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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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7 citations in Scopus®
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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:03 Faculty of Economics > Department of Banking and Finance
Dewey Decimal Classification:330 Economics
Language:English
Date:2012
Deposited On:22 May 2017 14:25
Last Modified:23 Nov 2017 06:09
Publisher:Wiley-Blackwell Publishing, Inc.
ISSN:0960-1627
Publisher DOI:https://doi.org/10.1111/j.1467-9965.2010.00471.x
Other Identification Number:merlin-id:14832

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