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

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Estimation The log-likelihood function is l (u; ρ) = − T 2 ln |ρ| − 1 2 T∑ ςt ⊤ t=1 ( ρ −1 − I ) ς t and the ML estimate of ρ is also ˆρ ML = 1 T T∑ t=1 ς ⊤ t ς t. Two-stage (Joe and Xu [1996]) and omnibus (Genest, Ghoudi and Rivest [1995]) estimators can then be obtained with ς t = ( Ψ [F 1 ] ( x t 1 ) , . . . , Ψ [FN ] ( x t N )) : 1. IFM estimate: F n = MLE of the n th marginal distribution. 2. Omnibus estimate : F n = n th empirical distribution. ⇒ The data are mapped to uniforms and transformed with the inverse gaussian distribution. The correlation parameter ρ of the Normal copula is then equal to the Pearson product moment of the transformed data. Modelling dependence in finance using copulas Copulas and multivariate financial models 2-7
Simulation • Generate a gaussian vector v of random variables with correlation ρ. • To simulate a vector x of random variables with marginals F 1 , . . . , F N and a Normal copula with parameters ρ, we use the following transformation x = ( Ψ −1 [F 1 ] (v 1 ) , . . . , Ψ −1 [F N ] (v N ) ) Application to marketing Segmentation is a useful tool for marketing (and scoring). Let Y be a random variable which corresponds to the target. The main idea is to define classes Class Definition 1 . Y ≤ y 1 M y M−1 ≤ Y ≤ y M Modelling dependence in finance using copulas Copulas and multivariate financial models 2-8
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: The copula function C (u; ρ) = Φ
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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