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Mathematical models and numerical methods for a capital valuation adjustment (KVA) problem

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  • Trevisani, Davide
  • López-Salas, José Germán
  • Vázquez, Carlos
  • García-Rodríguez, José Antonio

Abstract

In this work we rigorously establish mathematical models to obtain the capital valuation adjustment (KVA) as part of the total valuation adjustments (XVAs). For this purpose, we use a semi-replication strategy based on market theory. We formulate single-factor models in terms of expectations and PDEs. For PDEs formulation, we rigorously obtain the existence and uniqueness of the solution, as well as some regularity and qualitative properties of the solution. Moreover, appropriate numerical methods are proposed for solving the corresponding PDEs. Finally, some examples show the numerical results for call and put European options and the corresponding XVA that includes the KVA.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:apmaco:v:488:y:2025:i:c:s0096300324005666
    DOI: 10.1016/j.amc.2024.129105
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    References listed on IDEAS

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    1. 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.
    2. García Muñoz, Luis Manuel & Palomar Burdeus, Juan Esteban & de Lope Contreras, Fernando, 2016. "The recursive nature of KVA: KVA mitigation from KVA," MPRA Paper 70927, University Library of Munich, Germany.
    3. Wang, Haijin & Shu, Chi-Wang & Zhang, Qiang, 2016. "Stability analysis and error estimates of local discontinuous Galerkin methods with implicit–explicit time-marching for nonlinear convection–diffusion problems," Applied Mathematics and Computation, Elsevier, vol. 272(P2), pages 237-258.
    4. Simonella, Roberta & Vázquez, Carlos, 2023. "XVA in a multi-currency setting with stochastic foreign exchange rates," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 207(C), pages 59-79.
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