CDS Pricing with Fractional Hawkes Processes

We propose a fractional self-exciting model for the risk of corporate default. We study the properties of a time-changed version of an intensity based model. As a time-change, we use the inverse of an α-stable subordinator. Performing such a time-change allows to incorporate two particular features in the survival probability curves implied by the model. Firstly, it introduces random periods of time where the survival probability is frozen, thereby modeling periods of time where the viability of the company is not threatened. Secondly, the time-change implies possible sharp drops in the survival probability. This feature corresponds to the occurence of one-time events that threaten the creditworthiness of the company. We show that the joint probability density function and Laplace transform of the time-changed intensity and associate compensator are solutions of fractional Fokker-Planck equations. After a discussion regarding approximation of Caputo fractional derivatives, we describe a simple and fast numerical method to solve the Fokker-Planck equation of the Laplace transform. This Laplace transform is used to obtain the survival probabilities implied by our model. Finally, we use our results to calibrate the model to real market data and show that it leads to an improvement of the fit.