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

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Agenda 1. The Gaussian assumption in finance 2. Copulas and multivariate financial models 3. An open field for risk management • Market risk • Operational risk • Credit risk 4. New pricing methods with copulas • Multi-asset options • Credit derivatives Modelling dependence in finance using copulas 1
1 The Gaussian assumption in finance We consider the ‘universal’ financial model. Let (Ω, F, P) be the probability space. The asset prices processes S 1 (t) and S 2 (t) are given by the SDE representation { dS1 (t) = µ 1 S 1 (t) dt + σ 1 S 1 (t) dW 1 (t) dS 2 (t) = µ 2 S 2 (t) dt + σ 2 S 2 (t) dW 2 (t) where W 1 (t) and W 2 (t) are two F t –brownian motions with E [ W 1 (t) W 2 (t) | F t0 ] = ρ (t − t0 ) It comes that the logarithm return of assets is gaussian (= Gaussian assumption in finance). Empirical facts: Asset returns are not gaussian (see the financial econometric literature on ARCH, long-memory, Lévy processes, etc.). Problem: Univariate financial models are not gaussian, but multivariate financial models are gaussian! Modelling dependence in finance using copulas The Gaussian assumption in finance 1-1
Page 1: Modelling dependence in finance usi
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 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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