Time Changed Markov Processes in Unified Credit-Equity Modeling

Abstract

This paper develops a novel class of hybrid credit-equity models with state-dependent jumps, local-stochastic volatility, and default intensity based on time changes of Markov processes with killing. We model the defaultable stock price process as a time-changed Markov diffusion process with state-dependent local volatility and killing rate (default intensity). When the time change is a Levy subordinator, the stock price process exhibits jumps with state-dependent Levy measure. When the time change is a time integral of an activity rate process, the stock price process has local-stochastic volatility and default intensity. When the time change process is a Levy subordinator in turn time changed with a time integral of an activity rate process, the stock price process has state-dependent jumps, local-stochastic volatility, and default intensity. We develop two analytical approaches to the pricing of credit and equity derivatives in this class of models. The two approaches are based on the Laplace transform inversion and the spectral expansion approach, respectively. If the resolvent (the Laplace transform of the transition semigroup) of the Markov process and the Laplace transform of the time change are both available in closed form, the expectation operator of the time-changed process is expressed in closed form as a single integral in the complex plane. If the payoff is square integrable, the complex integral is further reduced to a spectral expansion. To illustrate our general framework, we time change the jump-to-default extended constant elasticity of variance model of Carr and Linetsky (2006) and obtain a rich class of analytically tractable models with jumps, local-stochastic volatility, and default intensity. These models can be used to jointly price equity and credit derivatives.