A Positivity-Preserving Improved Nonstandard Finite Difference Method to Solve the Black-Scholes Equation
Abstract
:1. Introduction
- K is the exercise price;
- T is the maturity;
- is the interest rate;
- is the reference volatility.
2. Finite Difference Approaches
2.1. The -Method
2.2. The Mixed Method
2.3. The Richardson Method
2.4. The Du Fort and Frankel Method
2.5. The MADE Method
3. Nonstandard Finite-Difference Strategy
- The function in the denominator of the approximation of the discrete derivative must be expressed in terms of a function of the step size, provided that (16) holds. This rule allows the introduction of a complex analytic function of h in the denominator with the condition thatExamples of functions that satisfy this condition are [40]:
4. Scheme Construction
5. Analysis of the New Scheme
6. Numerical Results with MMADE
7. Conclusions and Discussion
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Mehdizadeh Khalsaraei, M.; Shokri, A.; Ramos, H.; Mohammadnia, Z.; Khakzad, P. A Positivity-Preserving Improved Nonstandard Finite Difference Method to Solve the Black-Scholes Equation. Mathematics 2022, 10, 1846. https://doi.org/10.3390/math10111846
Mehdizadeh Khalsaraei M, Shokri A, Ramos H, Mohammadnia Z, Khakzad P. A Positivity-Preserving Improved Nonstandard Finite Difference Method to Solve the Black-Scholes Equation. Mathematics. 2022; 10(11):1846. https://doi.org/10.3390/math10111846
Chicago/Turabian StyleMehdizadeh Khalsaraei, Mohammad, Ali Shokri, Higinio Ramos, Zahra Mohammadnia, and Pari Khakzad. 2022. "A Positivity-Preserving Improved Nonstandard Finite Difference Method to Solve the Black-Scholes Equation" Mathematics 10, no. 11: 1846. https://doi.org/10.3390/math10111846
APA StyleMehdizadeh Khalsaraei, M., Shokri, A., Ramos, H., Mohammadnia, Z., & Khakzad, P. (2022). A Positivity-Preserving Improved Nonstandard Finite Difference Method to Solve the Black-Scholes Equation. Mathematics, 10(11), 1846. https://doi.org/10.3390/math10111846