HAL Id: tel-00082892 scientifjques de niveau recherche, publiés ou non, Arthur Charpentier. Dépendance et résultats limites, quelques applications en fjnance et assurance. To cite this version: Arthur Charpentier en fjnance et assurance Dépendance et résultats limites, quelques applications publics ou privés. recherche français ou étrangers, des laboratoires émanant des établissements d’enseignement et de destinée au dépôt et à la difgusion de documents https://tel.archives-ouvertes.fr/tel-00082892 L’archive ouverte pluridisciplinaire HAL , est abroad, or from public or private research centers. teaching and research institutions in France or The documents may come from lished or not. entifjc research documents, whether they are pub- archive for the deposit and dissemination of sci- HAL is a multi-disciplinary open access Submitted on 29 Jun 2006 Mathématiques [math]. Université Catholique de Louvain, 2006. Français. tel-00082892
Arthur CHARPENTIER - PhD Thesis Defense - Dependence structures and limiting results. Arthur Charpentier Dependence structures and limiting results, with applications in finance and insurance. Promoters: Jan Beirlant (KUL) & Michel Denuit (UCL) Katholieke Universiteit Leuven , June 2006. 1
Arthur CHARPENTIER - PhD Thesis Defense - Dependence structures and limiting results. Introduction Importance of tail dependence in risk management. In insurance, the cost of a claim is the sum of • the loss amount (paid to the insured), X 1 • the allocated expenses (lawyers, expertise...), X 2 Consider the following excess-of-loss reinsurance treaty, with payoff 0 , if x 1 ≤ d, g ( x 1 , x 2 ) = x 1 − d + x 1 − d x 2 , if x 1 > d. x 1 The pure premium is then E ( g ( X 1 , X 2 )) . 2
Arthur CHARPENTIER - PhD Thesis Defense - Dependence structures and limiting results. In life insurance, the payoff for a joint life insurance is ∞ � v k C k 1 ( x 1 > k and x 2 > k ) g ( x 1 , x 2 ) = k =1 (capital is due as long as the spouses are both still alive) and the pure premium is then E ( g ( T x , T y )) , where T x and T y denote the survival life lengths, of the man at age x and his wife y . In finance, the payoff of quanto derivatives is g ( x 1 , x 2 ) = x 2 ( x 1 − K ) + where X 2 is the exchange rate, and X 1 some overseas asset. 3
Arthur CHARPENTIER - PhD Thesis Defense - Dependence structures and limiting results. Modeling dependence with copulas Definition 1. A d -dimensional copula is a d -dimensional cumulative distribution function restricted to [0 , 1] d with standard uniform margins, d = 2 , 3 , . . . . Definition 2. Given F 1 , ..., F d some univariate distribution functions, the class of d -dimensional distribution functions F with marginal distributions F 1 , ..., F d respectively, is called a Fréchet class, denoted F ( F 1 , ..., F d ) . Definition 3. If U = ( U 1 , ..., U n ) has cdf C then the cdf of 1 − U = (1 − U 1 , ..., 1 − U n ) is also a copula, called survival copula of C , and denoted C ∗ . 4
Arthur CHARPENTIER - PhD Thesis Defense - Dependence structures and limiting results. Copula (cumulative distribution function) Level curves of the copula Figure 1: Copula, as a cumulative distribution function C ( u, v ) . 5
Arthur CHARPENTIER - PhD Thesis Defense - Dependence structures and limiting results. Copula density Level curves of the copula Figure 2: Density of a copula, c ( u, v ) = ∂ 2 C ( u, v ) /∂u∂v . 6
Arthur CHARPENTIER - PhD Thesis Defense - Dependence structures and limiting results. Theorem 4. Let C be a d -dimensional copula and F 1 , ..., F d be univariate distribution functions. Then, for x = ( x 1 , ..., x d ) ∈ R d , F ( x 1 , ..., x n ) = C ( F 1 ( x 1 ) , ..., F d ( x d )) (1) defines a distribution function with marginal distribution functions F 1 , ..., F d .Conversely, for a d -dimensional distribution function F with marginal distributions F 1 , ..., F d there is a copula C satisfying Equation (1) . This copula is not necessarily unique, but it is if F 1 , ..., F d are continuous, given by C ( u 1 , ..., u d ) = F ( F ← 1 ( u 1 ) , ..., F ← n ( x n )) , (2) for any u = ( u 1 , , ..., u d ) ∈ [0 , 1] d , where F ← 1 , ..., F ← denote the generalized d left continuous inverses of the F i ’s, i.e. F ← ( t ) = inf { x ∈ R , F i ( x ) ≥ t } for all 0 ≤ t ≤ 1 . i 7
Arthur CHARPENTIER - PhD Thesis Defense - Dependence structures and limiting results. Given, X with continuous marginals, the copula of X is the distribution of U = ( U 1 , . . . , U d ) = ( F 1 ( X 1 ) , . . . , F d ( X d )) . Hence, it is the distribution of the ranks. The survival copula of X is the distribution of 1 − U . Hence, P ( X ≤ x ) = C ( P ( X 1 ≤ x 1 ) , . . . , P ( X d ≤ x d )) , P ( X > x ) = C ∗ ( P ( X 1 > x 1 ) , . . . , P ( X d > x d )) , 8
Arthur CHARPENTIER - PhD Thesis Defense - Dependence structures and limiting results. Scatterplot of (Xi,Yi)’s Joint density Level curves of the copula 3 2 1 0 −1 −2 −3 −3 −2 −1 0 1 2 3 Scatterplot of (Ui,Vi)’s Copula density Level curves of the copula 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 3: Scatterplot and densities of ( X, Y ) and ( U, V ) . 9
Arthur CHARPENTIER - PhD Thesis Defense - Dependence structures and limiting results. In dimension 2 , consider the following family of copulae Definition 5. Let ψ denote a convex decreasing function (0 , 1] → [0 , ∞ ] such that ψ (1) = 0 . Define the inverse (or quasi-inverse if ψ (0) < ∞ ) as ψ − 1 ( t ) for 0 ≤ t ≤ ψ (0) ψ ← ( t ) = 0 for ψ (0) < t < ∞ . Then C ( u 1 , u 2 ) = ψ ← ( ψ ( u 1 ) + ψ ( u 2 )) , u 1 , u 2 ∈ [0 , 1] , is a copula, called an Archimedean copula, with generator ψ . 10
Arthur CHARPENTIER - PhD Thesis Defense - Dependence structures and limiting results. In higher dimension, one should add more conditions. Function f is d -completely monotonic if it is continuous and has derivatives which alternate in sign, i.e. for all k = 0 , 1 , ..., d , ( − 1) k d k f ( t ) /dt k ≥ 0 for all t . Definition 6. Assume further that ψ ← is d -completely monotonic, then C ( u 1 , ..., u n ) = ψ ← ( ψ ( u 1 ) + ... + ψ ( u d )) , u 1 , ..., u n ∈ [0 , 1] , is a copula, called an Archimedean copula, with generator ψ . 11
Arthur CHARPENTIER - PhD Thesis Defense - Dependence structures and limiting results. Archimedean copulas can also be characterized through Kendall’s cdf, K , K ( t ) = P ( C ( U 1 , ..., U d ) ≤ t ) , t ∈ [0 , 1] . where U = ( U 1 , ..., U d ) has cdf C . Note that K ( t ) = t − λ ( t ) where λ ( t ) = ψ ( t ) /ψ ′ ( t ) . And conversely ψ is �� u � 1 ψ ( u ) = ψ ( u 0 ) exp λ ( t ) dt pour 0 < u 0 < 1 . u 0 12
Arthur CHARPENTIER - PhD Thesis Defense - Dependence structures and limiting results. The frailty representation: if the X i ’s are E ( λ i Θ) distributed, and that, given Θ the X i ’s are independent, then F ( x ) = P ( X > x ) = E ( P ( X > x | Θ)) = E ( P ( X 1 > x 1 | Θ) · . . . · P ( X d > x d | Θ)) = E (exp( − Θ · (log P ( X 1 > x 1 ))) · . . . · exp( − Θ · (log P ( X d > x d )))) = φ ( − log P ( X 1 > x 1 ) − . . . − log P ( X d > x d )) φ ( φ ← ( F 1 ( x 1 )) , . . . , φ ← ( F d ( x d ))) , = where φ ( t ) = E ( e − t Θ ) is the Laplace transform of Θ . 13
Arthur CHARPENTIER - PhD Thesis Defense - Dependence structures and limiting results. Copula density Archimedean generator Laplace Transform 2.0 1.0 0.8 1.5 0.6 1.0 0.4 0.5 0.2 0.0 0.0 0.0 0.4 0.8 0 1 2 3 4 5 6 Level curves of the copula Lambda function Kendall cdf 0.0 1.0 0.8 −0.1 0.6 −0.2 0.4 −0.3 0.2 −0.4 0.0 0.0 0.4 0.8 0.0 0.4 0.8 Figure 4: (Independent) Archimedean copula ( C = C ⊥ , ψ ( t ) = − log t ). 14
Arthur CHARPENTIER - PhD Thesis Defense - Dependence structures and limiting results. If Θ is Gamma distributed, one gets Clayton’s copula (Figure 5), with parameter α ∈ [0 , ∞ ) has generator ψ ( x ; α ) = x − α − 1 α if 0 < α < ∞ , with the limiting case ψ ( x ; 0) = − log( x ) , for any 0 < x ≤ 1 . The associated copula is C ( u 1 , ..., u d ; α ) = ( u − α + ... + u − α − ( d − 1)) − 1 /α 1 d if 0 < α < ∞ , with the limiting case C ( u ; 0) = C ⊥ ( u ) , for any u ∈ (0 , 1] d . 15
Arthur CHARPENTIER - PhD Thesis Defense - Dependence structures and limiting results. Copula density Archimedean generator Laplace Transform 2.0 1.0 0.8 1.5 0.6 1.0 0.4 0.5 0.2 0.0 0.0 0.0 0.4 0.8 0 1 2 3 4 5 6 Level curves of the copula Lambda function Kendall cdf 0.0 1.0 0.8 −0.1 0.6 −0.2 0.4 −0.3 0.2 −0.4 0.0 0.0 0.4 0.8 0.0 0.4 0.8 Figure 5: Clayton’s copula. 16
Arthur CHARPENTIER - PhD Thesis Defense - Dependence structures and limiting results. If Θ is positive, stable, φ ( t ) = exp( − t − 1 /α ) , one gets Gumbel’s copula (Figure 6), with parameter α ∈ [1 , ∞ ) has generator ψ ( x ; α ) = ( − log x ) α if 1 ≤ α < ∞ , with the limiting case ψ ( x ; 0) = − log( x ) , for any 0 < x ≤ 1 . The associated copula is � � 1 + ( e − αu 1 − 1) ... ( e − αu d − 1) C ( u 1 , ..., u d ; α ) = − 1 α log , e − α − 1 if 1 ≤ α < ∞ , for any u ∈ (0 , 1] d . 17
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