Copulas: A Tool For Modelling Dependence In Finance

Transcription

1 Copulas: A Tool For Modelling Dependence In Finance Statistical Methods in Integrated Risk Management Frontières en Finance, Paris, 01/26/2001 Thierry Roncalli Groupe de Recherche Opérationnelle Crédit Lyonnais Joint work with a lot of people (see ref. further) The Working Paper Copulas For Finance is available on the web site:

2 1 Introduction Definition 1 A copula function C is a multivariate uniform distribution (a multivariate distribution with uniform margins). Theorem 1 Let F 1,..., F N be N univariate distributions. It comes that C (F 1 (x 1 ),..., F n (x n ),..., F N (x N )) defines a multivariate distributions F with margins F 1,..., F N (because the integral transforms are uniform distributions). F belongs to the Fréchet class F (F 1,..., F N ) F is a distribution with given marginals. Copulas are also a general tool to construct multivariate distributions, and so multivariate statistical models see for example Song [2000]. Introduction 1-1

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4 2 The dependence function Canonical representation Concordance order Measure of dependence From 1958 to 1976, virtually all the results concerning copulas were obtained in connection with the study and development of the theory of probabilistic metric spaces (Schweizer [1991]). Schweizer and Wolff [1976] = connection with rank statistics (see also Deheuvels [1979b]). The dependence function 2-1

5 2.1 Canonical representation Theorem 2 (Sklar s theorem) Let F be a N-dimensional distribution function with continuous margins F 1,..., F N. Then F has a unique copula representation F (x 1,..., x N ) = C (F 1 (x 1 ),..., F N (x N )) Copulas are also a powerful tool, because the modelling problem could be decomposed into two steps: Identification of the marginal distributions; Defining the appropriate copula function. In terms of the density, we have the following canonical representation f (x 1,..., x N ) = c (F 1 (x 1 ),..., F N (x N )) N n=1 f n (x n ). The dependence function 2-2

6 The copula function of random variables (X 1,..., X N ) is invariant under strictly increasing transformations ( x h n (x) > 0): C X1,...,X N = C h1 (X 1 ),...,h N (X N )... the copula is invariant while the margins may be changed at will, it follows that is precisely the copula which captures those properties of the joint distribution which are invariant under a.s. strickly increasing transformations (Schweizer and Wolff [1981]). Copula = dependence function of random variables. This property was already etablished by Deheuvels [1978,1979a]. The dependence function 2-3

7 2.2 Examples For the Normal copula, We have C (u 1,..., u N ; ρ) = Φ ρ ( Φ 1 (u1 ),..., Φ 1 (u N ) ) and c (u 1,..., u N ; ρ) = 1 ρ 1 2 exp ( 1 ( 2 ς ρ 1 I ) ) ς For the Gumbel copula, We have ( C (u 1, u 2 ) = exp ( ( ln u 1 ) δ + ( ln u 2 ) δ)1 δ ) Other copulas: Archimedean, Plackett, Frank, Student, Clayton, etc. The dependence function 2-4

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10 2.3 Concordance order The copula C 1 is smaller than the copula C 2 (C 1 C 2 ) if (u 1,..., u N ) I N, C 1 (u 1,..., u N ) C 2 (u 1,..., u N ) The lower and upper Fréchet bounds C and C + are C (u 1,..., u N ) = max N n=1 C + (u 1,..., u N ) = min (u 1,..., u N ) u n N + 1, 0 We can show that the following order holds for any copula C: C C C + The minimal and maximal distributions of the Fréchet class F (F 1, F 2 ) are then C (F 1 (x 1 ), F 2 (x 2 )) and C + (F 1 (x 1 ), F 2 (x 2 )). Example of the bivariate Normal copula (C (u 1, u 2 ) = u 1 u 2 ): C = C 1 C ρ<0 C 0 = C C ρ>0 C 1 = C + The dependence function 2-5

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13 Mikusiński, Sherwood and Taylor [1991] give the following interpretation of the three copulas C, C and C + : Two random variables X 1 and X 2 are countermonotonic or C = C if there exists a r.v. X such that X 1 = f 1 (X) and X 2 = f 2 (X) with f 1 non-increasing and f 2 non-decreasing; Two random variables X 1 and X 2 are independent if the dependence structure is the product copula C ; Two random variables X 1 and X 2 are comonotonic or C = C + if there exists a random variable X such that X 1 = f 1 (X) and X 2 = f 2 (X) where the functions f 1 and f 2 are non-decreasing; The dependence function 2-6

14 2.4 Measures of association or dependence If κ is a measure of concordance, it satisfies the properties: 1 κ C 1; C 1 C 2 κ C1 κ C2 ; etc. Schweizer and Wolff [1981] show that Kendall s tau and Spearman s rho can be (re)formulated in terms of copulas τ = 4 ϱ = 12 I 2 C (u 1, u 2 ) dc (u 1, u 2 ) 1 I 2 u 1u 2 dc (u 1, u 2 ) 3 The linear (or Pearson) correlation is not a measure of dependence. The dependence function 2-7

15 2.5 Some misinterpretations of the correlation The following statements are false: 1. X 1 and X 2 are independent if and only if ρ (X 1, X 2 ) = 0; 2. For given margins, the permissible range of ρ (X 1, X 2 ) is [ 1, 1]; 3. ρ (X 1, X 2 ) = 0 means that there are no relationship between X 1 and X 2. We consider the cubic copula of Durrleman, Nikeghbali and Roncalli [2000b] C (u 1, u 2 ) = u 1 u 2 + α [u 1 (u 1 1)(2u 1 1)] [u 2 (u 2 1)(2u 2 1)] with α [ 1, 2]. If the margins F 1 and F 2 are continous and symmetric, the Pearson correlation is zero. Moreover, if α 0, the random variables X 1 and X 2 are not independent. The dependence function 2-8

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18 Wang [1997] shows that the min. and max. correlations of X 1 LN (µ 1, σ 1 ) and X 2 LN (µ 2, σ 2 ) are ρ = ρ + = e σ 1σ 2 1 ( )1 ( 0 )1 e σ e σ e σ 1σ 2 1 ( )1 ( 0 )1 e σ e σ ρ and ρ + are not necessarily equal to 1 and 1. Example with σ 1 = 1 and σ 2 = 3: Copula ρ (X 1, X 2 ) τ (X 1, X 2 ) ϱ (X 1, X 2 ) C ρ = C ρ = C The dependence function 2-9

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20 Using an idea of Ferguson [1994], Nelsen [1999] defines the following copula C (u 1, u 2 ) = u 1 0 u u u u 2 u u 2 u 1 + u u 2 u 1 1 We have cov (U 1, U 2 ) = 0, but Pr {U 2 = 1 2U 1 1 } = 1, i.e. the two random variables can be uncorrelated although one can be predicted perfectly from the other. The dependence function 2-10

21 3 Understanding the dependence: three examples Mispecifications of marginals and dependence Dependence in multi-assets options Temporal dependence in Markov processes Understanding the dependence: three examples 3-1

22 3.1 Mispecifications of marginals and dependence Example of Costinot, Roncalli and Teïletche [2000] Normal copula + Gaussian marginals = Gaussian distribution. It means that ˆρ = C (CAC40,NIKKEI) C (CAC40,DowJones) C (NIKKEI,DowJones) Normal copula + Empirical marginals. In this case, we have ˆρ = C (CAC40,DowJones) C (CAC40,NIKKEI) C (NIKKEI,DowJones) Understanding the dependence: three examples 3-2

23 3.2 Dependence in multi-assets options Ref.: Bikos [2000], Cherubini and Luciano [2000], Durrleman [2001], Rosenberg [2000]. Vanilla options contain information on the future distribution of S (T ). This information (RND) is actually used in monetary policy (see BIS [1999]) (option prices = forward-looking indicators). Bikos [2000]: options on multi-assets contain information on the future distribution of S (T ) = ( ). S 1 (T ) S N (T ) Let Q n and Q be the risk-neutral probability distributions of S n (T ) and S (T ). With arbitrage theory, we can show that Q (+,..., +, S n (T ), +,..., + ) = Q n (S n (T )) The margins of Q are the RND Q n of Vanilla options. Understanding the dependence: three examples 3-3

24 How to build forward-looking indicators for the dependence? 1. estimate the univariate RND ˆQ n using Vanilla options; 2. estimate the copula Ĉ using multi-assets options by imposing that Q n = ˆQ n ; 3. derive forward-looking indicators directly from Ĉ. Breeden et Litzenberger [1978] remark that European option prices permit to caracterize the probability distribution of S n (T ) φ (T, K) := 1 + e r(t t C (T, K) 0) K = Pr {S n (T ) K} = Q n (K) Understanding the dependence: three examples 3-4

25 Durrleman [2000] extends this result in the bivariate case: 1. for a call max option, φ (T, K) is the diagonal section of the copula 2. for a spread option, we have φ (T, K) = C (Q 1 (K), Q 2 (K)) φ (T, K) = C (Q 1 (x), Q 2 (x + K)) dq 1 (x) Other results are derived in Durrleman [2001] (bounds, general kernel pricing, etc.) Understanding the dependence: three examples 3-5

26 Computation of the implied parameter ˆρ in a spread option: BS model: LN distribution calibrated with ATM options; Kernel pricing = LN distributions + Normal copula ˆρ 1 = Bahra model: mixture of LN distributions calibrated with eight European prices; Kernel pricing = MLN distributions + Normal copula ˆρ 2 = Remark 1 ˆρ 1 and ˆρ 2 are parameters of the Normal Copula. ˆρ 1 is a Pearson correlation, not ˆρ 2. BS model: negative dependence / Bahra model: positive dependence. Understanding the dependence: three examples 3-6

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28 3.3 Temporal dependence in Markov processes Markov operators and 2-copulas Markov operators and stochastic kernels Markov processes and product of 2-copulas The Brownian copula Understanding the temporal dependence structure of diffusion processes New interpretation of properties of Markov processes Understanding the dependence: three examples 3-7

29 3.3.1 Markov operators and 2-copulas Definition 2 (Brown [1966, p. 15]) Let (Ω, F, P ) be a probabilistic space. A linear operator T : L (Ω) L (Ω) is a Markov operator if (a) T is positive i.e. T [f] 0 whenever f 0; (b) 1 is a fixed point of T. (c) E [T [f]] = E [f] for every f L (Ω). Understanding the dependence: three examples 3-8

30 The relationship between Markov operator T and 2-copula functions C is given by the following two lemmas: Lemma 1 (Olsen, Darsow and Nguyen [1996, lemma 2.1, p. 249]) For a copula C, the operator defined by 1 T [f] (x) = d dx 2C (x, y) f (y) dy 0 is a Markov operator on L ([0, 1]). Lemma 2 (Olsen, Darsow and Nguyen [1996, lemma 2.2, p. 251]) Let T be a Markov operator on L ([0, 1]). The two place function defined by is a 2-copula. C (x, y) = x 0 T [ 1 [0,y] ] (s) ds Understanding the dependence: three examples 3-9

31 The multiplication product of copulas have been defined by Darsow, Nguyen and Olsen [1992] in the following manner I 2 I (x, y) (C 1 C 2 ) (x, y) = C 1 (x, s) 1 C 2 (s, y) ds The transposition of copula corresponds to the mapping function C (x, y) = C (y, x). The adjoint T of the Markov operator T is the operator such that we verify that (Brown [1966]) f 1 (x) T [f 2 ] (x) m (dx) = f 2 (x) T [f 1 ] (x) m (dx) [0,1] [0,1] In terms of copulas, we have T [f] (x) = d dy 1 0 1C (x, y) f (x) dx Understanding the dependence: three examples 3-10

32 3.3.2 Markov operators and stochastic kernels Using the Kantorovitch-Vulich-Ryff representation, Foguel [1969] shows that T K 1 T K 2 is a Markov operator with kernel K = K 1 K 2 defined by K (x, y) := (K 1 K 2 ) (x, y) = [0,1] K 1 (x, s) K 2 (s, y) ds Let P be the stochastic transition function of a Markov process, we can deduce that T P s,t = T Ps,θ P θ,t = T Ps,θ T Pθ,t with P s,t (x, A) := ( ) P s,θ P θ,t (x, A) = P s,θ (x, dy) P θ,t (y, A) Ω = P (x, A) this is the Chapman-Kolmogorov equation. Understanding the dependence: three examples 3-11

33 3.3.3 Markov processes and product of 2-copulas Darsow, Nguyen and Olsen [1992] prove the following theorem: Theorem 3 Let X = {X t, F t ; t 0} be a stochastic process and let C s,t denote the copula of the random variables X s and X t. Then the following are equivalent (i) The transition probabilities P s,t (x, A) = Pr {X t A X s = x} satisfy the Chapman-Kolmogorov equations P s,t (x, A) = P s,θ (x, dy) P θ,t (y, A) for all s < θ < t and almost all x R. (ii) For all s < θ < t, C s,t = C s,θ C θ,t (1) Understanding the dependence: three examples 3-12

34 In the conventional approach, one specifies a Markov process by giving the initial distribution µ and a family of transition probabilities P s,t (x, A) satisfying the Chapman-Kolmogorov equations. In our approach, one specifies a Markov process by giving all of the marginal distributions and a family of 2-copulas satisfying (1). Ours is accordingly an alternative approach to the study of Markov processes which is different in principle from the conventional one. Holding the transition probabilities of a Markov process fixed and varying the initial distribution necessarily varies all of the marginal distributions, but holding the copulas of the process fixed and varying the initial distribution does not affect any other marginal distribution (Darsow, Nguyen and Olsen [1992]). The Brownian copula C s,t (u 1, u 2 ) = u1 0 Φ ( tφ 1 (u 2 ) sφ 1 ) (u) t s Understanding the dependence: three examples 3-13 du

35 Understanding the dependence structure of diffusion processes The copula of a Geometric Brownian motion is the Brownian copula. The Ornstein-Uhlenbeck copula is ( u1 (t0 C s,t (u 1, u 2 ) = Φ, s, t) Φ 1 (u 2 ) (t 0, s, s) Φ 1 (u) 0 (s, s, t) with (t 0, s, t) = e 2a(t s) e 2a(s t 0) ) du Remark 2 A new interpretation of the parameter a follows. For physicists, a is the mean-reverting coefficient. From a copula point of view, this parameter measures the dependence between the random variables of the diffusion process. The bigger this parameter, the less dependent the random variables. Understanding the dependence: three examples 3-14

36 New interpretation of properties of Markov processes Theorem 4 (Darsow, Nguyen and Olsen [1992, theorem 5.1, p. 622]) The set C is a symmetric Markov algebra under and as previously defined. The unit and null elements are C and C +. C st is a left invertible copula the markov process is deterministic. C st is idempotent the Markov process is conditionally independent. Understanding the dependence: three examples 3-15

37 4 An open field for risk management Economic capital adequacy Market risk Credit risk Operational risk see BDNRR [2000] for a more detailed presentation (in particular for stress-testing, multivariate extreme value theory and operational risk) and DNRa [2000] for the problem of quantile aggregation. An open field for risk management 4-1

38 4.1 Economic capital adequacy With copulas, it appears that the risk can be splitted into two parts: the individual risks and the dependence structure between them. Coherent multivariate statistical model = Coherent model for individual risks + coherent dependence function Coherent model for individual risks = taking into account fat-tailed distributions, etc. coherent dependence function = understanding the aggregation of quantiles of the individual risks. An open field for risk management 4-2

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40 The influence of margins Rating VaR BBB A AA AAA α 99% 99.75% 99.9% 99.95% 99.97% Return time 100 days 400 days 4 years 8 years 13 years Φ 1 (α) Φ 1 (0.99) t 1 4 (α) t 1 4 (0.99) The influence of the dependence function: If a bivariate copula C is such that lim u 1 C (u, u) 1 u = λ exists, then C has upper tail dependence for λ (0, 1] and no upper tail dependence for λ = 0. C is the joint survival function, that is C (u 1, u 2 ) = 1 u 1 u 2 + C (u 1, u 2 ) An open field for risk management 4-3

41 Remark 3 The measure λ is the probability that one variable is extreme given that the other is extreme. Coles, Currie and Tawn [1999] define the quantile-dependent measure of dependence as follows λ (u) = Pr {U 1 > u U 2 > u} = C (u, u) 1 u 1. Normal copula extremes are asymptotically independent for ρ = 1, i.e λ = 0 for ρ < Student copula extremes are asymptotically dependent for ρ 1. An open field for risk management 4-4

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44 4.2 Market risk Copulas = a powerful tool for market risk measurement. Copulas have been already incorporated in some software solutions: SAS Risk Dimensions An open field for risk management 4-5

45 LME example: AL AL-15 CU NI PB P P P Gaussian margins and Normal copula 90% 95% 99% 99.5% 99.9% P P P Student margins (ν = 4) and Normal copula 90% 95% 99% 99.5% 99.9% P P P An open field for risk management 4-6

46 Gaussian margins and Student copula (ν = 1) 90% 95% 99% 99.5% 99.9% P P P Value-at-risk based on Student margins and a Normal copula (Gauss software, Pentium III 550 Mhz, simulations) Number of assets Computational time sc sc mn 7 sc mn 22 sc hr 44 mn 45 sc An open field for risk management 4-7

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49 5 Modelling credit risk Some articles are available, see for example Coutant, Martineu, Messines, Riboulet and Roncalli [2001], Li [2000], Lindskog [2000], Lindskog and McNeil [2000] and Nyfeler [2000]. Other articles are still in progress credit risk = one of the most important topic for financial applications of copulas. Dependence in credit risk models The credit migration approach (CreditMetrics) The actuarial approach (CreditRisk+) The survival approach Pricing of credit derivatives Modelling credit risk 5-1

50 5.1 Dependence in credit risk models Portfolio with liquids credits Portfolio with not liquids credits. downgrading risk default risk. What is the influence of introducing a dependence function? 1. impact on the joint migration probability distribution; 2. impact on the joint survival distribution. Modelling credit risk 5-2

51 5.1.1 The credit migration approach (CreditMetrics) Notations ( ) N = number of credits in the portfolio. πi,j = rating transition matrix from one state to another. S = set of the eight states {AAA, AA, A, BBB, BB, B, CCC, D}. S = {1,..., 8} with 1 D, 2 CCC, 3 B, etc. (8 AAA). π i = distribution of the initial rating i (R i is the corresponding random variable). π i = discrete probability distribution defined by {j, π i (j)} with π i (j) = Pr {R i j} = j k=1 π i,k and j S. P (i 1,..., i N ; j 1,..., j N ) = joint migration probability distribution P (i 1,..., i N ; j 1,..., j N ) = Pr { } R i1 j 1,..., R in j N p (i 1,..., i N ; j 1,..., j N ) = joint migration probability mass function p (i 1,..., i N ; j 1,..., j N ) = Pr { R i1 = j 1,..., R in = j N } Modelling credit risk 5-3

52 The bivariate migration probability distribution CreditMetrics uses a gaussian random variable Z i π i (j) = Pr { Z i z (i) j Gupton, Finger and Bhatia [1997] define then the joint migration probability distribution by { P (i 1, i 2 ; j 1, j 2 ) = Pr Z i1 z (i 1) j, Z 1 i2 z (i } ( 2) j = Φ z (i 1) 2 j, z (i ) 2) ; ρ 1 where ρ = ρ (C 1, C 2 ) is the asset return correlation of C 1 and C 2. Remark that π i (R i ) = Φ (Z i ). It comes that P (i 1, i 2 ; j 1, j 2 ) = Φ ( Φ 1 ( π i1 (j 1 )), Φ 1 ( ) ) π i2 (j 2 ) ; ρ We can then write P (i 1, i 2 ; j 1, j 2 ) as a function of the bivariate Normal copula C P (i 1, i 2 ; j 1, j 2 ) = C ( π i1 (j 1 ), π i2 (j 2 ) ; ρ ) } j 2 Modelling credit risk 5-4

53 The multivariate migration probability distribution Let C be a copula function. We have P (i 1,..., i N ; j 1,..., j N ) = C ( π i1 (j 1 ),..., π in (j N ) The probability mass function p (i 1,..., i N ; j 1,..., j N ) is given by the Radon-Nikodym density of the copula 1 k 1 =0 1 k N =0 ( 1) k 1+ +k N C ( ) π i1 (j 1 k 1 ),..., π in (j N k N ) Because a copula is a grounded function, we deduce that the joint default probability is It comes that p (i 1,..., i N ; D,..., D) = C ( π i1 (1),..., π in (1) C ( π i1 (1),..., π in (1)) p (i1,..., i N ; D,..., D) C + ( π i1 (1),..., π in (1) ) ) ) We use the convention ( 1) 0 = 1. Modelling credit risk 5-5

54 Some illustrations migration probabilities p (i 1, i 2 ; j 1, j 2 ) joint default probability p (i 1, i 2 ; D, D) discrete default correlation ρ D (C 1, C 2 ) = p (i 1, i 2 ; D, D) π i1,1π i2,1 ( ( π i1,1 1 πi1,1) πi2,1 1 πi2,1 ) Remark 4 The discrete default correlation is not a good measure of the dependence between defaults. We note that 1 < ρ D (C 1, C 2 ) ρ D (C 1, C 2 ) ρ + D (C 1, C 2 ) < 1 Even if the dependence is maximal (C = C + ), ρ D (C 1, C 2 ) can be very small (less than 10%). see CMMRR [2001] for the link between ρ D (C 1, C 2 ) and a 2 2 contingency table. Modelling credit risk 5-6

55 Ilqdo udwlqj Lqlwldo udwlqj DDD DD D EEE EE E FFF G DDD < ; 31; DD 3196 <41;: D 313; 5159 < EEE : 81;7 ;:1:7 71:7 31<; EE :1;8 ;4147 ;15: 31;< 4139 E :8 ;613: 61;9 817< FFF 314< ; 31: : G Wdeoh 4= V)S rqh0 hdu wudqvlwlrq pdwul{ +lq (, Lqlwldo udwlqj Ilqdo udwlqj DDD DD D EEE EE E FFF G < FFF ; 41<8 <168 ;7154 E ; <5175 <9157 EE ;< 9143 <4169 <<14: <;19; EEE ; <61;7 <<154 <<18< <<176 D 31<; :183 <:199 <<19; <<1;8 <<1;< <<1;4 DD :179 <<16: <<1<5 <<1<8 <<1< <<1;4 DDD Wdeoh 5= V)S rqh0 hdu pdujlqdo suredelolw glvwulexwlrqv +lq (,

56 Ilqdo udwlqj ri 2 Ilqdo udwlqj ri DDD DD D EEE EE E FFF G Z c DDD ; DD : 313; 319: D : ::1:< <4199 EEE EE ; E FFF G Z 2 c :8 ;613: 61;9 817< Wdeoh 6= Pljudwlrq suredelolwlhv R E c 2 c c 2 +lq (, zlwk ' $/ 2 ' % dqg 4 E c 2 'fdf Ilqdo udwlqj ri 2 Ilqdo udwlqj ri DDD DD D EEE EE E FFF G Z c DDD ; DD < 315: 31: D : 315: 8157 :: :7 <4199 EEE < EE E < FFF G Z 2 c :8 ;613: 61;9 817< Wdeoh 7= Pljudwlrq suredelolwlhv R E c 2 c c 2 +lq (, zlwk ' $/ 2 ' % dqg 4 E c 2 ' fdf Ilqdo udwlqj ri 2 Ilqdo udwlqj ri DDD DD D EEE EE E FFF G Z c DDD DD D < EEE EE 313: <; :8 E 313; < < :1;< ;613: FFF ; 41;7 41<6 61;9 G <; < Z 2 c 314< ; 31: : Wdeoh 8= Pljudwlrq suredelolwlhv R E c 2 c c 2 +lq (, zlwk ' %/ 2 ' &&& dqg 4 E c 2 ' fdf

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63 Computing the risk of a portfolio Credit risk measure can be obtained by Monte Carlo methods. To simulate the random variables R i1,..., R in, we consider the following two step algorithm: 1. we simulate the random numbers u 1,..., u N from the copula C; 2. the random variates r 1,..., r N are obtained by the inverse distribution function method: r n = π ( 1) i (u n ) Credit risk measure = credit value-at-risk with two components: downgrading risk and default risk. Some problems = credit modelling + recovery rates modelling. Modelling credit risk 5-7

64 Lqlwldo udwlqj DDD DD EEE EEE EE FFF 4 DDD DD EEE EEE EE FFF 5 DD DD EEE EEE EE FFF 6 DDD DD EEE EEE EE G 7 DDD DD EEE D EE FFF 8 DDD DD D EEE EE FFF 9 DDD DD EEE EEE EE FFF : DDD D EEE EEE EE G ; DDD DD EEE EEE EE EE < DDD DD EEE EEE EE FFF 43 DDD DD EEE EEE EE FFF Wdeoh 9= Vlpxodwlrq ri qdo udwlqjv zlwk Lqlwldo udwlqj DDD DD EEE EEE EE FFF 4 DDD DD EEE EEE E E 5 DD DD D D EE FFF 6 DDD DD EEE EEE EE G 7 DDD DD EEE D E FFF 8 DDD DD EEE EEE E FFF 9 DDD DD EEE EEE EE FFF : DDD DD EEE EEE EE G ; DDD DD EE EEE EE E < DDD DD EEE EE EE G 43 DDD DD EEE EEE EE FFF Wdeoh := Vlpxodwlrq ri qdo udwlqjv zlwk 2 5 ' 9 7 f2d f.d fd f2d f2 fd f2d f2d f f.d f2d f2 f2d fd fd 6 : ' 9 7 f2d f.d fd f2d f2 fd f2d f2d f f.d f2d f2 f2d fd fd 6 : 8

65 5.1.2 The actuarial approach (CreditRisk+) More results are in CMMRR [2001]. We just give here some ideas about the copula used by CreditRisk+. Notations T = time horizon. B n = Bernoulli random variable which indicates if the credit n has defaulted. p n = stochastic parameter of B n with p n = P n θn,m X m P n is estimated for each obligator via any credit rating system. X m are M independent Gamma distributed factors. Given those factors, the B n are assumed to be conditionally independent. Modelling credit risk 5-8

66 One common risk factor Let us denote X Γ (α, β), g (x) the pdf of X, F (t 1,..., t N ) the joint distribution of defaults, F n (t n ) the marginal distributions and F x n (t n ) the conditional distribution. We have It comes that F (t 1, t 2 ) = F (t 1,..., t N ) = 0 N n=1 F x n (t n ) g (x) dx ( ) F 1 (t 1 ) F 2 (t 2 ) 1 α α (F 1 (t 1 ) + F 2 (t 2 ) 1) + The copula is then C (u 1, u 2 ; α) = ( ) u 1 u 2 1 α α (u 1 + u 2 1) + Remark 5 The previous analysis is not exact. Only the subcopula can be defined for the four continuity points (see CMMRR [2001] for more details). Modelling credit risk 5-9

67 Approximation case We suppose that P n are small. In this case, we use a Poisson approximation ( ) X 1 P n αβ = exp X P n αβ We can show that with F (t 1,..., t N ) = φ F n (t n ) = N n=1 ln F n (t n ) { ( ) exp P n if 0 tn < 1 αβ 1 if t n > 1 φ is the Laplace transform associated with Γ (α, β). By remarking that ln F n (t n ) = P n αβ φ 1 (F n (t n )) Modelling credit risk 5-10

68 we have F (t 1,..., t N ) = φ = N n=1 φ 1 (F n (t n )) (2) ( F 1 (t 1 ) 1 α F N (t N ) 1 α N + 1 ) α The dependence function is then the Cook-Johnson copula. Note that equation (2) corresponds to Archimedean copulas. Moreover, the generator of the Archimedean copulas is the inverse of a Laplace transform. In this case, we can give a new probabilistic interpretation, because this model is a frailty model (see CMMRR [2001] for a discussion on the dependence in CreditRisk+). Extensions when the distribution of X is not Gamma can be found in CMMRR [2001]. This result comes from the fact that the random variables B n are assumed to be conditionally independent. Modelling credit risk 5-11

69 5.1.3 The survival approach see Coutant, Martineu, Messines, Riboulet and Roncalli [2001]. Modelling credit risk 5-12

70 5.2 Pricing of credit derivatives A default is generally described by a survival function S (t) = Pr {T > t}. Let C be a survival copula. A multivariate survival distributions S can be defined as follows S (t 1,..., t N ) = C (S 1 (t 1 ),..., S N (t N )) where (S 1,..., S N ) are the marginal survival functions. Nelsen [1999] notices that C couples the joint survival function to its univariate margins in a manner completely analogous to the way in which a copula connects the joint distribution function to its margins. Introducing correlation between defaultable securities can then be done using the copula framework (see Li [2000] and Maccarinelli and Maggiolini [2000]). Modelling credit risk 5-13

71 5.2.1 First-to-Default valuation Let us define the first-to-default τ as follows τ = min (T 1,..., T N ) Nelsen [1999] shows that the survival function of τ is given by the diagonal section of the survival copula. Let C be a copula. Its survival copula is given by the following formula C (u 1,..., u N ) = C (1 u 1,..., 1 u n,..., 1 u N ) with C (u 1,..., u n,..., u N ) = N n=0 ( 1) n u Z(N n,n) C (u) where Z (M, N) denotes the set { u [0, 1] N N n=1 X {1} (u n ) = M }. Modelling credit risk 5-14

72 When the copula is radially symmetric, we have C = C The survival distribution S of τ is S (t) = C (S 1 (t),..., S N (t)) It comes that the density of τ is given by f (t) = t S (t) = N n=1 n C (S 1 (t),..., S N (t)) f n (t) Modelling credit risk 5-15

73 5.2.2 Example N credit events, default of each credit event given by a Weibull survival function (the baseline hazard is constant and equal to 3% per year and the Weibull parameter is 2). C is a Normal copula of dimension N = very tractable (N can be very large) and n C is a Normal copula of dimension N 1. Modelling credit risk 5-16

74

75

76

77 5.2.3 Other credit derivatives see Coutant, Martineu, Messines, Riboulet and Roncalli [2001]. Modelling credit risk 5-17

78 6 Conclusion The study of copulas and the role they play in probability, statistics, and stochastic processes is a subject still in its infancy. There are many open problems and much work to be done (Nelsen [1999], page 4). In finance, the use of copulas is very recent (Embrechts, McNeil and Straumann [1999]). In one year, great progress have been made. Nevertheless, the finance industry needs more works on copulas and their applications. And there are many research directions to explore. Moreover, many pedagogical works have to be done in order to familiarize the finance industry with copulas. Conclusion 6-1

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