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

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Problem: the margins of the multivariate Risk-Neutral Distribution (RND) are not the univariate RND. ⇒ In this case, we may show that there exists arbitrage opportunities inside the same bank (see [10]). ⇒ Moreover, one-asset options could be viewed as limits of multi-asset options — see e.g. the Basket option G (T ) = (α 1 S 1 (T ) + α 2 S 2 (T ) − K) + . The copula construction Let Q be the multivariate RND of the random vector S (T ) | F t0 . In [10], we show that the margins of Q are necessarily univariate RND ∗ . Using Sklar’s theorem, it comes that Q admits the following canonical decomposition Q (S 1 (T ) , . . . , S N (T )) = C Q (Q 1 (S 1 (T )) , . . . , Q N (S N (T ))) C Q is called the risk-neutral copula (RNC). ∗ We prove this by using properties of the Girsanov theorem applied to multivariate probability measure. Modelling dependence in finance using copulas New pricing methods with copulas 4-3
Relationships between C Q and C P Let P be the objective (or historical) distribution. We denote by C P the objective copula. We can prove the following proposition: Proposition 1 If the drift and the diffusion of the asset prices vector S (t) are of the form µ (t) ⊙ S (t) and σ (t) ⊙ S (t) and if risk premiums are non stochastic, then the risk-neutral copula C Q is equal to the objective copula C P . Implication of this proposition: in this case, the univariate RND can be estimated using the options market whereas the RNC can be estimated using the spot market. So, the spot market contains useful information to price multi-asset options. Modelling dependence in finance using copulas New pricing methods with copulas 4-4
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: 4.1 Coherent valuation of multi-ass
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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