Slides euria-2

1. Arthur CHARPENTIER - cole dt EURIA. mesures de risques et dpendance Arthur Charpentier Universit de Rennes 1 & cole Polytechnique [email protected] http://blogperso.univ-rennes1.fr/arthur.charpentier/index.php/ 1

2. Arthur CHARPENTIER - cole dt EURIA. 3 0.9 V (rank of Y)Y 0.4 -1 (X i,Y i) (U i,V i) -3 -1 1 3 0.2 0.5 0.8 X U (rank of X) Density of the copula Isodensity curves of the density 2

3. Arthur CHARPENTIER - cole dt EURIA. Agenda General introductionModelling correlated risks A short introduction to copulas Quantifying dependence Statistical inference Agregation properties 3

5. Arthur CHARPENTIER - cole dt EURIA. Some references on large and correlated risksRank , J. (2006). Copulas: From Theory to Application in Finance. Risk Book ,Nelsen , R. (1999,2006). An introduction to copulas. Springer Verlag ,Cherubini , U., Luciano , E. & Vecchiato, W. (2004). Copula Methods inFinance. Wiley,Beirlant , J., Goegebeur , Y., Segers, J. & Teugels , J. (2004). Statistics ofExtremes: Theory and Applications. Wiley,McNeil , A. Frey , R., & Embrechts , P. (2005). Quantitative RiskManagement: Concepts, Techniques, and Tools. Princeton University Press, 5

7. Arthur CHARPENTIER - cole dt EURIA. Copulas, an introduction (in dimension 2)Denition 1. A copula C is a joint distribution function on [0, 1]2 , with uniformmargins on [0, 1].Set C(u, v) = P(U u, V v), where (U, V ) is a random pair with uniformmargins.C is a distribution function on [0, 1]2 , and thus C(0, v) = C(u, 0) = 0, C(1, 1) = 1.Furthermore C is increasing : since P is a positive measure, for all u1 u2 andv1 v2 , Copula, positive area 1.0 P(u1 < U u2 , v1 < V v2 ) 0, 0.8 0.6 thus 0.4 C(u2 , v2 ) C(u1 , v2 ) 0.2 C(u2 , v1 ) + C(u1 , v1 ) 0. 0.0 0.0 0.2 0.4 0.6 0.8 1.0 7

8. Arthur CHARPENTIER - cole dt EURIA.C has uniform margins, and thus C(u, 1) = P(U u, V 1) = P(U u) = u on [0, 1].Proposition 2. C is a copula if and only if C(0, v) = C(u, 0) = 0, C(u, 1) = uand C(1, v) = v for all u, v , with the following 2-increasingness property C(u2 , v2 ) C(u1 , v2 ) C(u2 , v1 ) + C(u1 , v1 ) 0,for any u1 u2 and v1 v2 . 8

9. Arthur CHARPENTIER - cole dt EURIA. Borders of the copula function !0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 !0.2 !0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Figure 1: Value of the copula on the border of the unit square. 9

10. Arthur CHARPENTIER - cole dt EURIA. Fonction de rpartition marges uniformes Z Y X Figure 2: Graphical representation of a copula. 10

11. Arthur CHARPENTIER - cole dt EURIA.If C is twice dierentiable, one can dene its density as 2 C(u, v) c(u, v) = . uv 11

12. Arthur CHARPENTIER - cole dt EURIA. Densit dune loi marges uniformes z x x Figure 3: Density of a copula. 12

13. Arthur CHARPENTIER - cole dt EURIA. Fonction de rpartition marges uniformes Densit dune loi marges uniformes Fonction de rpartition marges uniformes Densit dune loi marges uniformes Figure 4: Distribution functions and densities. 13

14. Arthur CHARPENTIER - cole dt EURIA. Fonction de rpartition marges uniformes Densit dune loi marges uniformes Fonction de rpartition marges uniformes Densit dune loi marges uniformes Figure 5: Distribution functions and densities. 14

15. Arthur CHARPENTIER - cole dt EURIA. Sklars theoremTheorem 3. (Sklar) Let C be a copula, and FX and FY two marginaldistributions, then F (x, y) = C(FX (x), FY (y)) is a bivariate distributionfunction, with F F(FX , FY ).Conversely, if F F(FX , FY ), there exists C such thatF (x, y) = C(FX (x), FY (y)). Further, if FX and FY are continuous, then C isunique, and given by C(u, v) = F (FX (u), FY (v)) for all (u, v) [0, 1] [0, 1] 1 1We will then dene the copula of F , or the copula of (X, Y ).In that case, the copula of (X, Y ) is the distribution function of (FX (X), FY (Y )).Proposition 4. If (X, Y ) has copula C , the copula of (g(X), h(Y )) is also C forany increasing functions g and h. 15

16. Arthur CHARPENTIER - cole dt EURIA. Copulas, an introduction (in dimension d 2)Denition 5. A copula C is a joint distribution function on [0, 1]d , withuniform margins on [0, 1].Let U = (U1 , ..., Ud ) denote a random pair with uniform margins.C is a distribution function on [0, 1]d , and thus C(u) = 0 if ui = 0 for somei {1, . . . , d}, and C(1) = 1.Furthermore C satises some increasing property since P is a positive measure(for all 0 u v 1, P(u < U v) 0), thus sign(z)C(z) 0, zwhere the sum is taken over all vertices of [u v], and where sign(z) is +1 ifzi = ui for an even number of i (and 1 otherwise, see Figure 6). And nally Chas uniform margins, and thus C(1, . . . , 1, ui , 1, . . . , 1) = ui on [0, 1]. 16

17. Arthur CHARPENTIER - cole dt EURIA. Increasing functions in dimension 3 Figure 6: The notion of 3-increasing functions. 17

18. Arthur CHARPENTIER - cole dt EURIA.Theorem 6. (Sklar) Let C be a copula, and F1 , . . . , Fd be d marginaldistributions, then F (x) = C(F1 (x1 ), . . . , Fd (xd )) is a distribution function, withF F(F1 , . . . , Fd ).Conversely, if F F(F1 , . . . , Fd ), there exists C such thatF (x) = C(F1 (x1 ), . . . , Fd (xd )). Further, if the Fi s are continuous, then C isunique, and given by C(u) = F (F1 (u1 ), . . . , Fd (ud )) for all (ui ) [0, 1] 1 1We will then dene the copula of F , or the copula of X .In that case, the copula of (X = (X1 , . . . , Xd ) is the distribution function ofU = (F1 (X1 ), . . . , Fd (Yd )).Again, if C is dierentiable, one can dene its density, d C(u1 , . . . , ud ) c(u1 , . . . , ud ) = . u1 . . . ud 18

19. Arthur CHARPENTIER - cole dt EURIA. Copulas in high dimension, a dicult problemIt is usually dicult to represent dependence in dimension d > 2, and it isusually studied by pairs.In dimension d = 2, one can dene the following Frchet class F(FX , FY , FZ )dened by its marginal distributions. But it can also be interested to studyF(FXY , FXZ , FY Z ) dened by it paired distributions.One of the problem that arises is the compatibility of marginals: one has toverify that CXY (x, y) = CX|Z (x|z)CY |Z (y|z)dz,for instance. 19

20. Arthur CHARPENTIER - cole dt EURIA. 1.0 0.8 1.0 0.6 0.8 p 0.6 0.4 1.0 0.4 0.8 0.2 0.6 0.2 0.4 0.2 0.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Composante 1 1.0 1.0 0.8 0.8 0.6 0.6 p p 0.4 0.4 0.2 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Composante 2 Composante 3 Figure 7: Scatterplot in dimension 3 including projections. 20

21. Arthur CHARPENTIER - cole dt EURIA. Copulas and ranksThe copula of X = (X1 , . . . , Xd ) is the distribution function ofU = (F1 (X1 ), . . . , Fd (Yd )).In practice, since marginal distributions are unknown, the idea is to substituteempirical distribution function, n #{observations Xi,j s lower than xi } 1 Fi (xi ) = = 1(Xi,j xi ). #{observations } n j=1Note that n #{observations Xi,j s lower than Xi,j0 } 1 Ri,j0Fi (Xi,j0 ) = = 1(Xi,j Xi,j0 ) = , #{observations } n j=1 nwhere Ri,j0 denotes the rank of Xi,j0 within {Xi,1 , ..., Xi,n }.On a statistical point of view, studying the copula means studying ranks. 21

22. Arthur CHARPENTIER - cole dt EURIA. Scatterplot of (X,Y) Scatterplot of the ranks of (X,Y) 9.0 20 8.5 8.0 15 Ranks of the Yis 7.5 Y (raw data) 10 7.0 6.5 5 6.0 5.5 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 5 10 15 20 X (raw data) Ranks of the Xis Scatterplot of the ranks of (X,Y), divided by n Scatterplot o+ ,-,/0, t1e copula!t3pe tran+orm o+ ,6,70 1.0 1.0 0.8 0.8 Vi=Ranks of the Yis/n+1 Ranks of the Yis/n 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Ranks of the Xis/n Ui=Ranks of the Xis/n+1Figure 8: Copulas, ranks and parametric inference, from (Xi , Yi ) to (Ui , Vi ). 22

23. Arthur CHARPENTIER - cole dt EURIA. Some very classical copulas The independent copula C(u, v) = uv = C (u, v).The copula is standardly denoted , P or C , and an independent version of L(X, Y ) will be denoted (X , Y ). It is a random vector such that X = X and LY =Y, with copula C .In higher dimension, C (u1 , . . . , ud ) = u1 . . . ud is the independent copula. The comonotonic copula C(u, v) = min{u, v} = C + (u, v).The copula is standardly denoted M, or C +, and an comonotone version of L(X, Y ) will be denoted (X + , Y + ). It is a random vector such that X+ = X and LY+ =Y, with copula C +.(X, Y ) has copula C+ if and only if there exists a strictly increasing function h L1 1such that Y = h(X), or equivalently (X, Y ) = (FX (U ), FY (U )) where U isU([0, 1]). 23

24. Arthur CHARPENTIER - cole dt EURIA.Note that for any u, v P(U u, V v) = P({U [0, u]} {V [0, v]}) min{P(U [0, u]), P(V [0, v])}thus, C(u, v) min{u, v} = C + (u, v). Thus, C+ is an upper bound for the set ofcopulas.In higher dimension, C + (u1 , . . . , ud ) = min{u1 , . . . , ud } is the comonotoniccopula. The contercomotonic copula C(u, v) = max{u + v 1, 0} = C (u, v).The copula is standardly denoted W, or C , and an contercomontone version of L(X, Y ) will be denoted (X , Y ). It is a random vector such that X = X and LY =Y, with copula C .(X, Y ) has copula C if and only if there exists a strictly decreasing function h L1 1such that Y = h(X), or equivalently (X, Y ) = (FX (1 U ), FY (U )) where U isU([0, 1]). 24

25. Arthur CHARPENTIER - cole dt EURIA.Note that for any u, v ,P(U u, V v) = P({U [0, u]} {V [0, v]}) = P(U [0, u]) + P(V [0, v]) P({U [0, u]} {V [0, v]})thus, C(u, v) u + v 1 since P({U [0, u]} {V [0, v]}) 1, and sinceC(u, v) 0, C(u, v) max{u + v 1, 0} = C (u, v). Thus, C is a lower boundfor the set of copulas.In higher dimension, C (u1 , . . . , ud ) = max{u1 + . . . + ud (d 1), 0} is not acopula: if (X, Y ) and (X, Z) are countercomonotonic, (Y, Z) is necessarilycomonotonic - it is not possible to have all component highly negativelycorrelated.Anyway, it is still the best pointwise lower bound. 25

26. Arthur CHARPENTIER - cole dt EURIA. 1 1 0.8 1 nd 0.8 0.2 0.4 0.6 0.8 la nd Independence copu Frechet lower bou Frechet upper bou0 0.2 0.4 0.6 0 0.2 0.4 0.6 0 0.8 0.8 0.8 0.6 0.6 0.8 0.8 0.6 0.8 u_ 0.4 0.6 u_ 0.4 0.6 2 2 u_ 0.4 0.6 2 0.4 0.4 u_1 0.2 u_1 0.2 0.2 0.4 u_1 0.2 0.2 0.2 Frchet Lower Bound Independent copula Frchet Upper Bound1.0 1.0 1.00.8 0.8 0.80.6 0.6 0.60.4 0.4 0.40.2 0.2 0.20.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Scatterplot, Lower Frchet!Hoeffding bound Scatterplot, Indepedent copula random generation Scatterplot, Upper Frchet!Hoeffding bound1.0 1.0 1.00.8 0.8 0.80.6 0.6 0.60.4 0.4 0.40.2 0.2 0.20.0 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 9: Contercomontonce, independent, and comonotone copulas. 26

27. Arthur CHARPENTIER - cole dt EURIA. Pitfalls on independence and comonotonicityThe following proposition is false,Uncorrect Proposition 7. If X and Y are independent, if Y and Z areindependent, then X and Z are independent.If (X, Y, Z) = (1, 1, 1) with probability 1/4, (1, 2, 1) with probability 1/4, (3, 2, 3) with probability 1/4, (3, 1, 3) with probability 1/4,then X and Y are independent, and Y and Z are independent, but X = Z. 27

28. Arthur CHARPENTIER - cole dt EURIA. X and Y independent Y and Z independent X and Z comonotonic 4 4 4 3 3 3Component Y Component Z Component Z 2 2 2 1 1 1 0 0 0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 Component X Component Y Component X Figure 10: Mixing independence and comonotonicity. 28

29. Arthur CHARPENTIER - cole dt EURIA. Pitfalls on independence and comonotonicityThe following proposition is false,Uncorrect Proposition 8. If X and Y are comonotonic, if Y and Z arecomonotonic, then X and Z are comonotonic.If (X, Y, Z) = (1, 1, 1) with probability 1/4, (1, 2, 3) with probability 1/4, (3, 2, 1) with probability 1/4, (3, 3, 3) with probability 1/4,then X and Y are comonotonic, and Y and Z are comonotonic, but X and Z areindependent. 29

30. Arthur CHARPENTIER - cole dt EURIA. X and Y comonotonic Y and Z comonotonic X and Z independent 4 4 4 3 3 3Component Y Component Z Component Z 2 2 2 1 1 1 0 0 0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 Component X Component Y Component X Figure 11: Mixing independence and comonotonicity. 30

31. Arthur CHARPENTIER - cole dt EURIA. Pitfalls on independence and comonotonicityThe following proposition is false,Uncorrect Proposition 9. If X and Y are comonotonic, if Y and Z areindependent, then X and Z are independent.If (X, Y, Z) = (1, 1, 3) with probability 1/4, (2, 1, 1) with probability 1/4, (2, 3, 3) with probability 1/4, (3, 3, 1) with probability 1/4,then X and Y are comonotonic, and Y and Z are independent, but X and Z areanticomonotonic. 31

32. Arthur CHARPENTIER - cole dt EURIA.If (X, Y, Z) = (1, 1, 1) with probability 1/4, (2, 1, 3) with probability 1/4, (2, 3, 1) with probability 1/4, (3, 3, 3) with probability 1/4,then X and Y are comonotonic, and Y and Z are independent, but X and Z arecomonotonic. 32

33. Arthur CHARPENTIER - cole dt EURIA. X and Y comonotonic Y and Z independent X and Z comonotonic 4 4 4 3 3 3Component Y Component Z Component Z 2 2 2 1 1 1 0 0 0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 Component X Component Y Component X Figure 12: Mixing independence and comonotonicity. 33

34. Arthur CHARPENTIER - cole dt EURIA. Elliptical (Gaussian and t) copulasThe idea is to extend the multivariate probit model, Y = (Y1 , . . . , Yd ) withmarginal B(pi ) distributions, modeled as Yi = 1(Xi ui ), where X N (I, ). The Gaussian copula, with parameter (1, 1), 1 (u) 1 (v) 1 (x2 2xy + y 2 ) C(u, v) = exp dxdy. 2 1 2 2(1 2 )Analogously the t-copula is the distribution of (T (X), T (Y )) where T is the t-cdf,and where (X, Y ) has a joint t-distribution. The Student t-copula with parameter (1, 1) and 2, t1 (u) t1 (v) ((+2)/2) 1 x2 2xy + y 2 C(u, v) = 1+ dxdy. 2 1 2 2(1 2 ) 34

35. Arthur CHARPENTIER - cole dt EURIA. Archimedean copulas Denition of Archimedean copulas Archimedian copulas C(u, v) = 1 ((u) + (v)), where is decreasing convex (0, 1), with (0) = and (1) = 0.Example 10. If (t) = [ log t] , then C is Gumbels copula, and if(t) = t 1, C is Claytons. Note that C is obtained when (t) = log t. How Archimedean copulas were introduced ?1. The frailty approach ( Oakes (1989)).Assume that X and Y are conditionally independent, given the value of anheterogeneous component . Assume further that P(X x| = ) = (GX (x)) and P(Y y| = ) = (GY (y))for some baseline distribution functions GX and GY .Then F (x, y) = P(X x, Y y) = E(P(X x, Y y| = )) 35

36. Arthur CHARPENTIER - cole dt EURIA.thus, since X and Y are conditionally independent, F (x, y) = E(P(X x| = ) P(Y y| = ))and therefore F (x, y) = E (GX (x)) (GY (y)) = ( log GX (x) log GY (y))where denotes the Laplace transform of , i.e. (t) = E(et ). Since FX (x) = ( log GX (x)) and FY (y) = ( log GY (y))and thus, the joint distribution of (X, Y ) satises F (x, y) = ( 1 (FX (x)) + 1 (FY (y))).Example 11. If is Gamma distributed, the associated copula is Claytons. If has a stable distribution, the associated copula is Gumbels. 36

37. Arthur CHARPENTIER - cole dt EURIA.Consider two risks, X and Y, such that X| = G E(G ) and Y | = G E(G ) are independent, X| = B E(B ) and Y | = B E(B ) are independent,(unobservable good (G) and bad (B ) risks).The following gures start from 2 classes of risks, then 3, and then a continuousrisk factor (0, ). 37

38. Arthur CHARPENTIER - cole dt EURIA. Conditional independence, two classes Conditional independence, two classes 20 3 2 15 1 10 0 !1 5 !2 !3 0 0 5 10 15 !3 !2 !1 0 1 2 3 Figure 13: Two classes of risks, (Xi , Yi ) and (1 (FX (Xi )), 1 (FY (Yi ))). 38

39. Arthur CHARPENTIER - cole dt EURIA. Conditional independence, three classes Conditional independence, three classes 3 40 2 30 1 0 20 !1 10 !2 !3 0 0 5 10 15 20 25 30 !3 !2 !1 0 1 2 3Figure 14: Three classes of risks, (Xi , Yi ) and (1 (FX (Xi )), 1 (FY (Yi ))). 39

40. Arthur CHARPENTIER - cole dt EURIA. Conditional independence, continuous risk factor Conditional independence, continuous risk factor 100 3 2 80 1 60 0 40 !1 20 !2 !3 0 0 20 40 60 80 100 !3 !2 !1 0 1 2 3Figure 15: Continuous classes of risks, (Xi , Yi ) and (1 (FX (Xi )), 1 (FY (Yi ))). 40

41. Arthur CHARPENTIER - cole dt EURIA.2. The survival approach: assume that there is a convex survival function S,with S(0) = 1, such that P(X > x, Y > y) = S(x + y),then the joint survival copula of (X, Y ) is S(S 1 (u) + S 1 (v)).Example 12. If S is the Pareto survival distribution, the associated copula isClaytons. If S is the Weibull survival distribution, the associated copula isGumbels. 41

42. Arthur CHARPENTIER - cole dt EURIA.3. The use of Kendalls distribution function K(t) = P(C(U, V ) t) where(U, V ) is a random pair with distribution function C .Then, for Archimedean copulas, (t) K(t) = t = t (t), (t)which can be inverted easily in 1 1 (t) = (t0 ) exp dt , t0 (t)for some 0 < t0 < 1 and 0 u 1. 42

43. Arthur CHARPENTIER - cole dt EURIA. Some more examples of Archimedean copulas (t) range (1) 1 (t 1) [1, 0) (0, ) Clayton, Clayton (1978) (2) (1 t) [1, ) 1(1t) (3) log [1, 1) Ali-Mikhail-Haq Gumbel Hougaard t (4) ( log t) [1, ) Gumbel, (1960), (1986) (5) log e t 1 e 1 (, 0) (0, ) Frank, Frank (1979), Nelsen(1987) (6) log{1 (1 t) } [1, ) Joe, Frank (1981), Joe(1993) (7) log{t + (1 )} (0, 1] (8) 1t [1, ) 1+(1)t (9) log(1 log t) (0, 1] Barnett (1980), Gumbel (1960) (10) log(2t 1) (0, 1] (11) log(2 t ) (0, 1/2] (12) ( 1 1) [1, ) t (13) (1 log t) 1 (0, ) (14) (t1/ 1) [1, ) (15) (1 t1/ ) [1, ) Genest & Ghoudi (1994) (16) ( + 1)(1 t) [0, ) t 43

44. Arthur CHARPENTIER - cole dt EURIA. Some characterizations of Archimedean copula L Frank copula is the only Archimedean such that (U, V ) = (1 U, 1 V ) (stability by symmetry), Clayton copula is the only Archimedean such that (U, V ) has the same copula as (U, V ) given (U u, V v) (stability by truncature), Gumbel copula is the only Archimedean such that (U, V ) has the same copula as (max{U1 , ..., Un }, max{V1 , ..., Vn }) for all n 1 (max-stability), 44

45. Arthur CHARPENTIER - cole dt EURIA. Extreme value copulas Extreme value copulas log u C(u, v) = exp (log u + log v) A , log u + log v where A is a dependence function, convex on [0, 1] with A(0) = A(1) = 1, et max{1 , } A () 1 for all [0, 1] .An alternative denition is the following: C is an extreme value copula if for allz > 0, 1/z 1/z C(u1 , . . . , ud ) = C(u1 , . . . , ud )z .Those copula are then called max-stable: dene the maximum componentwise ofa sample X 1 , . . . , Xn , i.e. Mi = max{Xi,1 , . . . , Xi,n }. 45

46. Arthur CHARPENTIER - cole dt EURIA.The joint distribution of M is P(M x) = C(F1 (x1 , . . . , Fd (xd ))n ,where C is the copula of the X i s. Since P(Mi xi ) = Fi (xi )n , it can be written P(M x) = C(P(M1 x1 )1/n , . . . , P(Md xd )1/n )n . 1/n 1/nThus, C(u1 , . . . , ud )n is the copula of the n maximum componentwise from asample with copula C .Example 13. : If A is constant (1 on [0, 1]), then X and Y are independent,and if A() = max {, 1 }, X and Y are comonotonic. Gumbels copula isobtained if ( A() = ((1 ) + + 1) 1/),for all 0 1 and 1. 46

47. Arthur CHARPENTIER - cole dt EURIA. Pickands dependence function A 1.0 0.9 0.8 0.7 0.6 0.5 0.0 0.2 0.4 0.6 0.8 1.0 Figure 16: Shape of Gumbels dependence function A(). 47

48. Arthur CHARPENTIER - cole dt EURIA. How to construct much more copulas ? Using geometric transformationsFrom a given copula C, cdf of random pair (U, V ), dene the copula of (U, 1 V ), C(U,1V ) (u, v) = u C(u, 1 v) the copula of (1 U, V ), C(1U,V ) (u, v) = v C(1 u, v) the copula of (1 U, 1 V ), the rotated or survival copula, C(1U,1V ) (u, v) = C (u, v) = u + v 1 + C(1 u, 1 v)Note that if P(X x, Y y) = C(P(X x), P(Y y)), then P(X > x, Y > y) = C (P(X > x), P(Y > y)). 48

49. Arthur CHARPENTIER - cole dt EURIA. 0.8 0.8 0.4 0.4 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 17: Using geometric transformation to generate new copulas. 49

53. Arthur CHARPENTIER - cole dt EURIA. Using mixture of copulasLemma 14. The set of copulas is convex, i.e. if {C , } is a collection ofcopulas, C(u, v) = C (u, v)d() Ris a copula, where is a distribution on Thus C = C1 + (1 )C2 denes a copula for all [0, 1].Example 15. Frchet (1951) suggested a mixture of the lower and the upperbound, C(u, v) = C (u, v) + (1 )C + (u, v), for some [0, 1].Example 16. Mardia (1970) suggested a mixture of the lower, the upperbound, and the independent copula 2 2 2 + C(u, v) = C (u, v) + (1 )C (u, v) + C (u, v), [0, 1]. 2 2 53

54. Arthur CHARPENTIER - cole dt EURIA. Using distortion functionsDenition 17. A distortion function is a function h : [0, 1] [0, 1] strictlyincreasing such that h(0) = 0 and h(1) = 1.The set of distortion function will be denoted H.Note that hH if and only if h1 H. Given a copula C, dene Ch (u, v) = h1 (C(h(u), h(v))).If h is convex, then Ch is a copula, called distorted copula.Example 18. if h(x) = x1/n , the distorted copula is Ch (u, v) = C n (u1/n , v 1/n ), for all n N, (u, v) [0, 1]2 .if the survival copula of the (Xi , Yi )s is C , then the survival copula of(Xn:n , Yn:n ) = (max{X1 , ..., Xn }, max{Y1 , ..., Yn }) is Ch . 54

55. Arthur CHARPENTIER - cole dt EURIA.Example 19. if C(u, v) = uv = C (u, v) (the independent copula), and() = log h(), then Ch (u, v) = h1 (h(u)h(v)) = 1 ((u) + (v)).Example 20. if h(x) = [1 ex ]/[1 e ] (an exponential distortion), and ifC = C , then 1 (eu 1)(ev 1) Ch (u, v) = log 1 + , e 1which is Frank copula. 55

56. Arthur CHARPENTIER - cole dt EURIA. Distorted Frank copula, h(x) = x Distorted Frank copula, h(x) = x(1 2) Distorted Frank copula, h(x) = x(1 3) Distorted Frank copula, h(x) = x(1 4) Figure 21: Distorted copula, from Frank copula. 56

57. Arthur CHARPENTIER - cole dt EURIA. Monte Carlo and copulasGeneration of independent variables can be done using a Random function.Denition 21. Function Random should satisfy the following properties (i) forall 0 a b 1, P (Random ]a, b]) = b a.(ii) successive calls of function Random should generate independent draws, i.e.0 a b 1, 0 c d 1 P (Random1 ]a, b] , Random2 ]c, d]) = (b a) (d c) ,or more generally, dene k-uniformity for all 0 ai bi 1, i = 1, ..., k, k P (Random1 ]a1 , b1 ] , ..., Randomk ]ak , bk ]) = (bi ai ) . i=1Thus, one can generate easily random vectors U = (U1 , ..., Ud ) with independentcomponent. 57

58. Arthur CHARPENTIER - cole dt EURIA.The idea to generate correlated vectors U = (U1 , ..., Ud ), the idea is to use rst P(U1 u1 , . . . , Ud ud ) = P(Ud ud |U1 u1 , . . . , Ud1 ud1 ) P(Ud1 ud1 |U1 u1 , . . . , Ud2 ud2 ) ... P(U3 u3 |U1 u1 , U2 u2 ) P(U2 u2 |U1 u1 ) P(U1 u1 ). 58

59. Arthur CHARPENTIER - cole dt EURIA.Starting from the end, P(U1 u1 ) = u1 since U1 is uniform, while P(U2 u2 |U1 = u1 ) = P(U2 u2 , U3 1, . . . Ud 1|U1 = u1 ) = lim P(U2 u2 , U3 1, . . . Ud 1|U1 [u1 , u1 + h]) h0 P(u1 U1 u1 + h, U2 u2 , U3 1, . . . Ud 1) = lim h0 P(U1 [u1 , u1 + h]) P(U1 u1 + h, U2 u2 , U3 1, . . . Ud 1) P(U1 u1 , U2 u2 , U3 1, . . . = lim h0 P(U1 [u1 , u1 + h]) C(u1 + h, u2 , 1, . . . , 1) C(u1 , u2 , 1, . . . , 1) C = lim = C(u1 , u2 , 1, . . . , 1). h0 h u1and more generally, k1 P(Uk uk |U1 = u1 , . . . , Uk1 = uk1 ) = C(u1 , . . . , uk , 1, . . . , 1). u1 . . . uk1 59

60. Arthur CHARPENTIER - cole dt EURIA.Thus, U = (U1 , .., Un ) with copula C could be simulated using the followingalgorithm, simulate U1 uniformly on [0, 1], u1 Random1 , simulate U2 from the conditional distribution 1 C(|u1 ), u2 [1 C(|u1 )]1 (Random2 ), simulate Uk from the conditional distribution 1,...,k1 C(|u1 , ..., uk1 ), uk [1,...,k1 C(|u1 , ..., uk1 )]1 (Randomk ),...etc, where the Randomi s are independent calls of a Random function.This is the underlying idea when using Cholesky decomposition. 60

61. Arthur CHARPENTIER - cole dt EURIA.Example: for Claytons copula, C(u, v) = (u + v 1)1/ , (U, V ) has jointdistribution C if and only if U is uniform on on [0, 1] and V |U = u hasconditional distribution P(V v|U = u) = 2 C(v|u) = (1 + u [v 1])11/ .The algorithm to generate Claytons copula is the simulate U1 uniformly on [0, 1], u1 Random1 , simulate U2 from the conditional distribution 2 C(|u), u2 [1 C(|u1 )]1 (Random2 ), i.e. u2 [(Random2 )/(1+ 1]u + 11/ . 1 61

62. Arthur CHARPENTIER - cole dt EURIA. 1.5 0.0 0.5 1.0 1.5 2.0 1.0 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Distribution of v given u=0.3 Distribution of v given u=0.5 Generation of Claytons copula 0.8 0.0 0.5 1.0 1.5 0.4 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Distribution of v given u=0.8 Figure 22: Simulation of Claytons copula. 62

63. Arthur CHARPENTIER - cole dt EURIA. 500 400 / / 300 q q 200 0 100 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 !i#tribution +e - !i#tribution +e 9 4 0.8 2 0 0.4 !4 !2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 !4 !2 0 2 4 -ni:orm mar=in# Stan+ar+ ?au##ian mar=in# Figure 23: Simulation of the independent copula. 63

64. Arthur CHARPENTIER - cole dt EURIA. 400 400 / / . . 200 200 0 0 010 012 014 014 015 610 010 012 014 014 015 610 !is$ri&u$i(n +e - !is$ri&u$i(n +e 7 4 015 2 0 014 !2 010 010 012 014 014 015 610 !2 0 2 4 -ni8(r9 9argins x)dx, one gets rT C(T, K) = e Q(ST > x)dx, Khence C P Q(ST x) = erT (T, x), or Q(ST x) = erT (T, x) K Kwhere P denotes the price of a put option. 1 2Consider an option on 2 assets, with payo h(ST , ST ). The price at time 0 iserT EQ (h(ST , ST )). 1 2 70

71. Arthur CHARPENTIER - cole dt EURIA. Copulas in nance: call on maximum 1 2 1 2Here the payo is h(ST , ST ) = (max{ST , ST } K)+ . The price is then C(T, K) = erT EQ ((max{ST , ST } K)+ ) 1 2 = erT EQ 1 2 1 1(max{ST , ST } x)dx K rT 1 2 = e 1 Q(max{ST , ST } x) dx, K 1 2 Q(ST x,ST x) 1 2hence, if (ST , ST ) has copula C (under Q), then P 1 P 2 C(T, K) = erT 1 C erT (T, x), erT (T, x) dx. K K K 71

72. Arthur CHARPENTIER - cole dt EURIA. Copulas in nance: call on spreads 1 2 1 2Here the payo is h(ST , ST ) = ([ST ST ] K)+ . The price is then rT 1 2 rT 2 1C(T, K) = e EQ ((ST ST K)+ ) = e EQ 1(ST + K x ST )dx = erT 2 2 1 Q(K + ST x) Q(ST + K x, ST x} x) dx, 1 2 Q(ST x,ST x+K) 1 2hence, if (ST , ST ) has copula C (under Q), then rT rT P 2 rT P 1 rT P 2C(T, K) = e e (T, xK)C e (T, x), e (T, x K) dx. K K K 72

73. Arthur CHARPENTIER - cole dt EURIA. Copulas in nance: bonds on option pricesUsing Tchens inequality, it is possible to derive bounds for options when thepayo is supermodular. 73

75. Arthur CHARPENTIER - cole dt EURIA. Natural properties for dependence measuresDenition 23. is measure of concordance if and only if satises1. is dened for every pair (X, Y ) of continuous random variables,2. 1 (X, Y ) +1, (X, X) = +1 and (X, X) = 1,3. (X, Y ) = (Y, X),4. if X and Y are independent, then (X, Y ) = 0,5. (X, Y ) = (X, Y ) = (X, Y ),6. if (X1 , Y1 ) P QD (X2 , Y2 ), then (X1 , Y1 ) (X2 , Y2 ),7. if (X1 , Y1 ) , (X2 , Y2 ) , ... is a sequence of continuous random vectors that converge to a pair (X, Y ) then (Xn , Yn ) (X, Y ) as n . 75

76. Arthur CHARPENTIER - cole dt EURIA.As pointed out in Scarsini (1984), most of the axioms are self-evident .If is measure of concordance, then, if f and g are both strictly increasing, then(f (X), g(Y )) = (X, Y ). Further, (X, Y ) = 1 if Y = f (X) with f almostsurely strictly increasing, and analogously (X, Y ) = 1 if Y = f (X) with falmost surely strictly decreasing (see Scarsini (1984)). 76

77. Arthur CHARPENTIER - cole dt EURIA. Association measures: Kendalls and Spearmans Rank correlations can be considered, i.e. Spearmans dened as 1 1 (X, Y ) = corr(FX (X), FY (Y )) = 12 C(u, v)dudv 3 0 0and Kendalls dened as 1 1 (X, Y ) = 4 C(u, v)dC(u, v) 1. 0 0 Historical version of those coecientsSpearmans rho was introduced in Spearman (1904) as (X, Y ) = 3[P((X1 X2 )(Y1 Y3 ) > 0) P((X1 X2 )(Y1 Y3 ) < 0)],where (X1 , Y1 ), (X2 , Y2 ) and (X3 , Y3 ) denote three independent versions of(X, Y ) (see Nelsen (1999)). 77

78. Arthur CHARPENTIER - cole dt EURIA.Similarly Kendalls tau was not dened using copulae, but as the probability ofconcordance, minus the probability of discordance, i.e. (X, Y ) = 3[P((X1 X2 )(Y1 Y2 ) > 0) P((X1 X2 )(Y1 Y2 ) < 0)],where (X1 , Y1 ) and (X2 , Y2 ) denote two independent versions of (X, Y ) (seeNelsen (1999)). 4QEquivalently, (X, Y ) = 1 2 1) where Q is the number of inversions n(nbetween the rankings of X and Y (number of discordance). 78

79. Arthur CHARPENTIER - cole dt EURIA. 1.5 Concordant pairs Discordant pairs 1.5 1.0 1.0 0.5 0.5 Y Y 0.0 0.0 !0.5 !0.5 !2.0 !1.5 !1.0 !0.5 0.0 0.5 1.0 !2.0 !1.5 !1.0 !0.5 0.0 0.5 1.0 X X Figure 30: Concordance versus discordance. 79

80. Arthur CHARPENTIER - cole dt EURIA. The case of the Gaussian random vectorIf (X, Y ) is a Gaussian random vector with correlation r, Kruskal then ( (1958)) 6 r 2 (X, Y ) = arcsin and (X, Y ) = arcsin (r) . 2 80

81. Arthur CHARPENTIER - cole dt EURIA. Link between Kendalls tau and Spearmans rhoNote that Kendalls tau and Spearmans are linked: it is impossible to have atthe same time 0.4 and = 0.Hence and satisfy 3 1 1 + 2 2 if 0 2 2 2 + 2 1 1 + 3 if 0. 2 2which yield the area given below. 81

82. Arthur CHARPENTIER - cole dt EURIA. 1.0 0.5 Rho de Spearman 0.0 -0.5 -1.0 -1.0 -0.5 0.0 0.5 1.0 Tau de Kendall Figure 31: Admissible region of and . 82

83. Arthur CHARPENTIER - cole dt EURIA. From Kendalltau to copula parameters Kendalls 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Gaussian 0.00 0.16 0.31 0.45 0.59 0.71 0.81 0.89 0.95 0.99 1.00 Gumbel 1.00 1.11 1.25 1.43 1.67 2.00 2.50 3.33 5.00 10.0 + Plackett 1.00 1.57 2.48 4.00 6.60 11.4 21.1 44.1 115 530 + Clayton 0.00 0.22 0.50 0.86 1.33 2.00 3.00 4.67 8.00 18.0 + Frank 0.00 0.91 1.86 2.92 4.16 5.74 7.93 11.4

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