뗚36뻭뗚12웚쫽톧뗄쪵볹폫죏쪶Vol136ꆡNo112ꆡ2006쓪12퓂MATHEMATICSINPRACTICEANDTHEORYDecem.,2006ꆡ쿠맘쾵쫽폫쿠맘탔뛈솿샮탣쏴,ꆡ붭컀뮪(뫓놱뿆벼듳톧샭톧풺,뫓놱쪯볒ꆡ050018)햪튪:ꆡ퇐뺿쇋뛈솿쿠맘탔뗄솽룶훷튪릤뻟:쿟탔쿠맘쾵쫽뫍캲늿쿠맘쾵쫽.쿟탔쿠맘쾵쫽랴펳쇋뇤솿볤뗄쿟탔쿠맘탔,헢뛔폚튻냣뗄췖풲탍럖늼쫇뫏쫊뗄.떫죧맻쯦믺뇤솿뻟폐늻뛔돆뗄캲늿뇤뮯쳘헷쪱,튪폃캲늿쿠맘쾵쫽쏨쫶쯼쏇횮볤뗄쿠맘탔.춨맽쿠맘몯쫽Copula,뛔뮦짮막쫐뗄캲늿쿠맘쾵쫽뷸탐쇋뚨솿럖컶.뷡맻뇭쏷:뮦짮막쫐뻟폐뷏잿뗄쿠맘탔.맘볼듊:ꆡ쿟탔쿠맘쾵쫽;캲늿쿠맘쾵쫽;쿠맘몯쫽Copula;럧쿕맜샭1ꆡ틽ꆡꆡ퇔ꆡꆡ뷼쪮쓪살,쿠맘탔뗄퇐뺿퓚뷰죚쇬폲틽웰쇋맣랺훘쫓.뒫춳틢틥쿂뗄쿠맘탔뛈솿횸뇪-쿟탔쿠맘쾵쫽뛔뷰죚닺쫕틦싊뗄쏨쫶틑늻퓙쫊폃.쿟탔쿠맘쾵쫽,뛔폚췖풲탍럖늼죧헽첬럖늼ꆢ톧짺t럖늼살쮵쫇뫏쫊뗄,틲캪췖풲탍럖늼뗄럖늼몯쫽쫇평쿟탔쿠맘쾵쫽캨튻좷뚨.좻뛸,뛔폚튻킩뻟폐볢럥ꆢ뫱캲럖늼쳘헷뗄맛닢퇹놾,퓙폃쿠맘쾵쫽쏨쫶뻍늻뫏쫊쇋.A[1]rtzner,Em[2][3]brechts,Patton뚼횸돶,캲늿쿠맘쾵쫽퓚쏨쫶뷰죚닺쫕틦싊,쳘뇰쫇벫뛋뇤틬쫂볾랢짺쪱뻟폐솼뫃쳘탔.놾컄뻍쪹폃쿟탔쿠맘쾵쫽펦힢틢컊쳢틔벰뮦짮막쫐캲늿쿠맘탔ퟶ짮죫첽쳖,쿠탅퓚쪵볊릤ퟷ훐믡뗃떽맣랺펦폃.2ꆡ쿟탔쿠맘쾵쫽쒿잰,맣랺쪹폃뗄쿟탔쿠맘쾵쫽,횻쫇뇤솿볤쿟탔쿠맘돌뛈뗄튻훖뛈솿.틲듋,뛔쿟탔쿠맘쾵쫽폐뇘튪ퟶ뷸튻늽샭뷢,틔쿂볙짨쯦믺뇤솿뻟폐폐쿞랽닮.1)죧맻쿟탔쿠맘쾵쫽쫇쇣,쮵쏷쯦믺뇤솿XꆢY횮볤늻뻟폐쿟탔쿠맘탔,떫솽뇤솿횮볤뿉쓜믡폐럇뎣쏷쿔뗄웤쯼맘쾵.샽죧,볙짨쯦믺뇤솿X럾듓뇪ힼ헽첬럖늼,쯦믺뇤솿Y=X2,죝틗쯣뗃쯼쏇횮볤뗄쿟탔쿠맘쾵쫽캪쇣,떫헢쪱뗄뇤솿X폫Y뻟폐쏷쿔뗄욽랽맘쾵.쿠맘쾵쫽캪쇣뗄풭틲쫇쪲쎴쓘?듓쿠맘쾵쫽뗄뚨틥살뾴:ΘCov(X,Y)E(XY)-EXõEY==(1)DXDYDXDY쿠맘쾵쫽Θ뗄볆쯣쫇뫍웚췻폐쏜쟐맘쾵.듓볆쯣E(XY)뗄릫쪽훐:2xE(XY)=ꇒ+ꇞ3-ꇞxõ1-e2dx(2)2Π컒쏇뿉틔뾴떽,믽럖쟸볤쫇(-ꇞ,+ꇞ).퓚쟸볤(-ꇞ,0)짏,뇤솿X폫뇤솿Y뻟폐뢺쿠맘맘쾵,뛸퓚(0,+ꇞ)짏,쯼쏇뻟폐헽쿠맘맘쾵,믽럖쪹뗃쯼쏇횮볤뗄헽뢺쿠뮥맘쾵뗖쿻,쫕룥죕웚:2005208217믹뷰쿮쒿:뫓놱쪡뿆벼쳼죭뿆톧퇐뺿훺쿮쒿(06457225)
12웚샮탣쏴,뗈:쿠맘쾵쫽폫쿠맘탔뛈솿189듓뛸뗃떽쇋뷏킡뗄Θ횵,짵훁캪쇣.헢뻍죝틗룸죋쏇퓬돉튻훖컳뷢,죏캪솽뇤솿뗄쿠맘탔뷏죵.쫂쪵짏,떱Θ=0쪱,늢늻틢캶솽뇤솿쿠뮥뛀솢.컄쿗[4]룸돶쇋Θ=0쪱,뇤솿XꆢY쿠뮥뛀솢뗄솽훖쳘쫢쟩뿶.2)죧맻쯦믺뇤솿X폫Y뗄쿟탔쿠맘쾵쫽뷓뷼폚1,벴ûΘûꇖ1,뇤솿XꆢY볤죔뿉쓜뻟폐럇쿟탔맘쾵.샽죧,짨쯦믺뇤솿X럾듓쟸볤[6,8]짏뗄뻹퓈럖늼,쯦믺뇤솿Y=X2,죝틗쯣뗃Θꇖ.헢쪱,컒쏇뿉틔죏캪뇤솿XꆢY횮볤뗄럇쿟탔맘쾵뿉틔폃쿟탔맘쾵쓢뫏.3)뛔폚룸뚨뗄쯦믺뇤솿XꆢY뗄뇟풵럖늼F(x)ꆢG(y),쿠맘쾵쫽Θ뗄횵늻튻뚨쓜좡떽[-1,1]짏뗄죎뫎횵.샽1ꆡ뛔폚럾듓뛾풪뇪ힼ헽첬럖늼뗄솽룶쯦믺뇤솿XꆢY,웤쿠맘쾵쫽뿉틔좡떽[-1,1]짏뗄죎뫎횵.쳘뇰뗘,죴Y=aX+b,a>0쪱,쿠맘쾵쫽캪+1,떱a<0쪱,쿠맘쾵쫽캪-1.떫뛔폚튻냣쟩뿶캴뇘죧듋,뾴샽2.샽2ꆡ짨쯦믺뇤솿XꆢY뻹럾듓뛔쫽헽첬럖늼,뻹횵랽닮럖뇰캪Λi,Ρi,i=1,2,뿉틔볆쯣뇤솿XꆢY뗄쿠맘쾵쫽캪[5]:-ΡΡΡΘe1Ρ2-1min=ꆡ2Θe1-1Ρ2max=Ρ21ö2Ρ2(e2öΦ0;2-11(e1-1))(e1öΕ0(3)2Ρ2-11)(e2-11ö2)Θmin폀풶듯늻떽-1,Θmax폀풶듯늻떽+)믘맩럖컶훐뗄쿠맘쾵쫽믘맩럖컶훐쪹폃뗄쿠맘쾵쫽쫇폃살뛈솿퇹놾훐x횵폫y횵횮볤쿟탔쿠맘맘쾵뗄튻룶횸뇪.웤뚨틥캪:nΧ=ꇆxy-ꇆxꇆy(4)222n(ꇆx-(ꇆx))n(ꇆ2y-(ꇆy))퓚쪵볊펦폃훐,폃퇹놾쿠맘쾵쫽Χ에뚨뇤솿X폫Y쫇럱뻟폐쿔훸뗄쿟탔쿠맘탔,늻쓜볲떥뗘죏캪ûΧû풽뷓뷼폚1,뻍쮵솽룶뇤솿볤뗄쿟탔쿠맘맘쾵뻍풽쿔훸.튻냣닉폃t춳볆솿[6]좷뚨솽룶뇤솿쫇럱듦퓚쿔훸뗄쿟탔쿠맘탔.Χt=(5)(1-Χ2)ö(n-2)볬퇩볙짨:H0:Θ=0(럇쿟탔쿠맘);H1:Θꇙ0(쿟탔쿠맘)퓚풭볙짨돉솢뗄쳵볾쿂,t춳볆솿럾듓ퟔ평뛈캪n-2뗄t럖늼.쒿잰쇷탐뗄춳볆죭볾훐,뛔믘맩쒣탍맀볆뗄략믘횵샯,폐튻룶춳볆횸뇪R2,튻냣돆캪뿉뻶쾵쫽.R2쫇붫쿟탔쿠맘쾵쫽뷸탐욽랽뗃떽뗄.뿉뻶쾵쫽쏨쫶쇋믘맩랽돌뛔쯦믺뇤솿Y뗄ퟜ뇤닮뗄뷢쫍돌뛈.샽죧,죴R2=,쮵쏷뇤솿Y뗄ퟜ뇤닮뗄◊뿉틔평뇤솿XꆢY횮볤뗄쿟탔쿠맘맘쾵살뷢쫍,뛸폠쿂뗄%쫇평웤쯼뗄늻뿉좷뚨뗄틲쯘쯹뷢쫍.3ꆡ캲늿쿠맘쾵쫽듓쿟탔쿠맘쾵쫽뗄뚨틥훐뿉틔뾴돶,뇤솿XꆢY뇘탫뻟폐폐쿞랽닮.좻뛸,뷰죚쫽뻝뗄럖늼뎣뎣믡돶쿖볢럥ꆢ뫱캲ꆢ컞쿞랽닮뗄쟩탎,헢쪱뗄쿟탔쿠맘쾵쫽뻍늻퓙쫊폃.늢쟒,컒쏇룼맘탄뗄쫇벫뛋뇤틬쫂볾돶쿖뗄룅싊.뇈죧막욱쫐뎡,떱튻훖막욱볛룱늨뚯뷏킡쪱,튻냣늻믡
190쫽ꆡ톧ꆡ뗄ꆡ쪵ꆡ볹ꆡ폫ꆡ죏ꆡ쪶36뻭펰쿬떽웤쯼막욱믲헻룶막쫐,떫떱튻훖막욱볛룱헇뗸떽튻뚨돌뛈쪱,뻍믡컼틽춶헟좥릺싲믲엗쫛,듓뛸믡펰쿬떽웤쯼막욱뗄볛룱,짵훁헻룶막쫐뗄뚯떴.틲듋,퇐뺿떱튻룶쯦믺뇤솿뗄좡횵듳폚믲킡폚쒳튻룶횵쪱,쇭튻룶쯦믺뇤솿믡랢짺퓵퇹뗄뇤뮯,믲쮵쯼뛔쇭튻룶쯦믺뇤솿좡듳(킡)횵뗄펰쿬돌뛈죧뫎,헢뻍쫇캲늿쿠맘뗄룅쓮.캲늿쿠맘쾵쫽뻍쫇뫢솿떱쯦믺뇤솿X듳럹뛈퓶볓믲헟듳럹뛈복짙쪱,쯦믺뇤솿Y튲랢짺듳럹뛈퓶볓믲헟듳럹뛈복짙뗄뛈솿횸뇪.폐킧뗘맀볆캲늿쿠맘쾵쫽쫇벫횵춳볆샭싛뗄훘튪쓚죝횮튻.ꆡ캲늿쿠맘쾵쫽뗄뚨틥뚨틥1짨쯦믺뇤솿(X,Y)뗄솪뫏럖늼몯쫽캪F,뇟풵럖늼몯쫽럖뇰캪F1,F2,죴벫쿞:limPr{X>F-11(Α)ûY>F-12(Α)}(6)Αꇺ1-듦퓚,퓲돆듋벫쿞횵캪쯦믺뇤솿(X,Y)뗄짏캲쿠맘쾵쫽,볇캪ΚU.춬샭뿉틔뚨틥쿂캲쿠맘쾵쫽:ΚL=limPr{X<F-11(Α)ûY<F-12(Α)}.(7)Αꇺ0+헢샯Α뇭쪾룅싊횵,F-1i(Α),i=1,2뇭쪾뛔펦폚Α뗄럖캻쫽.ꆡ탔훊뿉틔횤쏷,캲늿쿠맘쾵쫽뻟폐틔쿂탔훊[7]:1)캲늿쿠맘쾵쫽퓚[0,1]횮볤,쳘뇰뗘,떱Κ=0쪱,돆쯦믺뇤솿(X,Y)캲늿붥뷸뛀솢,럱퓲돆쯦믺뇤솿(X,Y)캲늿쿠맘.2)캲늿쿠맘쾵쫽쫇평쿠맘몯쫽Copula캨튻좷뚨.벴:ΚCθ(Α,Α)C(Α,Α)U=limΑꇺ10+Α(8)-1-Α,ꆡΚL=limΑꇺ헢샯Cθ(Α,Α)=1-2Α+C(Α,Α)쫇짺듦쿠맘몯쫽.쿠맘몯쫽뗄룅쓮퓚컄쿗[7]훐폐쿪쾸뷩짜.3)캲늿쿠맘쾵쫽퓚떥뗷뇤뮻좺뗄뇤뮻쿂쫇늻뇤뗄,헢튲쫇쟸뇰폚쿟탔쿠맘쾵쫽뗄훘튪튻뗣.4)뛔폚췖풲탍Copula,죧룟쮹쿠맘몯쫽,ퟔ평뛈쫇Μ뗄뛾풪t럖늼뗄쿠맘몯쫽,쯼쏇뗄짏캲쿠맘쾵쫽폫쿂캲쿠맘쾵쫽뻹쿠뗈,헢쪹뗃췖풲탍쿠맘몯쫽Copula퓚쏨쫶솽룶쯦믺뇤솿뻟폐늻춬뗄짏ꆢ쿂캲뇤뮯맦싉쪱,쫜떽쇋쿞훆.쳘뇰뗘,뛔폚룟쮹쿠맘몯쫽,평폚퓚쿟탔쿠맘쾵쫽Θ<1쪱,(X,Y)뗄캲늿쿠맘쾵쫽뗈폚쇣,틲듋춶ퟩ뫏뗄럧쿕맜샭훐,죧맻볙짨닺쫕틦싊럾듓룟쮹럖늼,쓇쎴뻍믡뗍맀럧쿕[8].뛔폚쿠맘몯쫽tCopula,떱쿟탔쿠맘쾵쫽Θ>-1쪱,(X,Y)뗄캲늿쫇붥뷸쿠맘뗄,평폚tCopula쓜릻쏨쫶뇤솿볤뗄캲늿쿠맘탔,쯹틔퓚쪵볊펦폃훐뎣뎣놻쪹폃.5)뛔폚ArchimedeanCopulaퟥ,튻냣짏캲쿠맘쾵쫽뫍쿂캲쿠맘쾵쫽늻쿠뗈.샽죧GumbelCopula짏캲쿠맘쾵쫽캪2-21öΑ,쿂캲쿠맘쾵쫽쫇쇣,헢쪹뗃쯼퓚쏨쫶뻟폐짏캲쿠맘탔뗄솽룶쯦믺뇤솿뗄뇤뮯맦싉쪱뻟폐뷏잿뗄뿌뮭쓜솦.FrankCopula쫇ArchimedeanCopulaퟥ훐뻟폐뛔돆탔뗄쿠맘몯쫽,짏ꆢ쿂캲쿠맘쾵쫽쿠뗈,뛸ClaytonCopula뗄쿂캲쿠맘쾵쫽쫇21öΑ,짏캲쿠맘쾵쫽쫇쇣.ꆡ뮦짮막쫐캲늿쿠맘탔럖컶캪뾼달짏몣ꆢ짮?솽막욱쫐뎡볤뗄쿠맘뇤뮯맦싉,톡퓱짏횤ퟛ뫏횸쫽(SH)뫍짮횤돉럖횸쫽(SZ)뗄쎿죕쫕엌볛캪퇹놾,붫볛룱P(t)뚨틥캪쫐뎡쎿죕쫕엌볛,쫕틦싊뚨틥캪Rt=log(Pt)-log(Pt-1),퇹놾쟸볤듓2000쓪1퓂4죕떽2005쓪3퓂25죕.쫽뻝살풴폚,샻폃춳볆죭볾R뷸탐럖컶.춼1쫇솽룶막욱쫐뎡쫕틦싊쿠뮥맘쾵뗄즢뗣춼.
12웚샮탣쏴,뗈:쿠맘쾵쫽폫쿠맘탔뛈솿191듓춼탎짏뿉틔뾴돶,솽쫐뎡쫕틦싊횮볤돊쿖쏷쿔뗄헽쿠맘맘쾵.캪쇋뚨솿뿌뮭솽쫐뎡뗄쿠맘뇤뮯쟩뿶,탨튪톡퓱잡떱뗄쿠맘몯쫽Copula.퓚헢샯,놸톡쿠맘몯쫽Ck럖뇰쫇룟쮹Copula,tCopula,GumbelCopula뫍ClaytonCopula.쿠맘몯쫽훐뗄닎쫽폃KendallΣ훈쿠맘쾵쫽살좷뚨,톡폅뇪ힼ쫇폫뺭퇩Copula(Cθ)뗄ퟮ뛌뻠샫살좷뚨,뻠샫릫쪽캪:d(Cθ(n),Ck)=ꆬCθ(n)-CkꆬL2=ꇆnꇆn122Cθijij춼1ꆡ짏횤ퟛ횸짮횤돉횸쿠맘춼(n),)-Ck(,.(9)nnnni=1j=1뇭1룸돶쇋늻춬뗄쿠맘몯쫽Copula폫KendallΣ훈쿠맘쾵쫽뗄횮볤뗄맘쾵.뇭1ꆡ쿠맘몯쫽Copula폫KendallΣ뗄맘쾵쿠맘몯쫽Copula쿠맘닎쫽폫KendallΣ뗄맘쾵GaussianCopula:ꇒ5-1(u1)x22-2Θxy+yexpΣ-xdytΘΠ2=sin-ꇞꇒ5-1(u2)1-ꇞ(1-Θ2d)2Π12-Θ2Copula:ꇒ-1Μ+2tΜ(u)1-ꇞꇒ-1tΜ(u)21x2-2Θxy+y2-21+-ꇞΜdxdyΘΠΣ=sin(1-Θ2)22Π1-Θ2GumbelCopula:exp{-[(-logu1)Α+(-logu2)Α1öΑ]}Α1=1-ΣClaytonCopula:(u-Η1+u-Η2-1)-1öΗΗ2Σ=1-Σ룹뻝컄쿗[7]룸돶뗄릫쪽,죝틗쯣뗃뮦짮막쫐KendallΣ훈쿠맘쾵쫽캪,퓙룹뻝Σ폫룷뇭2ꆡ룷쿠맘뷡릹Copula폫뺭퇩Copula뗄뻠샫쿠맘닎쫽뗄맘쾵뗃:Θ쿠맘뷡릹Co=01974,Α=61897,Ηpulad(Cθ(n),Ck)=ꆬCθ(n)-Ckꆬ=111793.좻뫳룹뻝(9)쪽,뗃뇭듓뇭2뿉틔뾴돶,ClaytonCopula쫇ퟮ폅뗄,tCopula듎횮,GaussianCopula쫇ퟮ닮뗄.퓚헢샯좡ClaytonCopula볆쯣뮦짮막쫐막욱쫐뎡캲늿쿠맘뇭3ꆡ뮦짮막쫐캲늿쿠맘쾵쫽쾵쫽룅싊ΑΚ.룹뻝릫쪽(8),뿉틔볆쯣퓚늻춬뗄룅싊횵Α쿂,솽막욱쫐뎡뗄캲늿쿠맘쾵쫽.뇭3훐잰솽룶쫇쿂캲쿠맘쾵쫽,뫳룶쫇짏캲쿠맘쾵쫽.듓뇭3뿉틔뾴돶쯦ퟅΑ뗄퓶듳,짏캲쿠맘쾵쫽쟷폚쇣.좻뛸쿂캲쿠맘쾵쫽퓚Α좡쪱,뻍틑뺭쫇Α쟷폚쇣쪱뗄벫쿞횵쇋.뇭3쮵쏷떱짏몣ퟛ횸쫕틦싊뗍폚럖캻쫽쪱,뿉틔풤닢짮횤돉횸쫕틦싊튲뗍폚뗄룅싊쫇%.떱짏몣ퟛ횸쫕틦싊룟폚럖캻쫽쪱,뿉틔풤닢짮횤돉횸쫕틦싊튲룟폚럖캻쫽뗄룅싊쫇%.헢쮵쏷솽막욱쫐뎡쿂뗸쪱뗄솪뚯쟷쫆룼볓쏷쿔.
192쫽ꆡ톧ꆡ뗄ꆡ쪵ꆡ볹ꆡ폫ꆡ죏ꆡ쪶36뻭4ꆡ뷡ꆡꆡ싛쿟탔쿠맘쾵쫽뫍캲늿쿠맘쾵쫽뚼쫇뛈솿뇤솿볤쿠맘탔뗄릤뻟.쿟탔쿠맘쾵쫽횻쓜뇭쪾뇤솿볤뗄쿟탔쿠맘맘쾵,뛸뛔폚뻟폐뫱캲럖늼쳘헷뗄뇤솿볤뗄쿠맘탔쏨쫶,뻍튪톰쟳웤쯼뗄뛈솿횸뇪.캲늿쿠맘쾵쫽쫇뛈솿쫂볾랢짺뗄룅싊뷏킡,좻뛸튻떩랢짺,듸살뗄뫳맻늻죝뫶쫓뗄튻훖쿠맘탔횸뇪,헢훖횸뇪퓚풤닢벫뛋볛룱뇤뚯탐캪쪱쿔뗃폈캪훘튪.쪵횤럖컶뷡맻뇭쏷,짏몣ꆢ짮?솽막욱쫐뎡듦퓚뷏잿뗄쿂캲쿠맘탔,벴튻룶쫐뎡돶쿖놩뗸,쇭튻룶쫐뎡튲돶쿖놩뗸뗄뿉쓜탔벫듳,좻뛸돶쿖춬쪱짏헇뗄뿉쓜탔쿠뛔뷏죵.캲늿쿠맘쾵쫽뗄틽죫,쪹벫뛋쫂볾뗄럧쿕뗃틔솿뮯,헢뇘붫퓚쪵볊릤ퟷ훐뗃떽맣랺펦폃.닎뾼컄쿗:[1]ꆡArtznerP,DelbaenF,EberJM,[J].MathematicalFinance,1999,9:203ꆪ228.[2]ꆡEmbrechtsP,LindskogF,[A].HandbookofHeavyTailedDistributionsinFinance,,Elsevier,2003,Chapter8:329ꆪ384.[3]ꆡ[J].JournalofFinancialEconometrics,2004,2(1):130ꆪ168.[4]ꆡ훓헲좨.솽뇤솿뛀솢탔폫늻쿠맘볤뗄맘쾵[J].쇸훝쪦톧놨,1997,(2):76ꆪ78.[5]ꆡEmbrechtsP,McNeilA,:Propertiesandpitfalls[J].In:RiskManagement:ValueatRiskandBeyond(),CambridgeUniversityPress,ꆪ223.[6]ꆡMarioFTriola,쇵탂솢틫.ElementaryStatistics[M].놱뺩:쟥뮪듳톧돶냦짧,2004.[7]ꆡ[M].Springer,1999.[8]ꆡEmbrechtsP,LindskogF,[M].inHandbookofHeavyTailedDistributionsinFinance,,Elsevier,,ꆡJIANGWei2hua(DepartmentofMathematics,HebeiUniversityofScienceandTechnology,Shijiazhuang050018,China))Abstract:ꆡ,moregenerally,:ꆡlinearcorrelation;tail2dependencecoefficient;copulafunction;riskmanagement
