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What happens after a default: The conditional density approach

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  • El Karoui, Nicole
  • Jeanblanc, Monique
  • Jiao, Ying

Abstract

We present a general model for default times, making precise the role of the intensity process, and showing that this process allows for a knowledge of the conditional distribution of the default only "before the default". This lack of information is crucial while working in a multi-default setting. In a single default case, the knowledge of the intensity process does not allow us to compute the price of defaultable claims, except in the case where the immersion property is satisfied. We propose in this paper a density approach for default times. The density process will give a full characterization of the links between the default time and the reference filtration, in particular "after the default time". We also investigate the description of martingales in the full filtration in terms of martingales in the reference filtration, and the impact of Girsanov transformation on the density and intensity processes, and on the immersion property.

Suggested Citation

  • El Karoui, Nicole & Jeanblanc, Monique & Jiao, Ying, 2010. "What happens after a default: The conditional density approach," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1011-1032, July.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:7:p:1011-1032
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    References listed on IDEAS

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    1. R. J. Elliott & M. Jeanblanc & M. Yor, 2000. "On Models of Default Risk," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 179-195, April.
    2. D. C. Brody & L. P. Hughston, 2002. "Entropy and information in the interest rate term structure," Quantitative Finance, Taylor & Francis Journals, vol. 2(1), pages 70-80.
    3. Axel Grorud & Monique Pontier, 2001. "Asymmetrical Information And Incomplete Markets," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 4(02), pages 285-302.
    4. Jeanblanc, Monique & Le Cam, Yann, 2009. "Progressive enlargement of filtrations with initial times," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2523-2543, August.
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