上财系列 金融风险控制与管理 qrm copulas and dependence

1、Quantitative Risk Management - Copulas and Dependence Ming-Heng Zhang RiskLab.CN Ri$kLab.CNTM Slide2 Out-Line nProperties of 2-Copulas nThe most important copulas nExtreme and singular copulas nDependency Bounds nApplication to VaR aggregation nApplication to two assets option pricing Ri$kLab.CNTM S

2、lide3 Chapter 1 Introduction nMotivations Multivariate Analysis Joint Distribution vs Marginal Distribution Joint and Marginal Distribution vs Risk Modelling For examples ?),( temultivaria offunction on distributi-joint theconstruct determine/ toHow ,cov and )( 2 1 exp 2 1 )(),(gievnFor 1 22 n ji ji

3、 ij i i i iii xxf DD x xfN i Ri$kLab.CNTM Slide4 Motivations nConsider a portfolio of financial risks X1,.,Xn a portfolio of traded financial assets e.g. equities; a credit portfolio of loans to various counter-parties; a portfolio of potential insurance losses in various lines of business or geogra

4、phical areas. nSampling i.i.d. nSuppose that the distribution of pay-off n-dimensional f(X1, . ,Xn) representing the risk of the portfolio or some contract written on the portfolio. nIts VERY difficulty to estimate joint distribution f()! nHow incorporate the marginal distribution of Xi to describe

5、the dependence structure of multivariate X1,.,Xn? Ri$kLab.CNTM Slide5 Bivariate Normal PDF nthe 2-dimension normal probability function )( )( 2)( )1 (2 1 exp 12 1 ),( )(cov( ),(),(gievnFor 22 2 22 y y yx yx x x yx yx yyxx yyyxxx xxx x yxf DD NN Ri$kLab.CNTM Slide6 Nearly Half-century of Copulas n195

6、9, M.Frchet and A.Sklar introduced the concepts of copulas Frchet,M.,1951, Sur Les tableaux de corrlation dont les marges sont donnes,Ann.Univ. Lyon 9,Sec.A,53-57 Frchet,M.,1957,Les tableaux de corrlation et les programmes lineaires, Revue Inst. Int. Statist. 25,23-40 Sklar,A.,1959, Fontions de repa

7、rtition a n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris 8, 229-231 Sklar,A.,1973, Random variables,joint distribution functions and copulas, Kybernetika 9, 449-460 Ri$kLab.CNTM Slide7 Gap between Risk Model and the World nTraditional Gaussian Hypothesis on the distribution of logari

8、thm returns or the complete market nTypical Model Markowitz Mean-variace portfolio, optimized, the variance corresponds to the risk measure but it implies the world is Gaussian nTwo difficulties Gaussian Assumption Joint Distribution Modelling nOne of the main issues of risk management is the aggreg

9、ation of individual risks. Ri$kLab.CNTM Slide8 Copulas and VaR Let X be random multivariate with distribution function Fi and Ui be standard-uniformly distributed at 0,1 nProbability-integral nQuantile Function nJoint Distribution nVaR of Portfolio Z at probability level iiii FUFUxF i )(;1 , 0)( 1 )

10、(|inf)()(|inf)( 11 xFxFxFxF iii )(:|inf)()( )(:|inf)()( 1 1 xFRxxFXVaR zFRzzFZVaR x n Z Z Z )(,),(),(),( 221121nnn xFxFxFxxxF Ri$kLab.CNTM Slide9 Functions of Copulas nRelationship between a multi-dimensional probability distribution function and its lower dimensional margins nFunctions of Copula Li

11、nk multivariate distributions to their on-dimensional margins nD.X. Li (1998, 2000) On default correlation: A copula function approach. Working paper 99-07, RiskMetrics Group. /picsresources/dlc.pdf nP.Embrechts, A.Mcneil and D.Straumann, Correlation and dependence in risk management:

12、 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 Applications nInsurance nLife (multi-life products) nNon-life (multi-line covers) nIntegrated risk management (Solvency 2/偿付能力) nDy

13、namic Financial analysis (assets and liabilities Model, ALM) nFinance nStress testing (Credit) nRisk aggregation nPricing/Hedging basket derivatives nRisk measure estimation under incomplete information nOther Fields nReliability, Survival analysis nEnvironmental science, Genetics nSee math.ethz.ch/

14、embrechts Ri$kLab.CNTM Slide11 Modelling by use of Copulas nModelling by use of Copulas Identification of the marginal distributions Selection/Definition/Construction of appropriate copula to presentation of the dependence structure nExample All assets are Gaussian Joint distribution is also Gaussia

15、n Ri$kLab.CNTM Slide12 Copulas and Multivariate Distributions nJoint Probability Distribution Function is of dependence structure of multivariate nTo Analyze without studying marginal distributions Linear correlation Rank Dependence Tail Dependence . Ri$kLab.CNTM Slide13 Chapter 2 Copulas nDef. Let

16、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 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=

17、a,b is the nth order difference of H on B where )()sgn()(cHcBVH s ofnumber oddan for if, 1 s ofnumber even an for if, 1 )sgn( kac kac c kk kk )()()( 1 1 1 1 tHtHBV b a b a b a b aH n n n n ),(),()( 111111nkkknkkk b a ttattHttbttHtH Ri$kLab.CNTM Slide14 Example of H-Volume nLet H be a 3-place real fu

18、nction with domain R3, and let B is the 3-box ,B=x1,x2y1,y2 z1,z2. Then, the H-Volume is nH-Volume can also be presented as ),(),(),(),( ),(),(),(),()( 111211121112 221212122222 zyxHzyxHzyxHzyxH zyxHzyxHzyxHzyxHBVH 0),() 1()( 2 1 2 1 1 21 21 ii iii iii H n n n uuuHBV Ri$kLab.CNTM Slide15 N-Increasin

19、g and Grounded Let H be An n-place real function and Dom H= S1S2Sn nbe n-increasing if VH(B)=0 for all n-boxes B whose vertices lie in Dom H nbe grounded if H(t)=0 for all t in Dom H such that tk=ak for at least one k nHas margins if each Sk is nonempty and has a greatest element bk Ri$kLab.CNTM Sli

20、de16 Example of n-increasing and grounded nLet 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) =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

21、 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) 12 sin) 1)(1( ),( y y ex zex zyxH Ri$kLab.CNTM Slide17 Def. of Copulas nDefinition - A copula function C is a multivariate uniform distribution (a multivariate distribution with uniform margins) or Dom C = I N =0,1N C

22、is grounded and N-increasing C has margins Cn which satisfy Cn (u)=C(1,.,1,u,1,.,1)=u for all u in 0,1 nDefinition - 2-Copula Dom C=0,10,1 C(0,u)=C(u,0) and C(u,1)=C(1,u)=u for all u in 0,1 C is 2-increasing : C(u1,v1) - C(u2,v2) - C(u1,v2) + C(u1,u2)0 whenever p2=(u2,v2) and p1=(u1,v1) in 0,12 such

23、 p2p1 Ri$kLab.CNTM Slide18 Alternative Definition of copula 0),() 1()( 2 1 2 1 1 21 21 ii iii iii H n n n uuuHBV nIn terms of a multivariate distribution function at 0,1n ,uniform random variables Ui: nIn terms of probability-integral transformation :Ui Fi(Xi) ,Pr),( 1 , 0),( 21121 21 nnnn n n uUuUu

24、UuuuC uuu )(,),(, )( ,Pr( )()(,),()(),()(Pr),( ,Pr),( 2211 2211 2222111121 221121 nn nn nnnn nnn xFxFxFC uUuUuU xFXFxFXFxFXFxxxF xXxXxXxxxF Ri$kLab.CNTM Slide19 Alternative Definition of Copula nDefinition. 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,

25、1 For all (a1,an), (b1,bn) in 0,1n with aibi, where uj1=aj and uj2=bj for all j in 1,2,n 0),() 1( 2 1 2 1 1 21 21 ii iii iii n n n uuuC Ri$kLab.CNTM Slide20 Explanations nCopula is a stochastic measurement on the volume of In= 0,1n nDependence function of random variables C(F1 (x1) ,Fn (xn) is multi

26、variate probability distribution function with margins F1 ,Fn(Fi单调不减） ),( )(,),(),( )()(,),()(),()(Pr ,Pr),( 21 2211 22221111 221121 n nn nnnn nnn uuuC xFxFxFC xFXFxFXFxFXF xXxXxXxxxF Ri$kLab.CNTM Slide21 Main Concepts of Copulas nJoint Distribution Function of Random Multivariate nGeneralized Inver

27、se of a Distribution Function nDomainC=A1A2 An(always0,1n ) Rn nGrounded C(a1,)=C(,a2,)=C(., an)=0 if ai=infAi(always 0) nIncreasing/H-Volume VH(B)0 means that the mass or area of the rectangle B where BDomainCRn nUp-Limits C(b1, bi-1,ui,bi+1,bn)=ui if bi=supAi(always 1) Ri$kLab.CNTM Slide22 Chapter

28、 3 Sklar theorem nLet F be an N-dimensional distribution function with margins F1 ,Fn . Then there exists an n-copula C such that for all x in Rn or If F1 ,Fn are all continuous, then C is unique; otherwise, C is uniquely determined on RanF1 RanF1RanFn. Conversely, if C is an n-copula and F1 ,Fn are

29、 distribution functions. Then the function H defined above is an n-dimensional distribution function with margins F1 ,Fn . Ri$kLab.CNTM Slide23 Concordance ordering nExamples Gumbel Copula Gaussian Copula/Normal Copula nN-copula C1 is smaller than n-copula C2 if for all (u1,u2,.,un) in the domain su

30、ch 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 2121 21 2 1 21 1 121 11 21 )(),(),( )1 ()(,)1 (),( 21 uuuu uu uFuFFuuC exFeexxF i x ii xx )(,),(),( ),( 1 1 1 1 2 nn ii uuuuC Nx i Ri$kLab.CNT

31、M Slide24 Frechet-Hoeffding Bounds Inequality nSpecific Copulas Note that Mn and n are n-copulas for all n=2 whereas the function Wn fails to be an n-copula for any n2 nFrechet-Hoeffding Bound nNormal Copula nBound of Copula See Prof. Embrechts, P. etal, as the fellowing Bounding Risk Measures for P

32、ortfolios with Known Marginal Risks Bounds for functions of multivariate risks Bounds for functions of dependent risks maximum0 , 1max)( production)( minimum,min)( 1 1 1 CnuuuW Cuuu CuuuM n n n n n n ( )( ) nn WuCMu CCCCCCCC 10001 Ri$kLab.CNTM Slide25 Probabilistic interpretation of the three copula

33、s nTwo 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; nTwo random variables X and Y are independent or have copula of if the dependence structure is the product copula -彼此独立

34、; nTwo 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; n C n MC n WC Ri$kLab.CNTM Slide26 Properties of Copula nFor n=2, let be continuous random variables. Then, are independ

35、ent iff the n-copula is . Each of the random variables is almost surely a strictly increasing function of any of others iff the n-copula of is . n C n WC Ri$kLab.CNTM Slide27 Properties of Copula nThe copula function of random variables is invariant under strictly increasing transformations (xi(xi)0

36、), 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|+. nLet k(xk) be strictly monotone and have copula C1(x1), ,n(xn), then If only 1(x1) is strictly decreasing, then If k(uk) are all strictly decreasing, then ),1 (),(),( 1)(,2)

37、(,),(1)(,),( 12111 nxxnxxnxx uuCuuCuuC nnnnnn )1 ,1 (1),( 1, 1 1)(,),( 111 nxx n k knxx uuCnuuuC nnn Ri$kLab.CNTM Slide28 Proof in Strictly Increasing nLet X1,.,Xn have margins F1,.,Fn and let 1(X1),. , n(Xn) have margins G1,.,Gn.(all i(Xi) strictly increasing). Then, )(,),(),( )(,),(),( )(,),(),(Pr

38、 )(,)(Pr)(, )( )()(Pr)(Pr)( 2211 1 2 1 21 1 11)(, )(, )( 1 2 1 21 1 1 11111)(,),( 11 22211 21 11 nn nnXXX nnn nnnnnxx kkkkkkkk xGxGxGC xFxFxFC xXxXxX xXxXxGxGC xFxXxXxG nnn nn kk Ri$kLab.CNTM Slide29 Proof in Strictly Decreasing nLet X1,.,Xn have margins F1,.,Fn and let 1(X1),. , n(Xn) have margins

39、G1,.,Gn.(only 1(X1) strictly decreasing). Then, )(,),(),(1 ( )(,),( )(,),(),( )(,),( )(,)(),(Pr )(,)(Pr )(,)(),(Pr )(,)(Pr)(, )( 2211)(, )(, 22)(, )( 221 1 1)(, )(, 22)(, )( 2221 1 1 222 2221 1 1 11111)(,),( 221 22 1221 22 1 1 11 nnXXX nnXX nnXXX nnXX nnn nnn nnn nnnnnxx xGxGxGC xGxGC xGxGxFC xGxGC

40、xXxXxX xXxX xXxXxX xXxXxGxGC nn nn nn nn nn Ri$kLab.CNTM Slide30 Survival Copula and Joint Survival Function nJoint survival probability and survival function nSurvival copula presents the joint survival probability in terms of the survival probabilities of the n components separately (under the Skl

41、ar theorem) 联结生存函数的函数 nRelationship with Copula where z(n-i,n,1) is the set of possible vectors with (n-i) components 111111 111111 ( ,)Pr,( ,)Pr, ( ,)Pr,( ,)Pr, nnnnnn nnnnnn F xxXxXxF xxXxXx C uuUuUuuuUuUu 11 11 0w(u)(, ,1) ( ,)( (),() ( , ,)(1,1)( 1)(1w) nn n i nn iZ n i n F xxC F xF x C uuuuC Ri

42、$kLab.CNTM Slide31 Survival Copula in 2-dimension n2-dimensional survival copula nSurvival copula vs Survival distributions function 12 12 1112 11 ( , )Pr,1( )( )( , ) 1( ) 1( )2 1( , ) ( )( ) 1(1( ),1( ) ( ( ),( ) ( , ) 1(1,1) F x yXx YyF xF yF x y F xF yF x y F xF xCF xF x C F x F x C u vuvCuv 3 0

43、w(u)(, ,1) 121323123 0 121323123 ( , , )( 1)(1w) 1( , )( , )( , )( , , ) ( , , ) (1,1,1) 2( )1( ) (1,1)(1,1)(1,1)(1,1,1) i iZ n i n iiii u v wC uvwCu vCu wCv wCu v w C u v wuvw uvwC uC uu CuvCuwCvwCuvw PrXx, YyPrXx, Yy X X Y Y P PPrX=x, Y=yPrX=x, Y=y Ri$kLab.CNTM Slide32 Conditional Distribution cmp

44、 Survival Copula nBayes Formula (joint probability ruler) nConditional distribution nTail dependence Measure Pr ,Pr |Pr yY yYxX yYxX Pr|Pr,PryYyYxXyYxX ) 1 , 0,;1 , 0Unioform,where 1 )1 ,1 (1 )(1 ),()()(1 1 )1 ,1 ( Pr ,Pr )( ),( Pr ,Pr )()(| )()(Pr|Pr 2 21 2 1111 1111 vuVU v vuCvu yF yxFyFxF v vuC v

45、V vVuU yF yxF yY yYxX vFyVFYuFxUFXyYxX Ri$kLab.CNTM Slide33 Survival Copula and Dual of a Copula Ri$kLab.CNTM Slide34 Confusion of “survival” nConfusion on Notation (R.B.Nelsen, pp.28-29) nThe survival copula of X and Y, the joint survival function to its univariate margins nThe joint survival funct

46、ion for two uniform (0,1) random variables whose joint distribution function is the copula C nRelationship ( , )1( )( )( , ) ( , ) 1(1,1) (Pr1,1) H x yF xG yH x y C u vuvCuv uuVv ( , )Pr,1( , )C u vUu VvuvC u v )1 ,1 ( ),(vuCvuC Ri$kLab.CNTM Slide35 Dual of Copula and co-copula nDual of Copula Some

47、of margins/components are less than or equal nCo-copula some of margins/components are greater than nNeither of these is a copula, but when C is the copula of, express a probability of 1211 1 11 1 12 ( ,)PrPr 1 Pr1 Pr, 1(1,1,1) n niinn i n iinn i n C u uuUuUuUu notUuUuUu Cuuu * 111 1 11 1 1 ( ,)PrPr

48、 1 Pr1 Pr, 1(1,1) n niinn i n iiin i n C uuUuUuUu notUuUuUu Cuu Ri$kLab.CNTM Slide36 Density of Copula nThe Density c(u1 , un) associated to the Copula C(u1 , un) is given by nThe density f(x1 , xn) of n-dimensional distribution F associated with the copula C(u1 , un) is presented as nExamples n n n

49、 n uu uuC uuc 1 1 1 ),( ),( n i iiniin xfxFxFcxxf 1 11 )()(,),(),( )(),( )1 ( )1 ( )()( )()( ),( ) 1(exp(),( 11 1 1 1 22 1 22 1 1 2 1 1 2 12 2 2 1 2 1 iiii n i T n n n n t T n utu uuc uuc i n Ri$kLab.CNTM Slide37 Copulas and Pearson Correlation nPearson Correlation nThe linear correlation is a measu

50、re of linear dependence If r.v. X and Y are independent, then (X,Y)=0; For r.v. X and Y, there exists Y=aX+b a.s., or PrY=aX+b=1 for aR-0, bR, (X,Y)=1 For any r.v. X and Y (aX+b, cY+d)= sign(ab)(X,Y) (AX+a, BY+b)= ACov(X,Y)Bt nExample )()( )( ),( 22 jjii jjii ji XEXE XXE XX 0 )()( 0 ),( ),1 , 0( 22

51、2 EE N ) 2 arcsin( 6 )(),( )()(set and), 0( yTxT xxTN y x Ri$kLab.CNTM Slide38 Drawbacks of linear correlation nLinear correlation of random variables X and Y is not defined if the variance of X or Y is infinite. Here exist examples of t- student distribution with =2, Non-life actuaries modelling (a

52、nd high-frequency data); nLinear correlation can easily be misinterpreted. That is, Independent NOT be equivalent to uncorrelated except Normal distribution; nLinear correlation is not invariant under non-linear strictly increasing transformations; nGiven distribution functions F and G for X and Y ,

53、 all linear correlations between -1 and 1 can in general not be obtained by a suitable choice of the joint distribution. nHot to determine the choice Copula and its unknown parameters under the given F and G and samples, i.e.,parameter estimation and infering; Ri$kLab.CNTM Slide39 Chapter 4 Dependen

54、ce nThe Real World is full of magic nCorrelation Dependence with Copula nConcordance nDependence nSummary of different copula functions “Dependence relations between random variables is one of the most widely studied subjects in probability and statistics. The nature of the dependence can take a var

55、iety 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 and N.L.Johanson, Ediotrs(John Weilry X,-X =-1 X,Y+1=X,X X,Y=Y,X if X a

56、nd Y are independent, then X,Y = =0 X,-Y= -X,Y=- X,Y If C1xj)and(yiyj) or (xixj)and(yixj)and(yiyj) or (xiyj) i.e., (xi-xj) (yi-yj)y|X x is a non- increasing function of x for all y, denoted by RTI(Y|X). nX is right tail increasing in Y if PrXx|Y y is a non- increasing function of y for all x, denote

57、d by RTI(X|Y). )(/ ),(|PrxFyxHxXyY )(/ ),(|PrxGyxHyYxX )(/ ),(|PrxFyxHxXyY )(/ ),(|PryGyxHyYxX Ri$kLab.CNTM Slide62 Properties of Tail Monotonicity/单调性 nIf X and Y satisfy each of the tail mono-tonicity conditions (i.e.,LTD or RTI), then X and Y are positively quadrant dependence. nTail Monotonicity

58、 presented in Copula LTD(Y|X) iff C(u,v)/u is non-increasing in u for any v; LTD(X|Y) iff C(u,v)/v is non-increasing in v for any u; RTI(Y|X) iff (1-u-v+C(u,v)/(1-u) is non-decreasing in u for any v; RTI(X|Y) iff (1-u-v+C(u,v)/(1-v) is non-decreasing in v for any u. Ri$kLab.CNTM Slide63 Definition o

59、f Stochastic Monotonicity nY is stochastically increasing in X if PrYy|X=x is a non- decreasing function of x for all y, denoted by SI(Y|X); nX is stochastically increasing in Y if PrXx|Y=y is a non- decreasing function of x for all y, denoted by SI(X|Y); nY is stochastically decreasing in X if PrYy

60、|X=x is a non- increasing function of x for all y, denoted by SD(Y|X); nX is stochastically decreasing in Y if PrXx|Y=y is a non- increasing function of x for all y, denoted by SD(X|Y). Ri$kLab.CNTM Slide64 Tail Dependence nMotivation - Particularly concerned with extreme values, EVT modelling nDefi