A simple tranformation of copulas

Transcription

1 A simple tranformation of copulas V. Durrleman, A. Nikeghbali & T. Roncalli Groupe e Recherche Opérationnelle Créit Lyonnais France July 31, 2000 Abstract We stuy how copulas properties are moifie after some suitable transformations. In particular, we show that using appropriate transformations permits to fit the epenence structure in a better way. 1 Introuction Copulas is one of the most promising tool for financial moelling. Bouyé, Durrleman, Nikeghbali, Riboulet an Roncalli [2000] review ifferent financial problems an show how copulas coul help to solve them. For example, they use copulas for operational risk measurement an the stuy of multiimensional stress scenarios. One of the ifficulty is in general the choice of the copula. Durrleman, Nikeghbali an Roncalli [2000] present some proceures to fin the optimal copula in a given class C. They are ifferent methos to construct the copula function (see the chapter 3 of Nelsen [1998]). The most famous construction is base on the following composition C (u, v) = ϕ 1 (ϕ (u) + ϕ (v)) (1) with ϕ a C 2 function an ϕ (1) = 0, ϕ (x) < 0 an ϕ (x) > 0 for all 0 x 1. Such copulas are calle Archimeean copulas (Genest an MacKay [1986]). They play an important role, because they are very tractable. Mikusiński an Taylor [2000] exten this construction to copulas with the form C (u, v) = ϕ 1 (ϕ (u) ϕ (v)) where enotes a continuous associative operation. The construction of new families of copulas is an important fiel of research. The aim of this article is to introuce a simple transformation of copulas which permits to generate new families this transformation has been previously presente by Christian Genest in the conference Distributions with Given Marginals an Statistical Moelling (Barcelona, July 17-20, 2000). In a first section, we fin necessary an sufficient conition for the transforme copula being a copula too. In the secon section, we explore the effect of the transformation on the epenence structure. An in the last section, we show how the transformation proceure coul be use to fit the empirical epenence structure in a better way. Corresponing author: Groupe e Recherche Opérationnelle, Bercy-Expo Immeuble Bercy Su 4è étage, 90 quai e Bercy Paris Ceex 12 France; aress: thierry.roncalli@creitlyonnais.fr 1

2 2 How to generate new families of copulas? In this section, we limit ourselves to the case of 2-imensional copulas. Let C be a copula. Given a bijection γ : [0, 1] [0, 1], we can now efine C γ on [0, 1] 2 by C γ (x, y) = γ 1 (C (γ (x), γ (y))) (2) We can fin necessary an sufficient conitions for C γ (u, v) being a copula by introucing stronger conitions for γ. Theorem 1 (Strong conitions) Assume that γ be a concave C 1 iffeomorphism from ]0, 1[ onto ]0, 1[, twice erivable an continous from [0, 1] onto [0, 1] such that γ (0) = 0 an γ (1) = 1, then C γ is a copula. Theorem 2 (Weak conitions) Assume that γ be a C 1 iffeomorphism from ]0, 1[ onto ]0, 1[, twice erivable on ]0, 1[ an continous from [0, 1] onto [0, 1] such that γ (0) = 0 an γ (1) = 1, then C γ is a copula if an only if 2 C x y for every (u, v) where the erivatives of C exist. ( γ γ 1 (C (u, v)) ) C C (u, v) [γ (γ 1 (C (u, v)))] 2 (u, v) (u, v) (3) x y Proof. It is sufficient to prove the secon result because in the case where γ is concave, we trivially have γ ( γ 1 (C (u, v)) ) C C [γ (γ 1 (C (u, v)))] 2 (u, v) (u, v) 0 (4) x y an therefore the conition is match as soon as C is a copula. In orer to prove the secon theorem, assume that C γ is a copula. By efinition, we have γ (C γ (x, y)) = C (γ (x), γ (y)) (5) an by oing (x, y) = (0, 0) an (x, y) = (1, 1), we obtain the necessary conition that γ (0) = 0 an γ (1) = 1. The continuity of γ an the assumption that γ be a C 1 ifféomorphism imply that γ > 0. At every (u, v) where the erivatives of C exist, we compute the cross-erivative of C γ 2 C γ x y (u, v) = γ (u) γ (v) γ (γ 1 (C (γ (u), γ (v)))) [ ( 2 C γ γ 1 (C (γ (u), γ (v))) ) C (γ (u), γ (v)) x y [γ (γ 1 (C (γ (u), γ (v))))] 2 x (γ (u), γ (v)) C y (γ (u), γ (v)) The conition is then necessary. On the other han, by integrating the cross-erivative between 0 < x 1 x 2 < 1 an 0 < y 1 y 2 < 1, we get Since γ (0) = 0 an γ (1) = 1, we obtain C γ (x 2, y 2 ) C γ (x 1, y 2 ) C γ (x 2, y 1 ) + C γ (x 1, y 1 ) 0 (7) C γ (x, 0) = 0 C γ (x, 1) = x C γ (0, y) = 0 C γ (1, y) = y (8) this shows us that inequality (7) hols for every 0 x 1 x 2 1 an 0 y 1 y 2 1, an this completes the proof. ] (6) 2

3 Remark 3 Conitions on γ seem very complicate but they enow us, as we shall see in the sequel, with a large family of feasible functions γ such as x n x or x sin ( π 2 x). Moreover, we coul euce new functions from existing ones, because γ (x) = (γ 1 γ 2 ) (x) an γ (x) = 1 γ1 1 (1 x) verify the assumptions of the theorem (1). Remark 4 Thanks to the expression (6), we euce that the ensity function c γ (u, v) associate to the copula C γ is [ ] c γ (u, v) = γ (u) γ (v) γ c (γ (u), γ (v)) γ (C γ (u, v)) C C (C γ (u, v)) [γ (C γ (u, v))] 2 (γ (u), γ (v)) (γ (u), γ (v)) (9) x y where c (u, v) is the ensity of C. Note also that the conitional istributions have the following form Pr {V v U = u} = γ (u) C γ (γ 1 (γ (u), γ (v)) (10) (C (γ (u), γ (v)))) x Simulation of variates with istribution C γ is then straightforwar thanks to a numerical root fining proceure. Figure 1: Contours of ensity for Frank copula Example 5 (Frank copula) The Frank copula has the following form ( C (u, v) = α 1 1 [( ln 1 e α ) 1 e α ( 1 e αu) ( 1 e αv)] ) 3

4 Figure 2: Contours of ensity for the transfome Frank copula with β = 3 Figure 3: Contours of ensity for the transfome Frank copula with β = 7 4

5 Assume that γ (x) = x 1 β with β 1. γ verifies the theorem (1) an it comes that Cγ (u, v) is a copula. The corresponing ensity function is then ( ) 2 c γ (u, v) = ū vc γ (u, v) e α(ū+ v) βuv ( C γ (u, v) ) 1 2 (1 e α ) exp ( α C γ (u, v) ) ( ( α 1 e α ) C γ (u, v) (1 β) ( 1 e α v) ( 1 e αū)) (11) with C γ (u, v) = ( ( α 1 1 [( ln 1 e α ) 1 e α ( 1 e αū) ( 1 e α v)] )) β (12) where ū = u 1 1 β, v = v β an C γ (u, v) = C γ (u, v) 1 β. In the figure 1, we have represente the ensity contours of ifferent multivariate istributions generate by the Frank Copula. We have set α equal to (the corresponing Kenall s tau is then equal to 0.5). When the margins are uniform (left an top quarant), we obtain irectly the ensity of the copula. In the other quarant, the margins are gaussian, Stuent or α stable istributions. We have reporte the contour plots of the transforme copula in the figure 2 an 3 in the cases β = 3 an β = 7. We remark clearly that this transform function has an important impact on the epenence structure. Note moreover that the transforme copula belongs then to a two-parameter family. In the appenix, we present another transform functions an illustrate graphically the impact on the epenence structure. We introuce now the notations G an G which represent the sets of the functions γ that verify respectiveley the hypothesis of the theorem (1) an (2). Before stuying the impact on the epenence measures, we can try to know more globally about this transformation. This seems to be a ifficult issue. We are only able to state the following remarks. Theorem 6 γ G but we also assume that γ is a C 2 ([0, 1], [0, 1]) function then C is an Archimeean copula if an only if C γ is an Archimeean copula. Proof. Suppose C is an Archimeean copula, then C (u, v) = ϕ 1 (ϕ (u) + ϕ (v)) for a given ϕ. We have C γ (u, v) = (ϕ γ) 1 ((ϕ γ) (u) + (ϕ γ) (v)) (13) Because γ G, ϕ γ has the same properties as ϕ (ϕ is a C 2 function with ϕ (1) = 0, ϕ (x) < 0 an ϕ (x) > 0 for all 0 x 1). It comes that ϕ γ is the Archimeean generator of C γ. On the other han suppose that C γ is an Archimeean copula, then for all (u, v) [0, 1], Because γ is a bijection for all (u, v) [0, 1], it comes that γ 1 (C (γ (u), γ (v))) = ϕ 1 (ϕ (u) + ϕ (v)) (14) C (u, v) = ( ϕ γ 1) 1 (( ϕ γ 1 ) (u) + ( ϕ γ 1) (v) ) (15) an C is an Archimeean copula with the generator ϕ γ 1. Corollary 7 Any Archimeean copula C (u, v) = ϕ 1 (ϕ (u) + ϕ (v)) can be rewritten as a transforme copula of the inepenant copula C with γ = exp ( ϕ 1). Corollary 8 Let C be a copula. Define C (C) = { C γ γ G an γ is a C 2 ([0, 1], [0, 1]) function }. If C is an Archimeean copula, then C (C) is the set of all Archimeean copulas. Lemma 9 Let C + be the upper Fréchet Boun. Then, C (C + ) = {C + }. Proof. This result is obvious because of the monotonicity of γ min(γ(u), γ(v)) = γ(min(u, v)). 5

6 3 Depenence properties of the transforme copula We are now intereste in knowing more about the tranforme copula. Let us look first at the Kenall s tau of the transforme copula. Theorem 10 γ G an assume that C x then we have C y is integrable an that for all x [0, 1], 0 < α γ (x) β < +, 1 + τ 1 α 2 τ γ 1 + τ 1 β 2 (16) Proof. We use the following efinition of Kenall s tau τ = 4 C(u, v) 2 C (u, v) u v 1 (17) [0,1] x y 2 Thanks to the Green s formula to obtain another tractable expression for τ ( τ = 4 C(u, v) C C C (u, v) σ (u, v) [0,1] x 2 [0,1] x y 2 an the first term can be evaluate (see Nelsen [1998]) C C τ = 1 4 (u, v) [0,1] x y 2 (u, v) u v ) 1 (18) (u, v) u v (19) With this expression we can easily erive an expression for τ γ by the the change of variable formula τ γ = 1 4 [0,1] 2 Uner the assumptions on γ, we have the following inequalities for all x [0, 1], C C x (u, v) y (u, v) [γ (γ 1 2 u v (20) (C (u, v)))] an the result is then easily erive. 1 β 1 γ (γ 1 (x)) 1 α x β γ 1 (x) x α (21) (22) Theorem 11 γ G an assume that for all x [0, 1], 0 < α γ (x) β < +, then we have ρ + 3 β 3 3 ρ γ ρ + 3 α 3 3 (23) Proof. We use the following efinition of Spearman s rho ρ = 12 C (u, v) u v 3 (24) [0,1] 2 With this expression we can easily erive an expression for ρ γ by the change of variable formula γ 1 (C (u, v)) ρ γ = 12 [0,1] γ (γ 1 (u)) γ (γ 1 u v 3 (25) (v)) 2 6

7 Using the inequalities (21) an (22), we obtain an this completes the proof. 1 β 3 (ρ + 3) ρ γ (ρ + 3) (26) α3 One interesting feature of our transformation is that it oes not change the tail epenences of the initial copula C. So, we can concentrate upon moifying the Kenall s tau or the Spearman s rho without changing the tail epenences. Let s first recall the efinition of the upper tail epenence measure (Joe [1997]). Let C.be a copula an λ (C) be efine as C (u, u) λ (C) = lim u 1 1 u = lim u 1 u C (u, u) (27) where C (u, v) = 1 u v + C (u, v). λ (C) is calle the upper tail epenence measure of C an C is sai to have upper tail epenence if λ (C) exists an is strictly greater than zero, this to say λ (C) ]0, 1]. If λ (C) = 0, we say that C has no upper tail epenence. Let s enote λ (C γ ) the upper tail epenence of the transforme copula C γ (x, y) = γ 1 (C (γ (x), γ (y))). We have the following interesting result 1 : Theorem 12 If the limit λ (C) exists, then we have an Proof. We have Note that an So, as we assume that lim u 1 of limit composition λ (C γ ) = λ (C) (28) λ (C) = 2 lim C (u, u) (29) u 1 u λ (C γ ) = 2 lim u 1 u C γ (u, u) C (u, u) = u u C (u, u) + C (u, u) (30) v u C γ [ (u) γ (u, u) = γ (γ 1 (C (u, u))) u C (γ (u), γ (u)) + ] C (γ (u), γ (u)) v hence we obtain the esire result. (31) u C (u, u) exists an as γ 1 (1) = γ (1) = 1, we have by the classical theorem lim u 1 u C γ (u, u) = lim u 1 C (u, u) (32) u To illustrate these properties, we consier the two-imensional stuent copula (Bouyé, Durrleman, Nikeghbali, Riboulet an Roncalli [2000]). Let us enote ρ an ν the parameters. We coul then show that (Embrechts, McNeil an Straumann [1999]) ( ) ν 1 ρ λ (C) = 2 t ν (33) 1 + ρ C. 1 We also have a similar result for the lower tail epenence measure, which is invariant uner the γ-transformation of the copula 7

8 where t is the tail of a univariate stuent istribution. We have represente in the figure 4 the corresponing upper tail epenence an the function λ (u) = Pr {U 1 > u U 2 > u} = (1 u) 1 C (u, u) for the stuent copula with ρ = 0 an ν = 1. For the transforme stuent copula, we have taken γ 1 (x) = x, γ 2 (x) = 5 x an γ 3 (x) = sin ( π 2 x). We remark effectively that the limits are the same (but its convergence to this limit is very ifferent 2 ). Figure 4: Quantile-epenent measure λ (u) 4 Fitting the epenence structure with the transforme copula In one previous work (Durrleman, Nikeghbali an Roncalli [2000]), we consier the problem to fin the optimal copula in a given class C. The iea is the following. Let Ĉ be the empirical copula (Deheuvels [1979]). Let C be finite subset of copulas ( C C), then we are intereste in knowing which one of the copulas in C fits best the epenence structure of ata in the sense of a given istance. The problem here is ifferent. Suppose that we have foun a copula C which coul be consiere as a goo caniate to match the epenence structure of the ata. How to fin a transforme function γ such that we coul improve the aequacy of C γ to the empirical copula Ĉ. We illustrate this problem by two examples. 4.1 The τ ϱ example Nelsen [1998] presents some relationships between the measures τ an ϱ. They coul be summarize by a bouning region B (τ, ϱ) efine by { 3τ 1 (τ, ϱ) B (τ, ϱ) 2 ϱ 1+2τ τ 2 2 τ 0 τ 2 +2τ 1 2 ϱ 1+3τ (34) 2 τ 0 2 We have inicate with the otte lines the quantile-epenent measure for the Fréchet bouns. 8

9 We have represente this region in the figure 5. Moreover, Nelsen [1998] presents some arguments to show that we coul think that the bouns coul not be improve. However, when we compute the τ ϱ region of a large number of families, it appears that it is ifficult to attain any points of the bounary region. For example, we have reporte in the figure the τ ϱ region of the 10 copula families 3 presente in Joe [1997]. These families are the most popular an the most use in copula moelling. Nevertheless, we remark that we coul not fin a copula which belongs to these families such that τ an ϱ correspons to the points A, B, C or D. In fact, the attainable region is very thin. Figure 5: Bouning region for τ an ϱ The question is now the following: coul we fin a copula C γ (u, v) such that we ( attain ) these points. The answer is partially yes 4 π. For example, if we use the gaussian copula with γ (x) = sin 2 x 1 β an β [1, + [, the τ ϱ region is shifte to the left an to the top (see the figure 6). We then obtain some copulas that verify τ 0 an ϱ > The τ/ ϱ λ example Let ˆτ an ˆλ be the empirical measures associate to the ata. Imagine that we want to fit both the Kenall s tau an the upper tail epenence. We coul efine a finite subset of copulas C an fin the copula which matches the two observe measures. However, that requires a lot of computations. Another possibility is to fin a copula C such that λ = ˆλ an then fin the function γ G such that we fit the observe Kenall s tau. This two steps moelling is possible because the transformation preserves the upper tail epenence (λ (C γ ) = λ (C)). 3 except the family B9. 4 In fact, there exist yet copulas which attain the bounary regions. They are the shuffles of Min (Mikusiński, Sherwoo an Taylor [1992]). Nevertheless, these copulas are very special because the support consists of line segments. Moreover, they are not parametric. 9

10 Figure 6: τ ϱ region of the transforme Normal copula We consier the example of the Marshall-Olkin copula, which is efine in the following way C (u, v) = u 1 α 1 v 1 α 2 min (u α 1, v α 2 ) (35) with (α 1, α 2 ) [0, 1] 2. We have τ = α 1 α 2 (α 1 α 1 α 2 + α 2 ) 1, ϱ = 3α 1 α 2 (2α 1 α 1 α 2 + 2α 2 ) 1 an λ = min (α 1, α 2 ) (Linskog [2000]). We have represente in the figure 7 the corresponing τ ϱ λ region an the transforme region with γ (x) = sin ( π 2 x) ( ) (. For τ τγ 0 = 1 3 an ϱ ϱ 0 γ 1 2), we coul now fin a copula which gives the same upper tail epenence, but with a smaller τ or a bigger ϱ. 5 Conclusion In this paper, we have investigate a new approach to construct copula families. This approach has an interesting property, because it changes epenence measures like the Kenall s tau or the Spearman s rho, without changing the upper (or lower) tail epenence. This γ-transformation coul then be use in a τ/ ϱ λ problematic, for example in financial moelling of asset returns. References [1] Bouyé, E., V. Durrleman, A. Nikeghbali, G. Riboulet an T. Roncalli [2000], Copulas for finance A reaing guie an some applications, Groupe e Recherche Opérationnelle, Créit Lyonnais, Working Paper [2] Deheuvels, P. [1979], La fonction e épenance empirique et ses propriétés Un test non paramétrique inépenance, Acaémie Royale e Belgique Bulletin e la Classe es Sciences 5e Série, 65,

11 Figure 7: τ ϱ λ region of the Marshall-Olkin an the γ -transforme copulas [3] Durrleman, V., A. Nikeghbali, an T. Roncalli [2000], Which copula is the right one?, Groupe e Recherche Opérationnelle, Créit Lyonnais, Working Paper [4] Embrechts, P., McNeil, A.J. an D. Straumann [1999], Correlation an epenency in risk management : properties an pitfalls, Departement of Mathematik, ETHZ, Zürich, Working Paper [5] Genest, C., K. Ghoui an L.P. Rivest [1998], Discussion of Unerstaning relationships using copulas, by Ewar Frees an Emiliano Valez, North American Actuarial Journal, 3, [6] Genest, C. an J. MacKay [1986], The joy of copulas: Bivariate istributions with uniform marginals, American Statistician, 40, [7] Joe, H. [1997], Multivariate Moels an Depenence Concepts, Monographs on Statistics an Applie Probability, 73, Chapmann & Hall, Lonon [8] Linskog, F. [2000], Moelling epenence with copulas an applications to risk management, RiskLab Research Paper [9] Mikusiński, P., H. Sherwoo an M.D. Taylor [1992], Shuffles of min, Stochastica, 13, [10] Mikusiński, P. an M.D. Taylor [2000], A remark on associative copulas, forthcoming in Commentationes Mathematicae Universitatis Carolinae [11] Nelsen, R.B. [1998], An Introuction to Copulas, Lectures Notes in Statistics, 139, Springer Verlag, New York 11

12 Appenix Some special functions γ (x) We present now some functions that belong to G. Then, we show graphically how these functions transform the repartition of the probability masses of the copula. Parameters γ (x) γ 1 (x) 1 β 1 x β 1 x β 2 sin π 2 x 2 arcsin (x) π 4 3 π arctan (x) tan π 4 x 4 (β 1, β 2 ) R 2 + (x) = (β 1 + β 2 ) x (β 1 x + β 2 ) 1 β 1 x (β 1 + β 2 β 1 x) 5 f L 1 (]0, 1[), f (x) 0, f (x) f (t) t 1 x 0 f (t) t Note that we coul obtain other functions by convex combination. For example, γ G with γ (x) = (γ 4 γ 1 ) (x) = (β 1 + β 2 ) x 1 β 3 ( ) 1. β 1 x 1 β 3 + β 2 We have reporte in the figures 8 14 the contour plots of the transforme Frank copula an the transforme istribution 5. We use the first example of the secon section. We remark that the repartition of the mass probability changes consequently thanks to the choice of the function γ. We coul explain that because the transformations will introuce some istortions. For example, they will put more masses on the lower or upper corner. Figure 8: Contours of ensity for the transfome Frank copula with ϱ (x) = sin ( π 2 x) 5 The parameters β 1 an β 2 are set to 1 an in figure 13, an 0.5 an 3 in figure

13 Figure 9: Contours of ensity for the transfome Frank copula with sin ( π 2 5 x ) Figure 10: Contours of ensity for the transfome Frank copula with 8 4 π arctan ( 2 x) 13

14 Figure 11: Contours of ensity for the transfome Frank copula with ϱ (x) = (x) (β 1 = 1, β 2 = Figure 12: Contours of ensity for the transfome Frank copula with ϱ (x) = ( x) (β 1 = 1, β 2 =

15 Figure 13: Contours of ensity for the transfome Frank copula with ϱ (x) = ( sin ( π 2 x)) Figure 14: Contours of ensity for the transfome Frank copula with ϱ (x) = 15 ( sin ( π 2 x))