Time-Changed Markov Processes in Unified Credit-Equity Modeling

43 Pages Posted: 27 Sep 2010

See all articles by Rafael Mendoza-Arriaga

Rafael Mendoza-Arriaga

University of Texas at Austin - Department of Information, Risk and Operations Management

Peter Carr

New York University Finance and Risk Engineering

Vadim Linetsky

Northwestern University - Department of Industrial Engineering and Management Sciences

Date Written: 2008-12

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 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 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.

Suggested Citation

Mendoza-Arriaga, Rafael and Carr, Peter P. and Linetsky, Vadim, Time-Changed Markov Processes in Unified Credit-Equity Modeling (2008-12). Mathematical Finance, Vol. 20, Issue 4, pp. 527-569, October 2010. Available at SSRN: https://ssrn.com/abstract=1681711 or http://dx.doi.org/10.1111/j.1467-9965.2010.00411.x

Rafael Mendoza-Arriaga (Contact Author)

University of Texas at Austin - Department of Information, Risk and Operations Management ( email )

CBA 5.202
Austin, TX 78712
United States
5126321860 (Phone)

HOME PAGE: http://rafaelmendoza.org

Peter P. Carr

New York University Finance and Risk Engineering ( email )

6 MetroTech Center
Brooklyn, NY 11201
United States
9176217733 (Phone)

HOME PAGE: http://engineering.nyu.edu/people/peter-paul-carr

Vadim Linetsky

Northwestern University - Department of Industrial Engineering and Management Sciences ( email )

Evanston, IL 60208-3119
United States

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