Modeling of Contagious Rating Changes and Its Application to Multi-Downgrade Protection∗

This paper presents a new modeling of the intensities of typical rating change events with a multivariate affine jump process. The main feature of our model is that the intensity model has a mutually exciting structure in the sense that an event occurrence can jump not only the intensity of the same type of events but those of another event types. Also, such a modeling with the multivariate affine jump process enables us to execute the maximum likelihood estimation of the model parameters relatively easily and to compute the expectation of the number of events without Monte Carlo simulation. We actually try the maximum likelihood estimation for a specific model with some historical record of rating changes for the corporate bond issued by Japanese companies. In addition, we propose a new credit derivative named multi-downgrade protection (MDP) as an application of our mutually contagious intensity model. Some numerical illustrations on the expectation of the number of events and the fair value of MDP are also presented.