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

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2.4 The Normal copula Two caracteristics in finance: High dimensional problems (e.g. a portfolio with 1000 securities) and probabilistic properties of the models (e.g. markovian property). All the copula functions are not good candidates for financial application in an industry point of view. The Normal copula has not yet been extensively studied (see however Song [2000]). Nevertheless, it may be an ‘industrial’ copula. Remark 1 The multivariate normal distribution is very tractable. It is very easy to estimate the parameters and simulation is straightforward. Moreover, this distribution has nice properties. Is it also the case for the Normal copula Modelling dependence in finance using copulas Copulas and multivariate financial models 2-5
The copula function C (u; ρ) = Φ ( Φ −1 (u 1 ) , . . . , Φ −1 (u N ) ; ρ ) The density is c (u; ρ) = |ρ| −1 2 exp ( − 1 ( 2 ς⊤ ρ −1 − I ) ) ς with ς = ( Φ −1 (u 1 ) , . . . , Φ −1 (u N )) . The Ψ transform We define the operator Ψ as follows Ψ [F] : R −→ R x ↦−→ Ψ [F] (x) = Φ −1 (F (x)) We note also Ψ −1 the (left) inverse operator (Ψ −1 ◦ Ψ = 1), i.e. Ψ −1 [F] (x) = F [−1] (Φ (x)). Modelling dependence in finance using copulas Copulas and multivariate financial models 2-6
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 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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