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Monotone methods in counterparty risk models with nonlinear Black–Scholes-type equations

Author

Listed:
  • Bénédicte Alziary

    (IMT - Institut de Mathématiques de Toulouse UMR5219 - UT Capitole - Université Toulouse Capitole - UT - Université de Toulouse - INSA Toulouse - Institut National des Sciences Appliquées - Toulouse - INSA - Institut National des Sciences Appliquées - UT - Université de Toulouse - UT2J - Université Toulouse - Jean Jaurès - UT - Université de Toulouse - UT3 - Université Toulouse III - Paul Sabatier - UT - Université de Toulouse - CNRS - Centre National de la Recherche Scientifique)

  • Peter Takáč

    (Universität Rostock)

Abstract

A nonlinear Black–Scholes-type equation is studied within counterparty risk models . The classical hypothesis on the uniform Lipschitz-continuity of the nonlinear reaction function allows for an equivalent transformation of the semilinear Black–Scholes equation into a standard parabolic problem with a monotone nonlinear reaction function and an inhomogeneous linear diffusion equation. This setting allows us to construct a scheme of monotone, increasing or decreasing, iterations that converge monotonically to the true solution. As typically any numerical solution of this problem uses most computational power for computing an approximate solution to the inhomogeneous linear diffusion equation, we discuss also this question and suggest several solution methods, including those based on Monte Carlo and finite differences/elements.

Suggested Citation

  • Bénédicte Alziary & Peter Takáč, 2022. "Monotone methods in counterparty risk models with nonlinear Black–Scholes-type equations," Post-Print hal-04888252, HAL.
    • Handle: RePEc:hal:journl:hal-04888252
    DOI: 10.1007/s40324-022-00306-0
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