Modelling Dependence with Copulas - IFOR

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8 Mixture of Extremal Distributions It then follows that the desired distribution is given by F (x 1 ,x 2 ,x 3 ,x 4 )=C(F 1 (x 1 ),F 2 (x 2 ),F 3 (x 3 ),F 4 (x 4 )). Random variate generation from the this distribution, a mixture of extremal distributions, is then straight forward according to the above algorithm. Extremal distributions are not only of theoretical interest as shown in the following example. Example 8.4. Consider prices on put and call options, with different exercise prices and a fixed time horizon T , on some underlying stock. At time t
9 Modelling Extremal Events in Practice 9.1 Pricing Risky Insurance Contracts Consider a portfolio consisting of n risks X 1 ,... ,X n which are positive. Let the risks represent potential losses in dependent lines of business for an insurance company. Suppose that losses not greater than k 1 ,... ,k n can be accepted, and that the insurance company wants to buy protection against the situation where l or more of the losses simultaneously exceed their respective thresholds k 1 ,... ,k n . Suppose that a reinsurance company offers to sell a contract which makes a payout only in thecasethatatleastl of the n risks exceeds their thresholds. To begin with we will look at the probability that a payout is made. Later we will look at the expected size of the payout under the assumption that losses which exceed their thresholds are paid in full. Assume historical data are available allowing estimation of Let • marginal distributions; • pairwise rank correlations. N = | { i ∈{1,...,n}|X i >k i } | be the number of losses exceeding their thresholds and let L l = 1 {N≥l} n ∑ i=1 ( Xi 1 {Xi>k i}) be the loss to the reinsurer. Suppose that the contract the insurance company is interested in is the one for which the losses are paid in full when all losses exceed their thresholds. If the insurance company holds this contract it has full protection against simultaneous big losses in all lines of business no matter how big the total loss is. The probability of payout is then given by P[N = n] =H(k 1 ,... ,k n ). Unfortunately a good estimation of the joint distribution, H, is not realistic due to lack of data and theoretical difficulties. However, data from the previous year(s) enable estimation of the Kendall’s tau rank correlation matrix via the sample version. Furthermore, estimation of univariate marginal distributions is well understood. Hence we can assume that rank correlations and margins are given. Since H(x 1 ,... ,x n )=Ĉ(F 1(x 1 ),... ,F n (x n )), the payout probability can not be calculated without choosing a suitable copula representing the dependence structure among the n risks. The standard approach might be to choose a Gaussian copula, given by Cρ Ga l (u) =Φ n ρ l (Φ −1 (u 1 ),... ,Φ −1 (u n )), where Φ n ρ l denotes the joint distribution function of the n-variate standard normal distribution function with linear correlation matrix ρ l and Φ −1 denotes the inverse of the distribution function of the univariate standard normal distribution. To parametrizise the Gaussian copula from the estimated Kendall’s tau rank correlation matrix the relation ρ l = sin(πτ/2) 67
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Eidgenössische Technische Hochschu
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Contents Contents 1 Motivation 1 2
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1 Motivation distributions. Hence w
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2 Copulas 1. DomC ′ = S 1 × S 2
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2 Copulas Theorem 2.3. If C ′ is
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2 Copulas Here C α1(X 1),α 2(X 2)
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2 Copulas . • Simulate a value u
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3 Dependence 3.2 Perfect Dependence
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3 Dependence Thus ∫∫ P[(X 1 −
Page 22 and 23: 3 Dependence Proof. For both Kendal
Page 24 and 25: 3 Dependence Definition 11. Let X a
Page 26 and 27: 3 Dependence Furthermore if C is a
Page 28 and 29: 4 Techniques for Construction of Mu
Page 35 and 36: 5 Archimedean Copulas In this chapt
Page 37 and 38: 5.2 Definitions (since ϕ and ϕ [
Page 39 and 40: 5.3 Properties 5.3 Properties For c
Page 41 and 42: 5.3 Properties Example 5.8. Conside
Page 43 and 44: 5.4 Order Theorem 5.7. Let ϕ be a
Page 45 and 46: 5.5 Multivariate Archimedean Copula
Page 47 and 48: 5.5 Multivariate Archimedean Copula
Page 49: 5.5 Multivariate Archimedean Copula
Page 52 and 53: 6 Elliptical Copulas Example 6.2. F
Page 54 and 55: 6 Elliptical Copulas We now address
Page 56 and 57: 6 Elliptical Copulas • • •
Page 59 and 60: 7 Extreme Value Copulas 7.1 Univari
Page 61 and 62: 7.2 Multivariate Extreme Value Theo
Page 63: 7.2 Multivariate Extreme Value Theo
Page 66 and 67: 8 Mixture of Extremal Distributions
Page 74 and 75: 9 Modelling Extremal Events in Prac
Page 85: 10 Conclusions In this paper we hav
Page 88 and 89: 11 Appendix { if(rcor[row,col] == 1
Page 90 and 91: 11 Appendix lognormal
Page 93: References References [1] Dhaene J.
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