| In the current literature, there are two distinct families of credit risk models: firm value model and intensity-based model. The first one makes explicit assumptions about the dynamics of firm value and default happens if the firm value is less than some threshold level. The second one treats default as a totally unexpected event, the stochastic structure is modeled by some intensity process. Is there any connection between these two classes of models?; Before answering the question, we introduce a class of stochastic processes named after the French scientist Paul Pierre Levy: Levy processes. In part of the dissertation, we analyze properties of Levy processes, which includes Levy densities, multivariate Levy processes and equivalent martingale measures for Levy processes. We model the firm value process as an exponential Levy process. Starting with the firm value models, we obtain the instantaneous default probability in form of the Levy measure. In this case, default may be caused by either the random jumps or the Brownian motion. The information of the instantaneous default probability is not enough to obtain the survival probability. Thus we provide PIDE (partial integro-differential equation) representations of the survival probability and the corresponding bond prices. Since default is a totally unexpected event in the intensity-based model, it need more assumptions to achieve the point. Under the assumption that the default is only caused by the random jumps in the firm value process, the default intensity is a decreasing function of the nature logarithm ratio of pre firm value to the default threshold, which is not given exogenously. And in this case, the survival probability has a closed-form expression as in the intensity-based model. It shows the connection between two default risk models. Several examples are shown to justify out setup.; The results of the equivalent martingale measures and the instantaneous default intensities based on the Levy processes can be extended to additive processes with local characteristics. |
