Pricing CDO tranches in an intensity based model with the mean reversion approach

We discuss the phenomenon of mean reversion in credit risk market and propose a class of models, in the framework of intensity based model, where the default intensity is composed of a common component and a idiosyncratic component which are specified by independent mean reverting stochastic processes of the following Markovian type dX(t)=(θ+σα(X(t),t))X(t)dt+σX(t)dW(t) where θ≥0θ≥0 is the long-term mean value, the parameter σ≥0σ≥0 stands for the scaling of the volatility, and α(X(t),t)α(X(t),t) is the mean correction with the function α:R×[0,∞)↦α(x,t)∈Rα:R×[0,∞)↦α(x,t)∈R being twice differentiable in xx and differentiable in tt, and W(t)W(t) is a Brownian motion. We demonstrate how this class of models can be used to price synthetic CDOs and present a closed-form solution of tranche spreads in synthetic CDOs.