# Hazard processes and martingale hazard processses

Coculescu, D; Nikeghbali, Ashkan (2012). Hazard processes and martingale hazard processses. 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 t is an honest time, providing an explicit expression for the difference. Eventually we provide a solution to another open problem: we show that if t is an arbitrary random (default) time such that its Azema's supermartingale is continuous, then t avoids stopping times.

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 t is an honest time, providing an explicit expression for the difference. Eventually we provide a solution to another open problem: we show that if t is an arbitrary random (default) time such that its Azema's supermartingale is continuous, then t avoids stopping times.

## Citations

4 citations in Web of Science®
5 citations in Scopus®

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English July 2012 24 Jan 2013 14:31 05 Apr 2016 16:19 Wiley-Blackwell 0960-1627 https://doi.org/10.1111/j.1467-9965.2010.00471.x
Permanent URL: https://doi.org/10.5167/uzh-70655