Hazard Processes and Martingale Hazard Processes

Coculescu, Delia; Nikeghbali, Ashkan (2012). Hazard Processes and Martingale Hazard Processes. Mathematical Finance, 22(3):519-537.

Abstract

In this paper, we build a bridge between different reduced-form approaches to pricing defaultable claims. In particular, we show how the well-known formulas by Duffie, Schroder, and Skiadas and by Elliott, Jeanblanc, and Yor are related. Moreover, in the spirit of Collin Dufresne, Hugonnier, and Goldstein, we propose a simple pricing formula under an equivalent change of measure.
Two processes will play a central role: the hazard process and the martingale hazard process attached to a default time. The crucial step is to understand the difference between them, which has been an open question in the literature so far. We show that pseudo-stopping times appear as the most general class of random times for which these two processes are equal. We also show that these two processes always differ when $\tau$ is an honest time, providing an explicit expression for the difference. Eventually we provide a solution to another open problem: we show that if $\tau$ is an arbitrary random (default) time such that its Azéma's supermartingale is continuous, then $\tau$ avoids stopping times.

Abstract

In this paper, we build a bridge between different reduced-form approaches to pricing defaultable claims. In particular, we show how the well-known formulas by Duffie, Schroder, and Skiadas and by Elliott, Jeanblanc, and Yor are related. Moreover, in the spirit of Collin Dufresne, Hugonnier, and Goldstein, we propose a simple pricing formula under an equivalent change of measure.
Two processes will play a central role: the hazard process and the martingale hazard process attached to a default time. The crucial step is to understand the difference between them, which has been an open question in the literature so far. We show that pseudo-stopping times appear as the most general class of random times for which these two processes are equal. We also show that these two processes always differ when $\tau$ is an honest time, providing an explicit expression for the difference. Eventually we provide a solution to another open problem: we show that if $\tau$ is an arbitrary random (default) time such that its Azéma's supermartingale is continuous, then $\tau$ avoids stopping times.

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