IDEAS home Printed from https://ideas.repec.org/p/hal/journl/hal-03910119.html
   My bibliography  Save this paper

When Capital Is a Funding Source: The Anticipated Backward Stochastic Differential Equations of X-Value Adjustments

Author

Listed:
  • Stéphane Crépey

    (LaMME - Laboratoire de Mathématiques et Modélisation d'Evry - ENSIIE - Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise - UEVE - Université d'Évry-Val-d'Essonne - Université Paris-Saclay - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

  • Wissal Sabbagh

    (LaMME - Laboratoire de Mathématiques et Modélisation d'Evry - ENSIIE - Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise - UEVE - Université d'Évry-Val-d'Essonne - Université Paris-Saclay - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

  • Shiqi Song

    (LaMME - Laboratoire de Mathématiques et Modélisation d'Evry - ENSIIE - Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise - UEVE - Université d'Évry-Val-d'Essonne - Université Paris-Saclay - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

Abstract

X-value adjustments (XVAs) refer to various financial derivative pricing adjustments accounting for counterparty risk and its funding (FVA) and capital (KVA) implications for a bank. In this paper we show that the XVA equations are well-posed, including in the realistic case where capital is deemed fungible as a source of funding for variation margin. This intertwining of capital at risk and the FVA, added to the fact that the KVA is part of capital at risk, leads to a system of backward SDEs (BSDEs) of the McKean type (anticipated BSDEs) for the FVA and the KVA, with coefficients entailing a conditional risk measure of the one-year-ahead increment of the martingale part of the FVA. This is first considered in the case of a hypothetical bank without debt. In the practical case of a defaultable bank, the resulting anticipated BSDEs, which are stopped before the default of the bank, are solved likewise after reduction to a reference market filtration.

Suggested Citation

  • Stéphane Crépey & Wissal Sabbagh & Shiqi Song, 2020. "When Capital Is a Funding Source: The Anticipated Backward Stochastic Differential Equations of X-Value Adjustments," Post-Print hal-03910119, HAL.
    • Handle: RePEc:hal:journl:hal-03910119
    DOI: 10.1137/19m1242781
    Note: View the original document on HAL open archive server: https://hal.science/hal-03910119v1
    as

    Download full text from publisher

    File URL: https://hal.science/hal-03910119v1/document
    Download Restriction: no
    File URL: https://libkey.io/10.1137/19m1242781?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. David Barrera & Stéphane Crépey & Babacar Diallo & Gersende Fort & Emmanuel Gobet & Uladzislau Stazhynski, 2019. "Stochastic Approximation Schemes for Economic Capital and Risk Margin Computations," Post-Print hal-01710394, HAL.
    2. Leif Andersen & Darrell Duffie & Yang Song, 2019. "Funding Value Adjustments," Journal of Finance, American Finance Association, vol. 74(1), pages 145-192, February.
    3. 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.
    4. Andrew Green & Chris Kenyon, 2014. "KVA: Capital Valuation Adjustment," Papers 1405.0515, arXiv.org, revised Oct 2014.
    5. repec:hal:wpaper:hal-01686952 is not listed on IDEAS
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Claudio Albanese & Stéphane Crépey & Rodney Hoskinson & Bouazza Saadeddine, 2021. "XVA analysis from the balance sheet," Quantitative Finance, Taylor & Francis Journals, vol. 21(1), pages 99-123, January.
    2. Lokman Abbas-Turki & St'ephane Cr'epey & Botao Li & Bouazza Saadeddine, 2024. "An Explicit Scheme for Pathwise XVA Computations," Papers 2401.13314, arXiv.org.
    3. Stéphane Crépey, 2022. "Positive XVAs," Post-Print hal-03910135, HAL.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Lokman A. Abbas-Turki & Stéphane Crépey & Babacar Diallo, 2018. "Xva Principles, Nested Monte Carlo Strategies, And Gpu Optimizations," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(06), pages 1-40, September.
    2. Brigo, Damiano & Francischello, Marco & Pallavicini, Andrea, 2019. "Nonlinear valuation under credit, funding, and margins: Existence, uniqueness, invariance, and disentanglement," European Journal of Operational Research, Elsevier, vol. 274(2), pages 788-805.
    3. Maxim Bichuch & Agostino Capponi & Stephan Sturm, 2020. "Robust XVA," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 738-781, July.
    4. Claudio Albanese & Stéphane Crépey & Rodney Hoskinson & Bouazza Saadeddine, 2021. "XVA analysis from the balance sheet," Quantitative Finance, Taylor & Francis Journals, vol. 21(1), pages 99-123, January.
    5. Alessandro Gnoatto & Athena Picarelli & Christoph Reisinger, 2020. "Deep xVA solver -- A neural network based counterparty credit risk management framework," Papers 2005.02633, arXiv.org, revised Dec 2022.
    6. Francesca Biagini & Alessandro Gnoatto & Immacolata Oliva, 2019. "Pricing of counterparty risk and funding with CSA discounting, portfolio effects and initial margin," Working Papers 04/2019, University of Verona, Department of Economics.
    7. Stéphane Crépey, 2022. "Positive XVAs," Post-Print hal-03910135, HAL.
    8. Yannick Armenti & Stéphane Crépey, 2017. "Central Clearing Valuation Adjustment," Working Papers hal-01169169, HAL.
    9. Claudio Albanese & Marc Chataigner & Stéphane Crépey, 2020. "Wealth Transfers, Indifference Pricing, and XVA Compression Schemes," Post-Print hal-03910047, HAL.
    10. Yannick Armenti & St'ephane Cr'epey, 2015. "Central Clearing Valuation Adjustment," Papers 1506.08595, arXiv.org, revised Feb 2017.
    11. Claudio Albanese & Simone Caenazzo & Stéphane Crépey, 2016. "Capital Valuation Adjustment and Funding Valuation Adjustment," Working Papers hal-01285363, HAL.
    12. Lixin Wu & Dawei Zhang, 2020. "xVA: DEFINITION, EVALUATION AND RISK MANAGEMENT," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 23(01), pages 1-24, February.
    13. Damiano Brigo & Marco Francischello & Andrea Pallavicini, 2017. "An indifference approach to the cost of capital constraints: KVA and beyond," Papers 1708.05319, arXiv.org.
    14. Matteo Bissiri & Riccardo Cogo, 2017. "Behavioral Value Adjustments," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(08), pages 1-37, December.
    15. Darrell Duffie & Michael Fleming & Frank Keane & Claire Nelson & Or Shachar & Peter Van Tassel, 2023. "Dealer capacity and US Treasury market functionality," BIS Working Papers 1138, Bank for International Settlements.
    16. Syrstad, Olav & Viswanath-Natraj, Ganesh, 2022. "Price-setting in the foreign exchange swap market: Evidence from order flow," Journal of Financial Economics, Elsevier, vol. 146(1), pages 119-142.
    17. Kathrin Glau & Daniel Kressner & Francesco Statti, 2019. "Low-rank tensor approximation for Chebyshev interpolation in parametric option pricing," Papers 1902.04367, arXiv.org.
    18. Adam Copeland & Darrell Duffie & Yilin (David) Yang, 2025. "Reserves Were Not So Ample After All," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 140(1), pages 239-281.
    19. Maxim Bichuch & Agostino Capponi & Stephan Sturm, 2018. "Robust XVA," Papers 1808.04908, arXiv.org, revised Feb 2020.
    20. Andrew Lesniewski & Anja Richter, 2016. "Managing counterparty credit risk via BSDEs," Papers 1608.03237, arXiv.org, revised Aug 2016.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:hal:journl:hal-03910119. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.