Credit risk valuation with rating transitions and partial information

This work intends to shed some light on a new use of Phase-type distributions in credit risk, taking into account different flows of information without huge numerical calculations. We consider credit migration models with partial information and study the influence of a deficit of information on prices of credit linked securities. The transitions through the various credit classes are modeled via a continuous time Markov chain but they are not directly observable by investors in the secondary market. We first consider the case of one bond issuer and study three settings of partial information. In a first model, information about ratings arrives at predetermined dates with delay periods. In a second model, information arrives randomly according to an exogenous Poisson process, whereas in a third model, information arrives randomly according to an endogenous rule (transitions are observed only when they lead the Markov chain to a class with a lower credit rating). We infer in the three settings bonds and options prices, and we provide an explicit description of the dynamics of bond prices under real and pricing measures. We also consider the case of several bond issuers. We first study a model for two different issuers and analyze the cross effects of deficit of information and contagion on bonds prices and correlation of default times. We then propose a model for several homogeneous issuers. Finally, numerical illustrations show the relevance of taking into account deficits of information on prices of credit linked securities.