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

Mohammad Mehdizadeh Khalsaraei, Ali Shokri, Higinio Ramos, Zahra Mohammadnia and Pari Khakzad
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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

Mathematics, 2022, vol. 10, issue 11, 1-16

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.

Keywords: Black–Scholes equation; MADE scheme; nonstandard finite differences; positivity-preserving scheme (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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