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

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2 Copulas and multivariate financial models 1. How to define multivariate financial models compatible with univariate non-Gaussian financial models Copula = a powerful tool 2. How to obtain tractable multivariate financial models (in terms of computational time) 3. How to specify multivariate financial models which may be understood/used by the finance industry Copula = a promising tool Copulas have been already incorporated in some software solutions: • SAS Risk Dimensions • Palisade @Risk Modelling dependence in finance using copulas Copulas and multivariate financial models 2-1
2.1 Pearson correlation and dependence Pearson correlation ρ = linear dependence measure. For two given asset prices processes S 1 (t) and S 2 (t) which are GBM, the range of the correlation is ρ − ≤ ρ (S 1 (t) , S 2 (t)) ≤ ρ + with ρ ± = exp (±σ 1 σ 2 (t − t 0 )) − 1 √ exp ( σ1 2 (t − t ) √ 0) − 1 · exp ( σ2 2 (t − t 0)) − 1 ρ (S 1 (t) , S 2 (t)) = ρ − (resp. ρ + ) ⇔ C 〈S 1 (t) , S 2 (t)〉 = C − (resp. C + ) ⇔ S 2 (t) = f (S 1 (t)) with f a decreasing (resp. increasing) function Perfect dependence ≠ |ρ| = 1 Modelling dependence in finance using copulas Copulas and multivariate financial models 2-2
Page 1 and 2: Modelling dependence in finance usi
Page 3: 1 The Gaussian assumption in financ
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 and 26: For a given target insolvency rate
Page 27: Correlated aggregate loss distribut
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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