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

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The case of the spread option that the price P (t 0 ) is In [12], Valdo Durrleman shows P (t 0 ) = S 2 (t 0 ) − S 1 (t 0 ) − Ke −r(T −t 0) + e −r(T −t 0) ∫ K −∞ 0 ∫ +∞ f 1 (x) · ∂ 1 C Q (F 1 (x) , F 2 (x + y)) dx dy A remark The copula construction implies that we can associate a risk-neutral copula to a multivariate risk-neutral distribution. But it does not mean that the combination of univariate RND with a copula define necessarily a multivariate risk-neutral distribution (see [10] for further details). Modelling dependence in finance using copulas New pricing methods with copulas 4-5
BS pricing in stochastic volatility environment We assume that the asset prices S n (t) are given by the Heston model ⎧ √ ⎨ dS n (t) = µ n S n (t) dt + V n (t)S n (t) dWn 1 (t) ⎩ dV n (t) = κ n (V n (∞) − V n (t)) dt + σ n √V n (t) dW 2 n (t) with E [ Wn 1 (t) Wn 2 ] (t) | F t0 = ρn (t − t 0 ), κ n > 0, V n (∞) > 0 and σ n > 0. The market prices of risk processes are λ 1 √ n (t) = (µ n − r) / V n (t) and λ 2 n (t) = λ n σn −1 √ V n (t). To compute prices of spread options, we consider that the RNC is the Normal copula with parameter ρ. We compare then the Heston prices with these given by the BS model using ATM implied volatilities. Numerical values (two assets with same characteristics except ρ n ): S n (t 0 ) = 100, τ = 1/12, r = 5%, V n (t 0 ) = V n (∞) = √ 20%, κ n = 0.5, σ n = 90% and λ n = 0. Modelling dependence in finance using copulas New pricing methods with copulas 4-6
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 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: Relationships between C Q and C P L
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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