TY - JOUR
T1 - Optimal non-uniform finite difference grids for the Black–Scholes equations
AU - Lyu, Jisang
AU - Park, Eunchae
AU - Kim, Sangkwon
AU - Lee, Wonjin
AU - Lee, Chaeyoung
AU - Yoon, Sungha
AU - Park, Jintae
AU - Kim, Junseok
N1 - Funding Information:
This work was supported by the Brain Korea 21 FOUR (BK 21 FOUR) from the Ministry of Education of Korea . The corresponding author (J.S. Kim) was supported by Korea University Research Fund . The authors appreciate the reviewers for their constructive comments, which have improved the quality of this paper.
Publisher Copyright:
© 2020 International Association for Mathematics and Computers in Simulation (IMACS)
PY - 2021/4
Y1 - 2021/4
N2 - In this article, we present optimal non-uniform finite difference grids for the Black–Scholes (BS) equation. The finite difference method is mainly used using a uniform mesh, and it takes considerable time to price several options under the BS equation. The higher the dimension is, the worse the problem becomes. In our proposed method, we obtain an optimal non-uniform grid from a uniform grid by repeatedly removing a grid point having a minimum error based on the numerical solution on the grid including that point. We perform several numerical tests with one-, two- and three-dimensional BS equations. Computational tests are conducted for both cash-or-nothing and equity-linked security (ELS) options. The optimal non-uniform grid is especially useful in the three-dimensional case because the option prices can be efficiently computed with a small number of grid points.
AB - In this article, we present optimal non-uniform finite difference grids for the Black–Scholes (BS) equation. The finite difference method is mainly used using a uniform mesh, and it takes considerable time to price several options under the BS equation. The higher the dimension is, the worse the problem becomes. In our proposed method, we obtain an optimal non-uniform grid from a uniform grid by repeatedly removing a grid point having a minimum error based on the numerical solution on the grid including that point. We perform several numerical tests with one-, two- and three-dimensional BS equations. Computational tests are conducted for both cash-or-nothing and equity-linked security (ELS) options. The optimal non-uniform grid is especially useful in the three-dimensional case because the option prices can be efficiently computed with a small number of grid points.
KW - Black–Scholes equations
KW - Equity-linked securities
KW - Finite difference method
KW - Optimal non-uniform grid
UR - http://www.scopus.com/inward/record.url?scp=85097454161&partnerID=8YFLogxK
U2 - 10.1016/j.matcom.2020.12.002
DO - 10.1016/j.matcom.2020.12.002
M3 - Article
AN - SCOPUS:85097454161
SN - 0378-4754
VL - 182
SP - 690
EP - 704
JO - Mathematics and Computers in Simulation
JF - Mathematics and Computers in Simulation
ER -