Distortion copula models

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This page relates to the paper "Using Distortions of Copulas to price CDOs". Glenis Crane and John van der Hoek. (pdf) University of Adelaide. Matlab code is available at Code:distortion_copula_toolbox.

[edit] Definition: Distortion of a Copula

Let C be a bivariate copula and \psi : [0,1] \rightarrow [0,1] be a bijective map with inverse ψ − 1 then C^{\psi}(u,v) = \psi^{-1}\left(C(\psi(u),\psi(v))\right),\ \ u,v \in I= [0,1] is a strict distortion of C. Technical conditions on ψ described in the paper (pdf) ensure Cψ(u,v) is also a copula, and we may generalize to bijective maps having only pseudo-inverses.

[edit] Density

The density of a distorted copula is given by

\frac{\partial^2}{\partial u \partial v} C^{\psi}(u,v) = \frac{\psi'(u) \psi'(v) }{[\psi'(C^{\psi})]^3 } \left\{ [\psi'(C^\psi)]^2 \frac{\partial^2}{\partial u \partial v} C(\psi(u),\psi(v)) - \psi''(C^{\psi}) \frac{\partial}{\partial u} C(\psi(u),\psi(v)) \frac{\partial}{\partial v} C(\psi(u),\psi(v)) \right\}

[edit] Application to Gaussian One Factor Model

Denoting the one and two dimensional cumulative normal distributions by Φ and Φ2 respectively we may relate two variables with cumulative marginals F1(z) and F2(y) by:

C^{\psi}(F_1(z),F_2(y)) = \psi^{-1} \left[ \Phi_2(\Phi^{-1}(\psi[F_1(z)]),\Phi^{-1}(\psi[F_2(y)];\rho) \right]

where as customary ρ is the common pairwise gaussian correlation. As an application, we may model the conditional probability that a firm defaults as:

p^{(V)} = \frac{ \Phi \left( \frac{\Phi^{-1} (\psi[F_1(z)]) - \rho \Phi^{-1}(\psi[F_2(y)])} { \sqrt{1-\rho^2} } \right) \psi'(F_2(y)) } { \psi'\left(\psi^{-1}\left[\Phi_2(\Phi^{-1}(\psi[F_1(z)), \Phi^{-1}(\psi[F_2(y)]);\rho)\right]\right) }

This reduces to the familiar expression

p^{(V)}(y) = \Phi \left( \frac{\Phi^{-1} (\bar{p}) - \rho y} { \sqrt{1-\rho^2} } \right)

in the absence of any distortion (ψ = identity) and normally distributed common factor y (i.e. F2 = Φ). Here \bar{p} = F_1(z) is the unconditional probability of default.

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