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BSDEs of Counterparty Risk

Author

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  • Stéphane Crépey

    (LaMME - Laboratoire de Mathématiques et Modélisation d'Evry - INRA - Institut National de la Recherche Agronomique - UEVE - Université d'Évry-Val-d'Essonne - CNRS - Centre National de la Recherche Scientifique)

  • Shiqi Song

    (LaMME - Laboratoire de Mathématiques et Modélisation d'Evry - INRA - Institut National de la Recherche Agronomique - UEVE - Université d'Évry-Val-d'Essonne - CNRS - Centre National de la Recherche Scientifique)

Abstract

We study a BSDE with random terminal time that appears in the modeling of coun-terparty risk in finance. We proceed by reduction of the original BSDE into a simpler BSDE posed with respect to a smaller filtration and a changed probability measure. We relax the basic immersion conditions of the classical reduced-form modeling approach in credit risk by modeling the default time as an invariant time, i.e. a time such that local martingales with respect to a reduced filtration and a possibly changed probability measure, once stopped right before that time, stay local martingales with respect to the original model filtration and probability measure. Using an Azéma supermartingale characterization of invariant times, we establish the equivalence between the original and the reduced BSDE.

Suggested Citation

  • Stéphane Crépey & Shiqi Song, 2014. "BSDEs of Counterparty Risk," Working Papers hal-01088941, HAL.
    • Handle: RePEc:hal:wpaper:hal-01088941
    Note: View the original document on HAL open archive server: https://hal.science/hal-01088941
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    References listed on IDEAS

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    1. P. Collin-Dufresne & R. Goldstein & J. Hugonnier, 2004. "A General Formula for Valuing Defaultable Securities," Econometrica, Econometric Society, vol. 72(5), pages 1377-1407, September.
    2. Damiano Brigo & Andrea Pallavicini, 2014. "Nonlinear consistent valuation of CCP cleared or CSA bilateral trades with initial margins under credit, funding and wrong-way risks," Journal of Financial Engineering (JFE), World Scientific Publishing Co. Pte. Ltd., vol. 1(01), pages 1-60.
    3. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    4. Damiano Brigo & Andrea Pallavicini, 2014. "CCP Cleared or Bilateral CSA Trades with Initial/Variation Margins under credit, funding and wrong-way risks: A Unified Valuation Approach," Papers 1401.3994, arXiv.org.
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    Cited by:

    1. Maxim Bichuch & Agostino Capponi & Stephan Sturm, 2016. "Arbitrage-Free XVA," Papers 1608.02690, arXiv.org.
    2. Maxim Bichuch & Agostino Capponi & Stephan Sturm, 2018. "Robust XVA," Papers 1808.04908, arXiv.org, revised Feb 2020.
    3. Maxim Bichuch & Agostino Capponi & Stephan Sturm, 2015. "Arbitrage-Free Pricing of XVA - Part II: PDE Representation and Numerical Analysis," Papers 1502.06106, arXiv.org, revised Aug 2016.
    4. Shiqi Song, 2014. "Local martingale deflators for asset processes stopped at a default time $S^\tau$ or right before $S^{\tau-}$," Papers 1405.4474, arXiv.org, revised Jul 2016.

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