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A Fast, Stable And Accurate Numerical Method For The Black–Scholes Equation Of American Options

Author

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  • MATTHIAS EHRHARDT

    (Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D–10117 Berlin, Germany)

  • RONALD E. MICKENS

    (Department of Physics, Clark Atlanta University, Atlanta, GA 30314, USA)

Abstract

In this work we improve the algorithm of Han and Wu [SIAM J. Numer. Anal. 41 (2003), 2081–2095] for American Options with respect to stability, accuracy and order of computational effort. We derive an exact discrete artificial boundary condition (ABC) for the Crank–Nicolson scheme for solving the Black–Scholes equation for the valuation of American options. To ensure stability and to avoid any numerical reflections we derive the ABC on a purely discrete level.Since the exact discrete ABC includes a convolution with respect to time with a weakly decaying kernel, its numerical evaluation becomes very costly for large-time simulations. As a remedy we construct approximate ABCs with a kernel having the form of a finite sum-of-exponentials, which can be evaluated in a very efficient recursion. We prove a simple stability criteria for the approximated artificial boundary conditions.Finally, we illustrate the efficiency and accuracy of the proposed method on several benchmark examples and compare it to previously obtained discretized ABCs of Mayfield and Han and Wu.

Suggested Citation

  • Matthias Ehrhardt & Ronald E. Mickens, 2008. "A Fast, Stable And Accurate Numerical Method For The Black–Scholes Equation Of American Options," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 11(05), pages 471-501.
    • Handle: RePEc:wsi:ijtafx:v:11:y:2008:i:05:n:s0219024908004890
    DOI: 10.1142/S0219024908004890
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    Cited by:

    1. Yasir Nawaz & Muhammad Shoaib Arif & Wasfi Shatanawi & Amna Nazeer, 2021. "An Explicit Fourth-Order Compact Numerical Scheme for Heat Transfer of Boundary Layer Flow," Energies, MDPI, vol. 14(12), pages 1-17, June.
    2. Company, Rafael & Jódar, Lucas & Pintos, José-Ramón, 2012. "A consistent stable numerical scheme for a nonlinear option pricing model in illiquid markets," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(10), pages 1972-1985.
    3. Alexander Buryak & Ivan Guo, 2014. "New analytic approach to address Put - Call parity violation due to discrete dividends," Papers 1407.7328, arXiv.org.

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