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 d X ( t ) = ( θ + σ α ( X ( t ) , t ) ) X ( t ) d t + σ X ( t ) d W ( t ) where θ ≥ 0 is the long-term mean value, the parameter σ ≥ 0 stands for the scaling of the volatility, and α ( X ( t ) , t ) is the mean correction with the function α : R × [ 0 , ∞ ) ↦ α ( x , t ) ∈ R being twice differentiable in x and differentiable in t , and 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.