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Arbitrage-Free Pricing of XVA - Part II: PDE Representation and Numerical Analysis

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  • Maxim Bichuch
  • Agostino Capponi
  • Stephan Sturm

Abstract

We study the semilinear partial differential equation (PDE) associated with the non-linear BSDE characterizing buyer's and seller's XVA in a framework that allows for asymmetries in funding, repo and collateral rates, as well as for early contract termination due to counterparty credit risk. We show the existence of a unique classical solution to the PDE by first proving the existence and uniqueness of a viscosity solution and then its regularity. We use the uniqueness result to conduct a thorough numerical study illustrating how funding costs, repo rates, and counterparty credit risk contribute to determine the total valuation adjustment.

Suggested Citation

  • 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.
    • Handle: RePEc:arx:papers:1502.06106
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    References listed on IDEAS

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    1. Jouini Elyes & Kallal Hedi, 1995. "Martingales and Arbitrage in Securities Markets with Transaction Costs," Journal of Economic Theory, Elsevier, vol. 66(1), pages 178-197, June.
    2. Tomasz R. Bielecki & Marek Rutkowski, 2014. "Valuation and Hedging of Contracts with Funding Costs and Collateralization," Papers 1405.4079, arXiv.org, revised Dec 2014.
    3. Stéphane Crépey & Shiqi Song, 2014. "BSDEs of Counterparty Risk," Working Papers hal-01088941, HAL.
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    5. Elyégs Jouini & Hédi Kallal, 1995. "Arbitrage In Securities Markets With Short‐Sales Constraints," Mathematical Finance, Wiley Blackwell, vol. 5(3), pages 197-232, July.
    6. Andrea Pallavicini & Daniele Perini & Damiano Brigo, 2012. "Funding, Collateral and Hedging: uncovering the mechanics and the subtleties of funding valuation adjustments," Papers 1210.3811, arXiv.org, revised Dec 2012.
    7. Maxim Bichuch & Agostino Capponi & Stephan Sturm, 2015. "Arbitrage-Free Pricing of XVA -- Part I: Framework and Explicit Examples," Papers 1501.05893, arXiv.org, revised Aug 2016.
    8. repec:dau:papers:123456789/5647 is not listed on IDEAS
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