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PDE models and numerical methods for total value adjustment in European and American options with counterparty risk

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  • Arregui, Iñigo
  • Salvador, Beatriz
  • Vázquez, Carlos

Abstract

Since the last financial crisis, a relevant effort in quantitative finance research concerns the consideration of counterparty risk in financial contracts, specially in the pricing of derivatives. As a consequence of this new ingredient, new models, mathematical tools and numerical methods are required. In the present paper, we mainly consider the problem formulation in terms of partial differential equations (PDEs) models to price the total credit value adjustment (XVA) to be added to the price of the derivative without counterparty risk. Thus, in the case of European options and forward contracts different linear and nonlinear PDEs arise. In the present paper we propose suitable boundary conditions and original numerical methods to solve these PDEs problems. Moreover, for the first time in the literature, we consider XVA associated to American options by the introduction of complementarity problems associated to PDEs, as well as numerical methods to be added in order to solve them. Finally, numerical examples are presented to illustrate the behavior of the models and numerical method to recover the expected qualitative and quantitative properties of the XVA adjustments in different cases. Also, the first order convergence of the numerical method is illustrated when applied to particular cases in which the analytical expression for the XVA is available.

Suggested Citation

  • Arregui, Iñigo & Salvador, Beatriz & Vázquez, Carlos, 2017. "PDE models and numerical methods for total value adjustment in European and American options with counterparty risk," Applied Mathematics and Computation, Elsevier, vol. 308(C), pages 31-53.
    • Handle: RePEc:eee:apmaco:v:308:y:2017:i:c:p:31-53
    DOI: 10.1016/j.amc.2017.03.008
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    Cited by:

    1. Salvador, Beatriz & Oosterlee, Cornelis W., 2021. "Total value adjustment for a stochastic volatility model. A comparison with the Black–Scholes model," Applied Mathematics and Computation, Elsevier, vol. 391(C).
    2. Trevisani, Davide & López-Salas, José Germán & Vázquez, Carlos & García-Rodríguez, José Antonio, 2025. "Mathematical models and numerical methods for a capital valuation adjustment (KVA) problem," Applied Mathematics and Computation, Elsevier, vol. 488(C).
    3. van der Zwaard, Thomas & Grzelak, Lech A. & Oosterlee, Cornelis W., 2021. "A computational approach to hedging Credit Valuation Adjustment in a jump-diffusion setting," Applied Mathematics and Computation, Elsevier, vol. 391(C).
    4. Arregui, Iñigo & Simonella, Roberta & Vázquez, Carlos, 2022. "Total value adjustment for European options in a multi‐currency setting," Applied Mathematics and Computation, Elsevier, vol. 413(C).
    5. Ludovic Goudenège & Andrea Molent & Antonino Zanette, 2025. "Computing XVA for American basket derivatives by machine learning techniques," Computational Management Science, Springer, vol. 22(2), pages 1-33, December.
    6. Beatriz Salvador & Cornelis W. Oosterlee & Remco van der Meer, 2020. "Financial Option Valuation by Unsupervised Learning with Artificial Neural Networks," Mathematics, MDPI, vol. 9(1), pages 1-20, December.

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