基于LevyCopula的组合信用衍生品定价模型

10(311)200910ResearchonFinancialandEconomicIssuesNumber10(GeneralSerialNo1311)October,2009LevyCopula,(/,116025):2007,,,,,VarianceGamma,LevyCopula,,:;VarianceGamma;LevyCopula;:F83015:A:10002176X(2009)1020076209,,,,,15,(ISDA),200712,62,20063482%,,,(single-name)(portfolio)((multi-name)),,(CDS)(TRORs)(CLN),,,N(Nth-to-defaultCDS)(CDOs)CDSNth-to-defaultCDSCDSISDA,1,m,AmC,R,,,AC:2009207226:(70871019;70671019);(06JJD910002);(2006T042);([2008]179090):(1968-),,,,,,E2mail:ydshi@2631net1CDS,A,1CDS:AB,m,Br,CC,BAm;C,BA,CDSA,,CDS,B1CDS,CDS,,CDSCDSNth-to-defaultCDS,1:M(NM),NBA,N,B,,N,AB,ACDSNth-to-defaultCDS,,,,Copula,Copula,Copula,,,,,,Copula:,,,:,,,,,Copula,Copula,,:,,,2008,47%,,,,,,77LevyCopula,,,[19][18]1923,,,170,500,,90%,,LevyCopula,,VarianceGamma,,,,CopulaLi(2000),,Copula,M,,:Vi(t)=iC+1-2iii=1,2,,M(1)(1),C,i,Ci,i=1,2,,M,:cov(Vi,Vj)=ijVar(Vi)=1i,j=1,2,,M(2),Vi(t)Kii,i,:P(it)=P(ViKi)(3)Vi(t),Ki,,Copula,,,,;Copula0,;,,Copula,DemartaMcNeil[1]t-,Copulat-Copula;FriendRogge[2]GregoryLaurent[3]ClaytonCopula,Copula,TotouomArmstrong[4]Copula,Copula;,[5-6],,[7],,,Levy,,KalemanovaSchmid[8](NIG)CopulaCDOs;MoosbruckerVarianceGamma[9];[10]Moosbrucker(2006)VarianceGamma;Baxter[11]GammaVarianceGammaGamma(catastrophe)Gamma,GammaVarianceGamma,,Levy,Levy,Levy,(Schoutens(2003))Levy(infinitelydivisible),,Levy-Khintchine,Levy:87200910311Xt()=E[eiXt]=e-tX(t)t0(4):X()=-i+1222++-(1-eiX+ix1|x|1)v(dx)(5)(5)Levy,,,Levy,CarrWu(2004)Levy,:LevyLevy,,,Tankov[12],,,,Levy,CopulaLevyCouplaKallsenTankov[13]Tankov(2003)LevyCoupla,CopulaLevyCoupla,LevyCouplaCoupla,,Coupla,LevyCouplaLevyCoupla,TankovLevyCoupla,LevyCoupla;Hunting[14]LevyCouplaLevyCoupla,,LevyCoupla,VarianceGamma,,LevyCoupla,,LevyCoupla,,LevyCoupla:Copula,,,,,,Copula,,,Copula,LevyCopula,Copula,ClaytonLevyCopula11CopulanŠRn,ŠR=[-,+],a,bŠRn,ab,abkakbk,nB=[a,b]=[a1,b1][a2,b2][an,bn],c=(c1,c2,,cn)B,ckakbk,H:ŠRnR,H-:1S1,S2,,SnŠRn,H:ŠRnR,DomH=S1S2Sn,B=[a,b]DomH,,BH-:VH(B)=sgn(c)H(c)(6),cB,sgn(c)=1ck=ak-1ck=ak2nHn-,DomHBVH(B)0,H(grounded),H(t)=0,tH,t1tk=ak,H,DomHskbk,HHk:xDomHk=Sk,:Hk(x)=H(b1,,bk-1,x,bk+1,,bn)(7)97LevyCopulaNelsen[15]KallsenTankov(2006)3nCnCopula,:(1)DomC=In,I=[0,1];(2)Cn-;(3)CCk,k=1,2,,n,uI,Ck(u)=u21LevyCopula(4)(5),Levy,Levy,Levyv(x),v(x)Copula,Copulav(x),:4XŠRnn-Levy,Levyv(x),XU:(ŠR\{0})nR:U(x1,x2,,xn)=ni=1sgn(xi)vnj=1(xj)(8)xR:(x)=[x,+)x0(-,x)x0(9)Tankov(2003),,LevynC,Copulann-,I{1,2,,n}CCI:ŠRkR(kI),:CI((ui)il)=limp(uj)jlc{-p,+}lcC(u1,u2,,un)jIcsgnuj(10),Ic{1,2,,n}\IIc1,Copula,Levy,LevyCopula:5C:ŠRnRLevyCopula,:(1){u1,u2,,un}(+,+,,+),C(u1,u2,,un)+;(2)Cn-,,C(u1,u2,,un)=0,i{1,2,,n},ui=0;(3)CCk,k=1,2,,n,uR,Ck(u)=u5,LevyCopulaCopula,[0,1],ŠR,,LevyLevyCopulaCopula,,LevyCopula,Sklar,CopulaClaytonLevyCopula31ClaytonLevyCopulaClaytonCopulaCopula,,Copula:1:[0,+)[0,+),(0)=+,(+)=0,(0,+),n(-1)ndn-1(t)dtn0,:C(x1,x2,,xn)=-1((x1)+(x2)++(xn))(11)nLevyCopula(11)(u)=u-,0,ClaytonCopula:C(x1,x2,,xn)=(x-1+x-2++x-n)-1/(12),,+,,0,,,,Levy08200910311LevyLevyCopulaLevyVarianceGamma11Levy(quasi-inverse),U,U-1:[0,+][0,+]:U(-1)=sup{x:U(x)t}t|RanUx,U(x)=ttRanU(13)LevyCopulaLevy:,,,,:2vRn+Levy,(|x|1)v(dx),vUi,i=1,2,,n,C(x1,x2,,xn)LevyCopula,x2,,xn:limX1C(x1,x2,,xn)=C(,x2,,xn){Vi}[0,1],{1i},{2i},,{ni}{Vi}:(1){1i}1Possion;(2){1i},({2i},,{ni}){j},ji,99x1C(x1,x2,,xn)X1={1i},zt=(z1t,z2t,,znt)Levy:eiu,zt=exp(tRn+(ei(u,z)-1)v(dz))(14):Zkt=i=1U(-1)i(ki)1[0,t](Vi),k=1,2,,n0t1i=1U(-1)i(ki),k=1,2,,nt1(15),Tankov(2003)21VarianceGammaLevy,(12)ClaytonCopula,VarianceGamma,VarianceGammab(t;9;)9,:b(t;9;)=9t+W(t)(16),W(t)(t;1;)1Gamma,VarianceGammaz(t):Z(t;;;9)=b((t;1;);9;)(17)(16),VarianceGammaGamma,Levy:v(x)=C|x|e--|x|1x0+cxe+X1x0(18),c=1/,+=92+22/2-92,-=92+22/2+92,(16)(12),-1(u)=u-1/,-1,n:-1(n)(u)=(-1)n(+1)(+2)(+n-1)nu-1-n(19)Cherubini(2004),{1i},2{2i},{3i},,{ni}:(1){Vi}n[0,1]v1,v2,,vn18LevyCopula{1i};(2)uk=ki,u1,uk,k=2,3,,n:uk=(u-1+u-2++u-n-1)-Vn1+(n-1)-1-1(20)(20)(15)VarianceGamma23,(12)=8=0183VarianceGamma,,(18),c=10,+=-=1,2000,(15)20002=833=018323,,=8,3,,;,=018,3,,,Nth-to-defaultCDS11Nth-to-defaultCDS,Nth-to-defaultCDS,,M,,1,1/MB,Nth-to-defaultCDS,PL,NA,DL,PL=DLS,,ti,i{1,2,,k},nn,n=N:PLN=S(ti)ki=11{Nti}(21),(ti)ti0N[ti-1,ti],N,N,DPN,[ti-1,N],APN,DLN=DPN-APN(4)4N28200910311,R,:DPN=1MNi=1(1-R)(i)1{NT}(22)APN=SN-ti-1ti-ti-1(N)(23)PLN=DLN:S=1MNi=1(1-R)(i)1{NT}(ti)ki=11{Nti}+N-ti-1ti-ti-1(N)(24)N,(24)21MVi(t),i=1,2,,MVarianceGamma,ClaytonCopula(12),VarianceGammaiKi,,:k=inf{t0:Vi(t)Ki}(25),LevyCopula:(1)(15)MVarianceGamma;(2)(25)k;(3)k,N,N,N;(4)(24)jSj,P,S=1/PPi=1Sj,1000,1,M=20,T=10,r=0105,R=014,=0125,Vi(0)=20,i=1,2,,M,Ki=0,i=1,2,,M,0r,ŽVi(t)=Vi(t)+Vi(0)(1+r)1,,Nth-to-defaultCDSN,,,,,5,NN=2,,1,,N,,,1Nth-to-defaultCDSN=019=1N=113251962010N=29271533613N=35621617512N=42941410213:,5,,38LevyCopula,VarianceGamma,ClaytonCopula,VarianceGamma,:,Nth-to-defaultCDSN,LevyCopula,LevyCopula,:,,:(1)Nth-to-defaultCDS,CDOs,;(2),,,LevyCopula,:[1]Demarta,S1,McNeil,A1J1ThetCopulaandRelatedCopulas[J]1InternationalStasticalReview,2005,73(11):111-1291[2]Friend,A1,E1Rogge1CorrelationatFirstSight[J]1EconomicNotes,2005,(34):