯싊맽돌ꆣ 믹폚틔짏죏쪶ꎬ퓚놾컄뗄쪵횤퇐뺿훐쪹폃죽훖믹폚늻춬탅쾢뗄쯦믺늨뚯싊쒣탍뛔늨뚯싊맽돌뷸탐췆뛏ꎬ웤훐ꎨ1ꎩ쫇쪹폃믘놨뗄탅쾢뛔늨뚯싊맽돌뷸탐췆 5
뛏ꎬꎨ2ꎩ쫇쪹폃죕쓚볛럹뗄탅쾢뛔늨뚯싊뷸탐췆뛏ꎬ뛸ꎨ3ꎩ쫇춬쪱쪹폃쇋믘놨뫍죕쓚볛럹솽훖탅쾢뛔늨뚯싊맽돌뷸탐췆뛏ꆣꎨ1ꎩ쫇뒫춳뗄쯦믺늨뚯싊쒣탍퇐뺿뗄쇬폲ꎬAlizadeh, Brandt, and Diebold (2001)뛔ꎨ1ꎩ폫ꎨ2ꎩ뗄쳘탔뷸[3]탐쇋쿪쾸뗄뇈뷏퇐뺿ꆣ뛸퓚ꎨ3ꎩ훐놾컄쫗듎쳡돶쇋춬쪱믹폚솽훖탅쾢뛔늨뚯싊맽돌뷸탐췆뛏ꆣ 볙뚨H=1ꎬ뇭쪾1룶붻틗죕뗄쪱볤뎤뛈캪떥캻1ꎻ뚨틥r=s−sꎬ i+1(i+1)id=sups−infsꆣ i+1tti<t≤(i+1)i<t≤(i+1)믹폚믘놨뗄쯦믺늨뚯싊쒣탍ꎨSV-Returnꎩꎺ (1)뻹횵랽돌ꎺr=σε i+1(i+1)(i+1) 늨뚯싊랽돌ꎺlnσ=lnσ+ρ(lnσ−lnσ)+βv ꎨ10ꎩ (i+1)Hi(i+1)(1)웤훐ε~N(0,1)ꎬv~N(0,1)ꆣ (i+1)(i+1)ꎨ10ꎩ훐뻹횵랽돌폫랽돌ꎺln|s−s|=−+lnσ+ε쫇뗈볛뗄ꆣ (i+1)ii(i+1)믹폚죕쓚볛럹뗄쯦믺늨뚯싊쒣탍(SV-Range)ꎺ (2)뻹횵랽돌ꎺln(d)=+lnσ+σε i+1i+1(i+1)늨뚯싊랽돌ꎺlnσ=lnσ+ρ(lnσ−lnσ)+βv ꎨ11ꎩ (i+1)i(i+1)(2)짏쪽훐ꎬ ε~N(0,1)ꎻv~N(0,1)ꆣ (i+1)(i+1)믹폚믘놨-죕쓚볛럹뗄쯦믺늨뚯싊쒣탍(SV-Return+Range)ꎺ 뻹횵랽돌ꎺ (1)r=σε (i+1)(i+1)(i+1)(2)ln(d)=+lnσ+σε i+1i+1(i+1)늨뚯싊랽돌ꎺlnσ=lnσ+ρ(lnσ−lnσ)+βv ꎨ12ꎩ (i+1)i(i+1)(1)(2)웤훐ε캪ꎨ1ꎩ훐쯹뇭쪾ꎬε캪ꎨ2ꎩ훐쯹뇭쪾ꎬv~N(0,1)ꆣ (i+1)(i+1)(i+1)2ꎮ2 쒣탍뗄맀볆 놾컄닉폃쇋믹폚MCMC뗄놴튶쮹럖컶랽램맀볆쇋쒣탍뗄닎쫽뫍탅쾢쇷맽돌ꎬ룃랽램뗄쿪쾸럖컶볻컄쿗[8]ꎬ헢샯틔SV-Return-Range쒣탍캪샽볲튪쮵쏷웤믹 6
놾쮼쿫ꆣ NN붫믘놨탲쇐뇭쪾캪R={r}ꎬD={d}ꆣ늨뚯싊랽돌훐뗄닎쫽뇭쪾캪Nii=1Nii=1'Nθ=(lnσ,ρ,β)ꆣ닉폃놴튶쮹럖컶훐뗄퓶닎쫽랽램ꎬ붫늨뚯싊탲쇐Σ={σ}퓶Nii=0볓캪튪맀볆뗄닎쫽ꆣ뿉붫쒣탍뾴돉평닣듎뷡릹뗄쳵볾럖늼ퟩ돉ꎺf(R,D|Σ,θ)ꎬf(Σ|θ)ꎬf(θ)ꆣ룹뻝놴튶쮹풭샭ꎬ닎쫽뫍늨뚯싊탲쇐뗄NNNN뫳퇩럖늼뿉틔뇭쪾죧쿂ꎺ f(θ,Σ|R,D)=K×f(θ)f(Σ,θ)f(R,D|Σ,θ) ꎨ13ꎩ NNNNNNT[10]ꆢ[11]틔폫Jacquier뗈쿠춬뗄랽쪽짨뚨닎쫽뗄쿈퇩럖늼쏜뛈ꎬ볙뚨닎쫽뗄쿈***퇩럖늼쏜뛈쫇뛀솢뗄ꆣ볙뚨δ=2δ−1ꎬ웤훐δ럾듓놴쯾럖늼δ~Beta(20,)ꎬ헢퇹δ뗄쿈퇩뻹횵쫇ꎻω럾듓헽첬럖늼ω~N(0,100)ꎬ볙뚨ln(v)뗄쿈퇩럖늼쏜뛈캪헽첬럖늼ln(v)~N(−,)ꎬ웤쿈퇩뻹횵뫍쿈퇩뇪ힼ닮럖뇰캪뫍ꆣ 듓틔짏뗄럖컶뿉횪ꎬ뛔SV쒣탍뫳퇩럖늼뗄볆쯣붫쫇튻룶룟캬뗄믽럖컊쳢ꎬ뛔폚헢퇹뗄룟캬믽럖컊쳢ꎬ폃쫽횵럖컶뗄랽램쫇늻뿉쓜쪵쿖뗄ꎬ뛸뒫춳뗄쒣쓢랽램뗄뺫뛈폫킧싊쫇벫닮뗄ꆣ믹폚십뛻뿉럲솴쏉쳘뾨실ꎨMarkov Chain Monte [12]Carloꎬ볲돆MCMCꎩ쒣쓢뗄랽램쫇뷼쓪살랢햹웰살룟킧뗄쒣쓢벼쫵ꎬ뿉틔맀볆돶쒣탍뗄닎쫽뫍늨뚯싊탲쇐ꆣ MCMC랽램ퟮ돵펦폃폚볆쯣컯샭(Metropolis뗈 1953)ꎬHastings(1970)뗄릤ퟷ쪹웤룼캪튻냣뮯ꎬ떫훷튪펦폃폚춳볆컊쳢ꆣGelfand뫍Smith(1990)뗄퇐뺿쿔쪾돶MCMC랽램퓚놴튶쮹볆쯣짏폐ퟅ뻞듳뗄잱솦ꎬ쯻쏇붫십뛻뿉럲솴뗄랽램폫Tanner뫍wong(1987)뗄퓶닎쫽랽램쿠뷡뫏ꎬ횤쏷쇋뛔몬폐잱퓚뚯첬뇤솿뗄볆솿쒣탍뗄맀볆쫇벫캪돉릦ꆣGibbs돩퇹뫍Hastings-metropolis쯣램쫇솽훖ퟮ캪뎣폃뗄MCMC랽램ꆣ놾컄쪹폃쇋Gibbs돩퇹뗄랽램쒣쓢SV쒣탍뗄뫳퇩럖늼ꆣ Gibbs돩퇹뗄랽램틑뺭돉릦뗘펦폃폚듳솿뗄볆솿쒣탍ꎬ웤쓚퓚뗄쮼쿫쫇춨맽퓚닎쫽쿲솿ψ뗄ힴ첬뿕볤Θ짏릹퓬돶튻룶뻟폐캨튻뗄만뚨럖늼쏜뛈f(ψ|R)뗄T(k)N(k)N십뛻뿉럲솴{ψ}ꎬ좡틑뺭뷸죫쫕솲뗄ힴ첬뗄{ψ}살뷼쯆볆쯣E(g(ψ))ꎬk=1k=m 7
벴 N1(k)E(g(ψ|R))≈g(ψ|R) T∑TN−m+1k=m웤훐ꎬE(g(ψ))뿉뚨틥캪닎쫽g믹폚퇹놾R뗄뫳퇩뻹횵뫍뫳퇩뇪ힼ닮뗈ꆣ TGibbs돩퇹쫗쿈붫닎쫽ψ럖돉B뿩:ψ=(ψ,...,ψ)ꆣ퓚펦폃훐ꎬ펦룃춨맽(1)(B)쫊떱뗄럖뿩쪹뗃듓쎿튻룶쳵볾럖늼쏜뛈f(ψ|ψ,...,ψ,ψ,...,ψ;R)훐(b)(1)(b−1)(b+1)(B)T뗄돩퇹돉캪뿉쓜ꆣ쿂쏦킴돶럖뿩Gibbs돩퇹뗄뻟쳥늽훨ꎺ (0)(0)(0) 1ꆢ좷뚨돵쪼뗣ψ=(ψ,...,ψ)ꎬ짨i=0ꎬ (1)(B) 2ꆢ내틔쿂뗄랽쪽듓쳵볾럖늼쏜뛈훐돩퇹 (i+1)(i)(i) ψ~f(ψ|ψ,...,ψ;R) (1)(1)(2)(B)T ꆭ (i+1)(i+1)(i+1)(i+1)(i)(i)ψ~f(ψ|ψ,ψ,...,ψ,ψ,...,ψ;R) (b)(b)(1)(2)(b−1)(b+1)(B)T ꆭ (i)(i+1)(i+1)(i+1)ψ~f(ψ|ψ,ψ,...,ψ;R) (B)(B)(1)(2)(B−1)T(i)(i+1)헢퇹뻍뚨틥쇋튻룶듓ψ떽ψ뗄틆맽돌ꆣ 3ꆢi=i+1,떽뗚2ꎩ늽ꆣ (0)(1)(i)짏쫶뗄쯣램닺짺쇋튻룶쿲솿탲쇐ψ,ψ,...,ψ,...,ퟷ캪십뛻뿉럲솴뗄튻룶(i)(i+1)쪵쿖맽돌ꎬ듓ψ떽ψ뗄틆쏜뛈쫇 B(i)(i+1)(i+1)(i+1)(i+1)(i+1)(i)(i) K(ψ,ψ)=f(ψ|ψ,ψ,...ψ,ψ,...,ψ) G∏(b)(1)(2)(b−1)(b+1)(B)b=1 퓚놾컄훐ꎬ짏쫶뗄뫳퇩럖늼릹퓬뫍MCMC쒣쓢뚼쫇믹폚튻룶뷏캪돉쫬뗄죭[12]볾냼WINBUGSꎨGilks, Spiegelhalter 뗈1996ꎩ짏쪵쿖뗄ꆣ 3 쪵횤뷡맻럖컶 퇹놾쫽뻝쳘탔쏨쫶 퓚쪵횤퇐뺿훐쪹폃뗄퇹놾쫽뻝캪1997쓪10퓂-2004쓪5퓂뗄짏몣횤좯쫐뎡 8
뗄짏횤횸쫽ꎬ퇹놾솿캪1600ꆣ뇭2룸돶쇋쏨쫶뛔쫽뻸뛔횵믘놨뫍뛔쫽볛룱럹뛈뗄쳘헷뗄춳볆솿ꆣ뇈뷏뇭2폫뇭1뗄뷡맻뿉틔뾴돶ꎬ쯤좻듦퓚튻뚨뗄욫닮ꎬ떫쏨쫶솽룶늨뚯싊듺샭횸뇪쳘헷뗄춳볆솿ꎨꆢSkewness뫍Kurtosisꎩ폫웤샭싛횵믹놾쫇튻훂뗄ꆣ듓듺샭횸뇪뗄ퟔ쿠맘쾵쫽뾴ꎬ뛔쫽볛룱럹뛈뇈뛔쫽뻸뛔횵믘놨뻟폐룼룟뗄쿠맘탔ꆣ 뇭2 퇹놾춳볆쳘탔 춳볆솿 Mean SkewnessKurtosisퟔ쿠맘쾵쫽 1뷗2뷗 5뷗 10뷗 20뷗뇤솿 |r| (d) i3ꎮ2 쒣탍맀볆뷡맻폫헯뛏럖컶 캪쇋쒣쓢닎쫽뗄뫳퇩럖늼ꎬ뷸탐쇋20000듎뗄MCMC쒣쓢돩퇹.쒣쓢쫕솲ힴ첬볠닢쿔쪾ꎬ떱뷸탐쇋4000듎돩퇹쪱ꎬ쯹폐뗄닎쫽뚼틑뺭뷸죫쇋쫕솲ힴ첬ꎬ놾컄톡좡쇋5001ꆪ20000릲볆15000룶돩퇹ꆣ뇭4캪룹뻝헢킩돩퇹볆쯣뗄닎쫽뫳퇩뻹횵뫍웤쯼뗄튻킩춳볆솿ꆣ뇭4뗄뷡맻뇭쏷ꎬ쿠뛔폚뫳퇩뻹횵,쒣쓢닎쫽뗄뇪ힼ욫닮sd뫍MC컳닮뚼쿠떱뗄킡ꆣ틲듋,MCMC뫳퇩뻹횵맀볆뗄뷡맻쫇쿠떱뺫좷뗄ꆣ 듓쒣탍닎쫽맀볆뗄뷡맻뾴ꎬ돖탸탔닎쫽ρ뚼듳폚ꎬ뷡맻뇭쏷늨뚯싊돥H믷쫇룟뛈돖탸뗄ꆣ뇈뷏늻춬쒣탍뗄닎쫽맀볆뷡맻ꎬ뿉틔뾴돶ꎺSV- Return+Range쒣탍뗄늨뚯싊랽돌뗄돖탸탔닎쫽ρퟮ듳ꎬ웤듎쫇SV-Range쒣탍ꎬퟮ킡뗄쫇SV- HReturnꎻSV-Range쒣탍뗄늨뚯싊랽돌뗄뇤틬탔닎쫽βퟮ짙ꎬ웤듎쫇SV- Return+Range쒣탍ꎬퟮ듳뗄쫇SV- Returnꆣ돖탸탔닎쫽ρ풽듳ꆢ뇤틬탔닎쫽βH풽짙뇭쏷늨뚯싊뗄돖탸풽듳뫍뇤틬탔풽킡ꎬ뛸헢폖틢캶ퟅ쒣탍붫뻟폐룼잿뗄늨뚯싊풤닢쓜솦ꆣ듓쒣탍뗄AIC횸뇪뾴ꎬSV- Return+Range쒣탍쿠뛔폚SV- Return쒣탍뫍SV-Range쒣탍뚼폐룼뫃뗄쓢뫏킧맻ꆣ (1)(2)뇭5룸돶쇋룹뻝쯦믺늨뚯싊쒣탍뗄뻹횵랽돌뗄닐닮탲쇐εꆢε뗄헯뛏(i+1)H(i+1)H럖컶뗄뷡맻ꆣ럖컶늻춬쒣탍뗄닐닮뗄춳볆쳘탔ꎬ뿉틔뗃떽틔쿂뷡맻ꎺꎨ1ꎩ듓닐닮 9
뗄뇪ힼ욫닮ꆢ욫뛈ꆢ럥뛈ꆢ헽첬춳볆솿ꎨJ-Bꎩ뾴ꎬ쯹폐쒣탍뗄닐닮뚼믹놾짏럻뫏뇪ힼ헽첬럖늼ꎻꎨ2ꎩSV- Return쒣탍뗄닐닮믹놾짏늻듦퓚ퟔ쿠맘ꎬ떫닐닮욽랽듦퓚ퟔ쿠맘ꎻꎨ3ꎩSV-Range쒣탍뗄닐닮듦퓚ퟔ쿠맘ꎬ떫닐닮욽랽늻듦퓚ퟔ쿠맘ꎻꎨ4ꎩ뛸뛔폚SV- Return+Range쒣탍ꎬ믘놨뻹횵랽돌폫볛룱럹뛈뻹횵랽돌뗄닐닮폫닐닮욽랽뚼늻듦퓚ퟔ쿠맘ꆣ틲듋ꎬ쮫뇤솿뗄늨뚯싊쒣탍쓜릻룄짆쒣탍닐닮뗄쳘탔ꆣ 뇭4 늨뚯싊랽돌닎쫽맀볆뷡맻 ρ σ AIC lnσ β SV- Return () () () [] [] [] SV-Range () () () ꎨꎩ [] [] [-4] [-4] SV- Return+Range () () () ꎨꎩ [] [] [-4] [-4] 힢:ꎨꎩ훐캪뗣맀볆뗄뇪ힼ닮ꎬ[ ]훐캪캪뗣맀볆뗄MC퇹놾뇪ힼ닮ꆣ 뇭5 SV쒣탍뗄뻹횵랽돌닐닮뗄헯뛏럖컶 SV- Return+Range 쒣탍 SV- Return SV-Range 춳볆솿 Return Range 뇪ힼ욫닮 욫뛈 럥뛈 헽첬춳볆솿ꎨ J-Bꎩ () () () () Q(1) () () () () Q(2) () () Residuals () () Q(5) () () () () Q(20) () () () () Q(1) 춳() () 볆Squared () () 솿residuals Q(2) () () () () Q(5) () () () () Q(20) () () () () 힢:ꎨꎩ쓚캪쿔훸룅싊쮮욽. 3ꎮ3 늻춬듺샭횸뇪쒣탍뗄뇈뷏 10Box-Pierce Q:
(1) 늨뚯싊풤닢쓜솦뇈뷏 움볛늨뚯싊쒣탍탔쓜뗄튻룶훘튪랽쏦쫇늨뚯싊쒣탍뛔캴살늨뚯싊뗄풤N−1222닢쓜솦ꆣ뛔폚틔믘놨늨뚯싊뗄풤닢쓜솦ꎬ컒쏇쪹폃MSE=Nr−σˆꆢ()∑iii=1TN−1222−1222MAE=Nꆢ뫍r−σˆ||[LE]=N{ln(r)−σˆln()}∑ii∑iit=1i=1N−122헢쯄룶횸뇪살뛈솿ꆣ뛸뛔폚뛔쫽볛룱뇤뮯럹뛈ꎬ쪹폃|LE|=N|ln()−n(ˆrlσ)|∑iii=1NN−12−1MSE뫍=N(ln(d)−−n(σˆl))MAE=Nd−−ln(σˆꆣ |ln())|∑∑iiiii=1i=1뇭6룸돶쇋늻춬쒣탍뗄룷룶횸뇪뗄볆쯣뷡맻ꆣ듓뷡맻뿉틔뾴돶ꎬ늻싛믘놨늨뚯싊뮹쫇뛔쫽볛룱뇤뮯럹뛈ꎬVol-Return-Range쒣탍볈폐ퟮ뫃뗄풤닢탔쓜ꎬ웤듎쫇Vol-Range쒣탍ꎬ뛸ퟮ닮뗄쫇Vol-Return쒣탍ꆣ 뇭6 늻춬쒣탍뗄늨뚯싊퇹놾쓚풤닢쓜솦뇈뷏 2횸뇪 MSE MAE [LE] |LE| 쒣탍 믘놨늨뚯Vol-Return -07 싊뗄풤닢 Vol-Range E-07 Vol-Return-Range E-07 뛔쫽볛룱Vol-Return 0. 2279 0. 3903 뇤뮯럹뛈풤닢 Vol-Return-Range ퟛ짏쯹쫶ꎬ춨맽늻춬쒣탍뗄AIC횸뇪ꆢ쒣탍닐닮쓢뫏돌뛈ꆢ늨뚯싊뗄풤닢쓜솦뗈랽쏦뗄ퟛ뫏뇈뷏럖컶ꎬ뿉틔뗃돶뷡싛ꎺ죕쓚볛룱뇤뮯럹뛈횸뇪쿠뛔폚죕믘놨횸뇪냼삨쇋룼뛠뗄막쫐늨뚯싊뗄탅쾢ꎬ쓜릻룄짆막쫐늨뚯싊뗄맀볆ꆣ (2) 믹폚늻춬듺샭횸뇪쒣탍뗄VaR뇈뷏 붫늨뚯탔탲쇐뫳퇩럖늼뗄쒣쓢뻹횵쫓캪늨뚯탔탲쇐뗄맀볆횵,늢룹뻝헢튻탲쇐볆쯣쇋샺쪷VaR=Uσꎬ웤훐U캪룸뚨럖늼퓚훃탅쮮욽1−α짏뗄럖캻쫽ꆣ t1−αt1−α쿖춨맽컳닮싊살쳖싛VaR뛔럧쿕닢뛈뗄ힼ좷탔ꆣ퓚95ꎥ뗄훃탅쮮욽쿂ꎬ컳닮싊뚨틥캪쪵볊믘놨뗍폚ꎨ-VaRꎩ뗄쫽쒿돽틔퇹놾쫽뗃떽뗄뇈싊ꆣ평폚톡좡훃쮮 11
욽캪95ꎥꎬ죴쒣탍샭쿫ꎬ컳닮싊펦뗈폚5ꎥꎬ죴컳닮싊맽뛈킡폚5ꎥ퓲쮵쏷샻폃쒣탍맀볆뗄늨뚯탔듳폚쪵볊쟩뿶뛸떼훂VaR횵뷏룟ꎬ룟맀쇋쫐뎡럧쿕ꎻ죴컳닮싊맽뛈듳폚5ꎥ퓲쮵쏷쒣탍맀볆뗄늨뚯탔킡폚쪵볊쟩뿶뛸떼훂횵욫뗍ꎬ뗍맀쇋쫐뎡럧쿕ꆣ 뇭7룸돶쇋늻춬쒣탍뗄볬퇩뷡맻ꎨ95%뫍99%훃탅뛈ꎩꆣ듓뷡맻뿉틔뾴돶ꎬ쯤좻죽룶쒣탍뚼뇈뷏뫃뗘맀볆쇋쫐뎡럧쿕ꎬ떫뇈뷏뛸퇔ꎬ믹폚SV- Return볆쯣뗄VaR튻뚨돌뛈짏룟맀쇋쫐뎡럧쿕ꎬ믹폚SV-Range볆쯣뗄VaR튻뚨돌뛈짏뗍맀쇋쫐뎡럧쿕ꎬ뛸믹폚Vol-Return-Range볆쯣뗄VaR룼캪ힼ좷뗘랴펳쇋쫐뎡럧쿕ꆣ틲듋ꎬ믹폚솽룶탅쾢횸뇪뗄쯦믺늨뚯싊쒣탍쓜릻룼뫃뗘닢솿컒맺막쫐뗄럧쿕ힴ뿶ꆣ 뇭7 늻춬쒣탍뗄컳닮싊 쒣탍 SV- Return SV-Range Vol-Return-Range 훃탅뛈 % 95% % % % 99% %%4 뷡싛 놾컄틽죫믹폚죕쓚볛룱럹뛈폫믘놨뗄솽룶닢솿횸뇪뗄쯦믺늨뚯싊쒣탍쏨쫶컒맺막욱쫐뎡뗄늨뚯싊ꎬ늢폫떥닢솿횸뇪뗄쯦믺늨뚯싊쒣탍뷸탐뛔뇈럖컶ꆣ샻폃훐맺막쫐쫽뻝뷸탐뗄쪵횤뷡맻뇭쏷ꎬ폫떥닢솿횸뇪뗄쯦믺늨뚯싊쒣탍쿠뇈ꎬ믹폚솽룶닢솿횸뇪뗄쯦믺늨뚯싊쒣탍쓜룼뫃뗘쏨쫶막욱쫐뎡늨뚯싊ꆣ 닎뾼컄쿗ꎺ [1] Engle R, Gallo G. A multiple indicators model for volatility using intra-daily data[R]. NBER Working Paper 10117, 2003 [2] Andersen TG, T Bollerslev, F X Diebold. The distribution of realized stock return volatility[J]. Journal of Financial Economics, 2001, 61: 43–76 [3] Alizadeh S, M W Brandt, F X Diebold. Range Based Estimation of Stochastic Volatility Models[J]. Journal of Finance, 2002, 57: 1047-1091 [4] Chou R Y. Forecasting financial volatilities with extreme values: The conditional autoregressive range (CARR) model[R]. Adademia Sinica Working Paper, 2001 [5] Brand M W, Christofer S J.. “Volatility forecasting with range-based EGARCH models[R], Wharton School University of Pennsylvania, 2002 12
[6] Parkinson M. The Extreme Value Method for Estimating the Variance of the Rate of Return[J]. Journal of Business, 1980, 53: 61-66. [7] Celso Brunetti. Relative efciency of return-based and range-based volatility estimators[R]. Department of Finance, Johns Hopkins University ,2004 [8] 췵뒺럥ꎬ붯쿩쇖. 믹폚쯦벴늨뚯싊뗄훐맺막쫐늨뚯싊맀볆[J] . 맜샭뿆톧톧놨ꎬ2003ꎬ6(4)ꎺ63-72 [9] Jacquier E, Polson G, Rossi E. Bayesian analysis of stochastic volatility model with fat Heavy-tailed Distributions Working paper. 2001 [10] Jacquier E, Polson G, Rossi E. Bayesian Analysis of Stochastic Volatility Models[J]. Journal of Business & Economic Statistics, 1994, 12:371-390 [11] Gilks W R, Richardson S. Spiegelhalter, . Markov Chain Monte Carlo in Practice[M], London: Chapman & Hall Press, 1998 춨톶뗘횷ꎺ짏몣쫐 몪떦슷220뫅 뢴떩듳톧뷰죚퇐뺿풺ꎬ붯쿩쇖ꎬ 폊뇠ꎺ200433ꎻ TELꎺ021-65100059ꎬ 13818512973 EMAILꎺjiangxianglin@, cianglin@ ퟷ헟볲뷩ꎺꎨ1ꎩ붯쿩쇖ꎨ1974ꆪꎩꎬ쓐ꎬ맣컷죋ꎬ늩쪿ꆢ뢴떩듳톧뷰죚퇐뺿풺붲쪦ꎬ퇐뺿랽쿲캪뷰죚릤돌ꆢ놾쫐뎡뗄샭싛폫쪵횤퇐뺿ꎬ emailꎺjiangxianglin@ꎻ ꎨ2ꎩ컢쿾쇘ꎨ1974-ꎩꎬ얮ꎬ룊쯠죋ꎬ짏몣샭릤듳톧맜샭톧풺늩쪿짺ꎬ퇐뺿랽쿲뷰죚릤돌ꆢ릫쮾닆컱ꎬemailꎺwuxiaolin1234@ꆣ Price-Range and Return Based Stochastic Volatility Model ---- with an Application to Chinese Stock Market Volatility [1][2]Jiang Xiang-linꎬWU Xiao-lin ([1] Institute for Financial Studies, Fudan Unversity, Shanghai 200433, Chinaꎻ [2] Management School, University of Shanghai for Science and Technology, Shanghai 200093, China) AbstractꎺMeasurement and modeling of financial asset volatility is an important problem in financial theory and practice. Many ways exist to measure and model financial asset volatility. In this paper we propose to a stochastic volatility model based on daily returns and intra-daily high-low price range jointly. Empirical results on Chinese stock market indicate that stochastic volatility model based on the two index outperforms those based on one index in capturing volatility character and market risk . Keywords: volatility modeling, intra-daily high-low price range, inter-daily returns, stochastic volatility model, value at risk 13
