Time Changed Markov Processes in Unified Credit - Equity Modeling ∗

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 Lévy subordinator, the stock price process exhibits jumps with state-dependent Lévy 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 Lévy 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 CEV model (JDCEV) 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. ∗This research was supported in part by the grants from the Federal Deposit Insurance Corporation, Moody’s Investor Services, and the National Science Foundation under grant DMS-0802720. †Department of Industrial Engineering and Management Sciences, McCormick School of Engineering and Applied Sciences, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, E-mail: [email protected]. ‡Bloomberg LP and NYU Courant Institute, 251 Mercer Street, New York, NY, 10012, USA, Phone: (212) 617-5056, E-mail: [email protected]. §Department of Industrial Engineering and Management Sciences, McCormick School of Engineering and Applied Sciences, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, Phone: (847) 491 2084, E-mail: [email protected], Web: http://users.iems.northwestern.edu/∼linetsky.