Affine Credit Risk Models under Incomplete Information

We consider the problem of computing some basic quantities such as defaultable bond prices and survival probabilities in a credit risk model according to the intensity based approach. We let the default intensities depend on an external factor process that we assume is not observable. We use stochastic filtering to successively update its distribution on the basis of the observed default history. On one hand this allows us to capture aspects of default contagion (information-induced contagion). On the other hand it allows us to evaluate the above quantities also in our incomplete information context. We consider in particular affine credit risk models and show that in such models the nonlinear filter can be computed via a recursive procedure. This then leads to an explicit expression for the filter that depends on a finite number of sufficient statistics of the observed interarrival times for the defaults provided one chooses an initial distribution for the factor process that is of the Gamma type.