A recursive method for fractional Hawkes intensities and the potential approach of credit risk

Abstract

This article explores the potential approach for credit risk, which is an alternative to structural and reduced models. In the context of credit risk, it consists in assuming that the survival probability of a company is equal to the ratio of the expected value of a supermartingale divided by its initial value. This approach, that was previously used for modeling the term structure of interest rates, is extended by the use of a fractional self-exciting process, or fractional Hawkes process. This is a self-exciting process that is time-changed by the inverse of an alpha-stable subordinator. A self-exciting process allows to take into account the clustering of economic events such as defaults, whereas the time-change allows to properly fit highly convex survival probability curves. We derive a new recursive method that allows to compute all the moments of a self-exciting process intensity, as well as its time-changed counterpart. We show that this method can be used to approximate the survival probabilities in the potential approach. More specifically, we prove that the approximation converges and we provide a bound on the numerical error. Finally, we calibrate the model and show that it allows to properly fit survival probability curves that are highly convex.