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

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 dr...

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Bibliographic Details
Published in:Mathematics (Basel) 2022-06, Vol.10 (11), p.1846
Main Authors: Mehdizadeh Khalsaraei, Mohammad, Shokri, Ali, Ramos, Higinio, Mohammadnia, Zahra, Khakzad, Pari
Format: Article
Language:English
Subjects:
Accuracy
Approximation
Black-Scholes equation
Boundary conditions
Finite difference method
Food science
Initial conditions
MADE scheme
Mathematical analysis
Methods
nonstandard finite differences
Numerical analysis
Numerical methods
Oscillations
Partial differential equations
positivity-preserving scheme
Production methods
Put & call options
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math10111846