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A Positivity-Preserving Improved Nonstandard Finite Difference Method to Solve the Black-Scholes Equation

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  • Mohammad Mehdizadeh Khalsaraei

    (Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh 83111-55181, Iran)

  • Ali Shokri

    (Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh 83111-55181, Iran)

  • Higinio Ramos

    (Scientific Computing Group, Universidad de Salamanca, Plaza de la Merced, 37008 Salamanca, Spain)

  • Zahra Mohammadnia

    (Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh 83111-55181, Iran)

  • Pari Khakzad

    (Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh 83111-55181, Iran)

Abstract

In this paper, we evaluate and discuss different numerical methods to solve the Black–Scholes equation, including the θ -method, the mixed method, the Richardson method, the Du Fort and Frankel method, and the MADE (modified alternating directional explicit) method. These methods produce numerical drawbacks such as spurious oscillations and negative values in the solution when the volatility is much smaller than the interest rate. The MADE method sacrifices accuracy to obtain stability for the numerical solution of the Black–Scholes equation. In the present work, we improve the MADE scheme by using non-standard finite difference discretization techniques in which we use a non-local approximation for the reaction term (we call it the MMADE method). We will discuss the sufficient conditions to be positive of the new scheme. Also, we show that the proposed method is free of spurious oscillations even in the presence of discontinuous initial conditions. To demonstrate how efficient the new scheme is, some numerical experiments are performed at the end.

Suggested Citation

  • Mohammad Mehdizadeh Khalsaraei & Ali Shokri & Higinio Ramos & Zahra Mohammadnia & Pari Khakzad, 2022. "A Positivity-Preserving Improved Nonstandard Finite Difference Method to Solve the Black-Scholes Equation," Mathematics, MDPI, vol. 10(11), pages 1-16, May.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:11:p:1846-:d:825882
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    References listed on IDEAS

    as
    1. Mohammad Mehdizadeh Khalsaraei & Ali Shokri & Zahra Mohammadnia & Hamid Mohammad Sedighi & Efthymios G. Tsionas, 2021. "Qualitatively Stable Nonstandard Finite Difference Scheme for Numerical Solution of the Nonlinear Black–Scholes Equation," Journal of Mathematics, Hindawi, vol. 2021, pages 1-12, May.
    2. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    3. Darae Jeong & Minhyun Yoo & Junseok Kim, 2018. "Finite Difference Method for the Black–Scholes Equation Without Boundary Conditions," Computational Economics, Springer;Society for Computational Economics, vol. 51(4), pages 961-972, April.
    4. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    5. M. Mehdizadeh Khalsaraei & R. Shokri Jahandizi, 2016. "A family of positivity preserving schemes for numerical solution of Black–Scholes equation," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 3(04), pages 1-8, December.
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