Optimal non-uniform finite difference grids for the Black–Scholes equations

Jisang Lyu; Eunchae Park; Sangkwon Kim; Wonjin Lee; Chaeyoung Lee; Sungha Yoon; Jintae Park; Junseok Kim

doi:10.1016/j.matcom.2020.12.002

Optimal non-uniform finite difference grids for the Black–Scholes equations

Jisang Lyu, Eunchae Park, Sangkwon Kim, Wonjin Lee, Chaeyoung Lee, Sungha Yoon, Jintae Park, Junseok Kim

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

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.

Original language	English
Pages (from-to)	690-704
Number of pages	15
Journal	Mathematics and Computers in Simulation
Volume	182
DOIs	https://doi.org/10.1016/j.matcom.2020.12.002
Publication status	Published - 2021 Apr

Bibliographical note

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)

Keywords

Black–Scholes equations
Equity-linked securities
Finite difference method
Optimal non-uniform grid

ASJC Scopus subject areas

Theoretical Computer Science
General Computer Science
Numerical Analysis
Modelling and Simulation
Applied Mathematics

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10.1016/j.matcom.2020.12.002

Cite this

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title = "Optimal non-uniform finite difference grids for the Black–Scholes equations",

abstract = "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.",

keywords = "Black–Scholes equations, Equity-linked securities, Finite difference method, Optimal non-uniform grid",

author = "Jisang Lyu and Eunchae Park and Sangkwon Kim and Wonjin Lee and Chaeyoung Lee and Sungha Yoon and Jintae Park and Junseok Kim",

note = "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: {\textcopyright} 2020 International Association for Mathematics and Computers in Simulation (IMACS)",

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doi = "10.1016/j.matcom.2020.12.002",

language = "English",

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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

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ER -

Optimal non-uniform finite difference grids for the Black–Scholes equations

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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