Time-Changed CIR Default Intensities with Two-Sided Mean-Reverting Jumps

Abstract

The present paper introduces a jump-diffusion extension of the classical diffusion default intensity model by means of subordination in the sense of Bochner. We start from the bi-variate process of the diffusion state variable and default indicator process (X,D) in the diffusion intensity model and time change it with a Levy subordinator T. We then give a detailed characterization of the resulting time changed process (Xφ,Dφ)=(X(T),D(T)) as a Markovian Ito semimartingale and, in particular, show from the Doob-Meyer decomposition of Dφ that the default time in the time-changed model has a jump-diffusion or a pure jump intensity. In particular, when X is a CIR diffusion, the default intensity of the subordinate model (SubCIR) turns out to be a jump-diffusion or a pure jump process with two-sided mean-reverting jumps that stays non-negative. The SubCIR default intensity model is fully analytically tractable by means of the explicitly computed eigenfunction expansion of the relevant semigroups. This allows explicit closed-form pricing of credit-sensitive securities.