Credit Risk Models with Incomplete Information

Incomplete information is at the heart of information-based credit risk models. In this paper, we rigorously define incomplete information with the notion of "delayed filtrations." We characterize two distinct types of delayed information, continuous and discrete: the first generated by a time change of filtrations and the second by finitely many marked point processes. This notion unifies the noisy information in Duffie and Lando [Duffie, D., D. Lando. 2001. Term structures and credit spreads with incomplete accounting information. Econometrica 69 633-664] and the notion of partial information in Collin-Dufresne et al. [Collin-Dufresne, P., R. Goldstein, J. Helwege. 2003. Is credit event risk priced? Modeling contagion via the updating of beliefs. Working paper, Carnegie Mellon University, Pittsburgh], under which structural models are translated into reduced-form intensity-based models. We illustrate through a simple example the importance of this notion of delayed information, as well as the potential pitfall for abusing the Laplacian approximation techniques for calculating the intensity process in an information-based model.