Modelling dependence in finance using copulas - Thierry Roncalli's ...

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Let ϑ 1 and ϑ 2 be two losses with distributions G 1 and G 2 . We denote ϑ the total loss with distribution G. We have CaR (α) = G −1 (α) = CaR 1 (α) + CaR 2 (α) = G −1 1 (α) + G−1 2 (α) It is equivalent to assume that C 〈ϑ 1 , ϑ 2 〉 = C + . In this case, we have ϑ 2 = G (−1) 2 (G 1 (ϑ 1 )). Let us denote ϖ the function x ↦→ x + G (−1) 2 (G 1 (x)). We have It comes that CaR (α) = ϖ α = Pr {ϑ 1 + ϑ 2 ≤ CaR (α)} = E [ 1 [ϖ(ϑ1 )≤CaR(α)] ( ) = G 1 ϖ −1 (CaR (α)) ) 1 (α) ( G (−1) CaR (α) = G (−1) 1 (α) + G (−1) ( 2 (G 1 G (−1) )) 1 (α) ] and we obtain the relationship = CaR 1 (α) + CaR 2 (α) Modelling dependence in finance using copulas An open field for risk management 3-8
Correlated aggregate loss distributions or correlated frequencies The total loss distribution ϑ for the bank as whole is defined by ϑ = I∑ J∑ i=1 j=1 ϑ (i, j) In this case, we could introduce the dependence directly between the aggregate loss distributions. Or, we could introduce the dependence indirectly between the frequency distributions: ϑ = I∑ J∑ i=1 j=1 N(i,j) ∑ n=0 ζ n (i, j) For example, we could use multivarariate Poisson distributions generated from the Normal copula (Song [2000]). Modelling dependence in finance using copulas An open field for risk management 3-9
Page 1 and 2: Modelling dependence in finance usi
Page 3 and 4: 1 The Gaussian assumption in financ
Page 5: 2.1 Pearson correlation and depende
Page 8: 2.3 Copulas in a nutshell A copula
Page 11 and 12: The copula function C (u; ρ) = Φ
Page 13 and 14: Simulation • Generate a gaussian
Page 15 and 16: We can show that the solution of th
Page 19 and 20: 3 An open field for risk management
Page 21 and 22: Let consider an example of equity p
Page 23 and 24: 3.2 Operational risk Industry defin
Page 25: For a given target insolvency rate
Page 31: The limit case λ = lim α→1 λ (
Page 35 and 36: 4.1 Coherent valuation of multi-ass
Page 37 and 38: Relationships between C Q and C P L
Page 39: BS pricing in stochastic volatility
Page 46: An example of the computation of th
Page 49 and 50: Some examples ∗ The Default Digit
Page 51: Numerical illustrations ϖ = 1 —
Page 57 and 58: 6 References (Copulas and Finance)
Page 59 and 60: [18] Georges, P., A-G. Lamy, E. Nic
Page 61: 7 Other references [1] Deheuvels, P

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