1、Quantitative Risk Management- Copulas and Dependence,Ming-Heng ZhangRiskLab.CN,Ri$kLab.CNTM,Slide2,Out-Line,Properties of 2-CopulasThe most important copulasExtreme and singular copulasDependency BoundsApplication to VaR aggregationApplication to two assets option pricing,Ri$kLab.CNTM,Slide3,Chapter
2、 1 Introduction,MotivationsMultivariate AnalysisJoint Distribution vs Marginal DistributionJoint and Marginal Distribution vs Risk ModellingFor examples,Ri$kLab.CNTM,Slide4,Motivations,Consider a portfolio of financial risks X1,.,Xna portfolio of traded financial assets e.g. equities;a credit portfo
3、lio of loans to various counter-parties;a portfolio of potential insurance losses in various lines of business or geographical areas.Sampling i.i.d.Suppose that the distribution of pay-off n-dimensional f(X1, . ,Xn) representing the risk of the portfolio or some contract written on the portfolio.Its
4、 VERY difficulty to estimate joint distribution f()!How incorporate the marginal distribution of Xi to describe the dependence structure of multivariate X1,.,Xn?,Ri$kLab.CNTM,Slide5,Bivariate Normal PDF,the 2-dimension normal probability function,Ri$kLab.CNTM,Slide6,Nearly Half-century of Copulas,19
5、59, M.Frchet and A.Sklar introduced the concepts of copulasFrchet,M.,1951, Sur Les tableaux de corrlation dont les marges sont donnes,Ann.Univ. Lyon 9,Sec.A,53-57Frchet,M.,1957,Les tableaux de corrlation et les programmes lineaires, Revue Inst. Int. Statist. 25,23-40Sklar,A.,1959, Fontions de repart
6、ition a n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris 8, 229-231Sklar,A.,1973, Random variables,joint distribution functions and copulas, Kybernetika 9, 449-460,Ri$kLab.CNTM,Slide7,Gap between Risk Model and the World,Traditional Gaussian Hypothesis on the distribution of logarithm
7、returns or the complete marketTypical Model Markowitz Mean-variace portfolio, optimized, the variance corresponds to the risk measure but it implies the world is Gaussian Two difficultiesGaussian AssumptionJoint Distribution ModellingOne of the main issues of risk management is the aggregation of in
8、dividual risks.,Ri$kLab.CNTM,Slide8,Copulas and VaR,Let X be random multivariate with distribution function Fi and Ui be standard-uniformly distributed at 0,1Probability-integralQuantile FunctionJoint DistributionVaR of Portfolio Z at probability level ,Ri$kLab.CNTM,Slide9,Functions of Copulas,Relat
9、ionship between a multi-dimensional probability distribution function and its lower dimensional marginsFunctions of Copula Link multivariate distributions to their on-dimensional marginsD.X. Li (1998, 2000) On default correlation: A copula function approach. Working paper 99-07, RiskMetrics Group. w
10、ww.gloriamundi.org/picsresources/dlc.pdfP.Embrechts, A.Mcneil and D.Straumann, Correlation and dependence in risk management: properties and pitfalls. In: M.A.H.Dempster(Ed.),Value at risk and beyond, Cambridge University Press, Cambridge,1999,pp.176-223.,Ri$kLab.CNTM,Slide10,Potential Copula Applic
11、ations,InsuranceLife (multi-life products)Non-life (multi-line covers)Integrated risk management (Solvency 2/偿付能力)Dynamic Financial analysis (assets and liabilities Model, ALM)FinanceStress testing (Credit)Risk aggregationPricing/Hedging basket derivativesRisk measure estimation under incomplete inf
12、ormationOther FieldsReliability, Survival analysisEnvironmental science, GeneticsSee www.math.ethz.ch/embrechts,Ri$kLab.CNTM,Slide11,Modelling by use of Copulas,Modelling by use of CopulasIdentification of the marginal distributionsSelection/Definition/Construction of appropriate copula to presentat
13、ion of the dependence structure ExampleAll assets are GaussianJoint distribution is also Gaussian,Ri$kLab.CNTM,Slide12,Copulas and Multivariate Distributions,Joint Probability Distribution Function is of dependence structure of multivariateTo Analyze the dependence structure of multivariate distribu
14、tions without studying marginal distributionsthe dependence structureLinear correlationRank DependenceTail Dependence .,Ri$kLab.CNTM,Slide13,Chapter 2 Copulas,Def. Let S1,S2,Sn be nonempty subsets of R, and let H be an n-place real function such that Dom H=S1S2Sn. Let B=a,b be an n-box all of whose
15、vertices are in Dom H. Then the H-volume of B is given by where the sum is taken over all vertices c of B, and sgn is given by Equivalently, the H-Volume of an n-box B=a,b is the nth order difference of H on B where,Ri$kLab.CNTM,Slide14,Example of H-Volume,Let H be a 3-place real function with domai
16、n R3, and let B is the 3-box ,B=x1,x2y1,y2 z1,z2. Then, the H-Volume isH-Volume can also be presented as,Ri$kLab.CNTM,Slide15,N-Increasing and Grounded,Let H be An n-place real function and Dom H= S1S2Snbe n-increasing if VH(B)=0 for all n-boxes B whose vertices lie in Dom Hbe grounded if H(t)=0 for
17、 all t in Dom H such that tk=ak for at least one kHas margins if each Sk is nonempty and has a greatest element bk,Ri$kLab.CNTM,Slide16,Example of n-increasing and grounded ,Let H be the function with domain -1,+10,0, /2 given by Then, (should verify H is n-increasing) H is grounded since H(-1,y,z)
18、=H(x,0,z)=H(x,y,0)=0; H is one-dimensional margins: H1(x)=H(x, /2) = (x+1)/2 H2(y)=H(1, y, /2) = 1-e-y H3(z)=H(1,z) = sinz and also 2-dimensional margins H1,2(x,y)=H(x,y, /2) H2,3(y,z)=H(1,y, z) H1,3(x,z)=H(x, z),Ri$kLab.CNTM,Slide17,Def. of Copulas,Definition - A copula function C is a multivariate
19、 uniform distribution (a multivariate distribution with uniform margins) orDom C = I N =0,1NC is grounded and N-increasingC has margins Cn which satisfy Cn (u)=C(1,.,1,u,1,.,1)=u for all u in 0,1Definition - 2-CopulaDom C=0,10,1C(0,u)=C(u,0) and C(u,1)=C(1,u)=u for all u in 0,1C is 2-increasing : C(
20、u1,v1) - C(u2,v2) - C(u1,v2) + C(u1,u2)0 whenever p2=(u2,v2) and p1=(u1,v1) in 0,12 such p2p1,Ri$kLab.CNTM,Slide18,Alternative Definition of copula,In terms of a multivariate distribution function at 0,1n ,uniform random variables Ui:In terms of probability-integral transformation :Ui Fi(Xi),Ri$kLab
21、.CNTM,Slide19,Alternative Definition of Copula,Definition.C(u1,un) is increasing in each components;C(1,1,ui,1,1) = ui for all i in 1,2,n and ui in 0,1For all (a1,an), (b1,bn) in 0,1n with aibi, where uj1=aj and uj2=bj for all j in 1,2,n,Ri$kLab.CNTM,Slide20,Explanations,Copula is a stochastic measu
22、rement on the volume of In= 0,1nDependence function of random variables C(F1 (x1) ,Fn (xn) is multivariate probability distribution function with margins F1 ,Fn(Fi单调不减）,Ri$kLab.CNTM,Slide21,Main Concepts of Copulas,Joint Distribution Function of Random MultivariateGeneralized Inverse of a Distributi
23、on FunctionDomainC=A1A2 An(always0,1n ) RnGrounded C(a1,)=C(,a2,)=C(., an)=0 if ai=infAi(always 0)Increasing/H-VolumeVH(B)0 means that the mass or area of the rectangle B where BDomainCRnUp-LimitsC(b1, bi-1,ui,bi+1,bn)=ui if bi=supAi(always 1),Ri$kLab.CNTM,Slide22,Chapter 3 Sklar theorem,Let F be an
24、 N-dimensional distribution function with margins F1 ,Fn . Then there exists an n-copula C such that for all x in Rn H(x1 , xn) = C(F1 (x1) ,Fn (xn) or H(F-11 (u1) , F-1n (un) = C(u1 , un) If F1 ,Fn are all continuous, then C is unique; otherwise, C is uniquely determined on RanF1 RanF1RanFn. Conver
25、sely, if C is an n-copula and F1 ,Fn are distribution functions. Then the function H defined above is an n-dimensional distribution function with margins F1 ,Fn .,Ri$kLab.CNTM,Slide23,Concordance ordering,ExamplesGumbel CopulaGaussian Copula/Normal CopulaN-copula C1 is smaller than n-copula C2 if fo
26、r all (u1,u2,.,un) in the domain such that C1(u1,u2,.,un)C2(u1,u2,.,un) ,denoted by C1C2 The order “”called the concordance order for distributions or the stochastic order for random variables,Ri$kLab.CNTM,Slide24,Frechet-Hoeffding Bounds Inequality,Specific Copulas Note that Mn and n are n-copulas
27、for all n=2 whereas the function Wn fails to be an n-copula for any n2Frechet-Hoeffding Bound Normal CopulaBound of Copula See Prof. Embrechts, P. etal, as the fellowingBounding Risk Measures for Portfolios with Known Marginal RisksBounds for functions of multivariate risksBounds for functions of de
28、pendent risks,Ri$kLab.CNTM,Slide25,Probabilistic interpretation of the three copulas,Two random variables X and Y are counter-monotonic, or have copula of if there exists a r.v. Z such that X=fx(Z) and Y=fY(Z) with fX non-increasing and fY non-decreasing -反向单调;Two random variables X and Y are indepe
29、ndent or have copula of if the dependence structure is the product copula -彼此独立;Two random variables X and Y are co-monotonic, or have copula of if there exists a r.v. Z such that X=fX (Z) and Y=fY(Z) where the functions fX and fY are non-decreasing -同向单调;,Ri$kLab.CNTM,Slide26,Properties of Copula,F
30、or n=2, let x1 , xn be continuous random variables. Then,x1 , xn are independent iff the n-copula is .Each of the random variables x1 , xn is almost surely a strictly increasing function of any of others iff the n-copula of x1 , xn is .,Ri$kLab.CNTM,Slide27,Properties of Copula,The copula function o
31、f random variables is invariant under strictly increasing transformations (xi(xi)0), i.e. C(1(x1), n(xn)= C(x1,xn) where, transformation function i(xi) are strictly increasing such as log, Exp, and |f(x)-K|+.Let k(xk) be strictly monotone and have copula C1(x1), ,n(xn), then If only 1(x1) is strictl
32、y decreasing, then If k(uk) are all strictly decreasing, then,Ri$kLab.CNTM,Slide28,Proof in Strictly Increasing,Let X1,.,Xn have margins F1,.,Fn and let 1(X1),. , n(Xn) have margins G1,.,Gn.(all i(Xi) strictly increasing). Then,Ri$kLab.CNTM,Slide29,Proof in Strictly Decreasing,Let X1,.,Xn have margi
33、ns F1,.,Fn and let 1(X1),. , n(Xn) have margins G1,.,Gn.(only 1(X1) strictly decreasing). Then,Ri$kLab.CNTM,Slide30,Survival Copula and Joint Survival Function,Joint survival probability and survival functionSurvival copula presents the joint survival probability in terms of the survival probabiliti
34、es of the n components separately (under the Sklar theorem) 联结生存函数的函数Relationship with Copula where z(n-i,n,1) is the set of possible vectors with (n-i) components,Ri$kLab.CNTM,Slide31,Survival Copula in 2-dimension,2-dimensional survival copulaSurvival copula vs Survival distributions function,PrXx
35、, Yy,X,Y,PPrX=x, Y=y,Ri$kLab.CNTM,Slide32,Conditional Distribution cmp Survival Copula,Bayes Formula (joint probability ruler)Conditional distributionTail dependence Measure,Ri$kLab.CNTM,Slide33,Survival Copula and Dual of a Copula,Ri$kLab.CNTM,Slide34,Confusion of “survival”,Confusion on Notation (
36、R.B.Nelsen, pp.28-29)The survival copula of X and Y, the joint survival function to its univariate marginsThe joint survival function for two uniform (0,1) random variables whose joint distribution function is the copula CRelationship,Ri$kLab.CNTM,Slide35,Dual of Copula and co-copula,Dual of Copula
37、Some of margins/components are less than or equalCo-copula some of margins/components are greater thanNeither of these is a copula, but when C is the copula of, express a probability of ,Ri$kLab.CNTM,Slide36,Density of Copula,The Density c(u1 , un) associated to the Copula C(u1 , un) is given byThe
38、density f(x1 , xn) of n-dimensional distribution F associated with the copula C(u1 , un) is presented asExamples,Ri$kLab.CNTM,Slide37,Copulas and Pearson Correlation,Pearson CorrelationThe linear correlation is a measure of linear dependenceIf r.v. X and Y are independent, then (X,Y)=0;For r.v. X an
39、d Y, there exists Y=aX+b a.s., or PrY=aX+b=1 for aR-0, bR, (X,Y)=1For any r.v. X and Y (aX+b, cY+d)= sign(ab)(X,Y) (AX+a, BY+b)= ACov(X,Y)BtExample,Ri$kLab.CNTM,Slide38,Drawbacks of linear correlation,Linear correlation of random variables X and Y is not defined if the variance of X or Y is infinite
40、. Here exist examples of t-student distribution with =2, Non-life actuaries modelling (and high-frequency data);Linear correlation can easily be misinterpreted. That is, Independent NOT be equivalent to uncorrelated except Normal distribution;Linear correlation is not invariant under non-linear stri
41、ctly increasing transformations;Given distribution functions F and G for X and Y , all linear correlations between -1 and 1 can in general not be obtained by a suitable choice of the joint distribution.Hot to determine the choice Copula and its unknown parameters under the given F and G and samples,
42、 i.e.,parameter estimation and infering;,Ri$kLab.CNTM,Slide39,Chapter 4 Dependence,The Real World is full of magicCorrelation Dependence with CopulaConcordance DependenceSummary of different copula functions “Dependence relations between random variables is one of the most widely studied subjects in
43、 probability and statistics. The nature of the dependence can take a variety of forms and unless some specific assumptions are made about the dependence, no meaningful statistical model can be contemplated.” Jogdeo (1982) ,“Concept of dependence,” in Encyclopedia of Statistical Science, Vol.1,S.Kotz
44、 and N.L.Johanson, Ediotrs(John Weilry & Sons, New York),Ri$kLab.CNTM,Slide40,Definition of Concordance/和谐性,Measure of Concordance A numeric measure of association between two continuous random variables X and Y whose copula is C is a measure of concordance if it satisfies the following properties(N
45、elsen(1998),p136) (X,Y =C) is defined for every pair x,y of continuous r.v.;X,-X =-1 X,Y+1=X,XX,Y=Y,Xif X and Y are independent, then X,Y = =0X,-Y= -X,Y=- X,YIf C1C2, then C1 C2If Xn,Yn is a sequence of continuous r.v. with copula Cn and Cn converges point-wise to C, then limits of Cn is equal to C
46、when n goes to infinite .,Ri$kLab.CNTM,Slide41,Explanation on Definition of Concordance,Fact that the copula function of random variables (X1,X2,.,Xn) is invariant under strictly increasing transformationsThe famous measures in term of copulaKendalls tauSpearmans rhoGini indice,Ri$kLab.CNTM,Slide42,
47、Definition of Dependence/非独立性/相关性,Measure of Dependence A numeric measure of association between two continuous random variables X and Y whose copula is C is a measure of concordance if it satisfies the following properties(Nelsen(1998),p170) (X,Y = C) is defined for every pair x,y of continuous r.v
48、.;X,Y= Y,X0X,Y1X,Y=0 if and only if X and Y are independentX,Y=1 if and only if each of X and Y is almost surely a strictly monotone function of the otherIf and are almost surely monotone functions on Ran X and Ran Y respectively, then (X),(Y)= X,YIf Xn,Yn is a sequence of continuous r.v. with copula Cn and Cn converges point-wise to C, then limits of Cn is equal to C when n goes to infinite.,
