A Lattice Model with Incomplete Information: A Credit Risk Application

We design a discrete time arbitrage-free model under incomplete information for application to credit risk models in the spirit of Duffie and Lando (2001). We assume a fundamental value process evolving according to a complete market model and a sequence of imperfect signals conveying information on the state of the fundamental value. The market value is computed by conditioning on the available information. The model shows that if the information setting is built consistently, the market remains complete even in the face of incomplete information. If insiders are not allowed into the market, the requirement is that the equivalent martingale measure of market prices have to coincide with the martingale measure of the fundamental price in the case of perfect information (Information Consistency). Such martingale measure can be recovered for arbitrary dependence structures between the fundamental process and the signal. If insiders are allowed into the market, the Information Consistency requirement has to be made more stringent imposing the H-condition. We show that in this case only a subset of signals are consistent with no arbitrage, and are those that are not-Granger caused by the fundamental process. The model is finally applied to the study of the dynamics of the expected loss of a BBB rated firm in presence of incomplete information.