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

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

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.

Suggested Citation

  • Lyu, Jisang & Park, Eunchae & Kim, Sangkwon & Lee, Wonjin & Lee, Chaeyoung & Yoon, Sungha & Park, Jintae & Kim, Junseok, 2021. "Optimal non-uniform finite difference grids for the Black–Scholes equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 690-704.
    • Handle: RePEc:eee:matcom:v:182:y:2021:i:c:p:690-704
    DOI: 10.1016/j.matcom.2020.12.002
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    8. Min Huang & Guo Luo, 2019. "A simple and efficient numerical method for pricing discretely monitored early-exercise options," Papers 1905.13407, arXiv.org, revised Jun 2019.
    9. Yingzi Chen & Wansheng Wang & Aiguo Xiao, 2019. "An Efficient Algorithm for Options Under Merton’s Jump-Diffusion Model on Nonuniform Grids," Computational Economics, Springer;Society for Computational Economics, vol. 53(4), pages 1565-1591, April.
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    Cited by:

    1. Chaeyoung Lee & Soobin Kwak & Youngjin Hwang & Junseok Kim, 2023. "Accurate and Efficient Finite Difference Method for the Black–Scholes Model with No Far-Field Boundary Conditions," Computational Economics, Springer;Society for Computational Economics, vol. 61(3), pages 1207-1224, March.
    2. Černá, Dana & Fiňková, Kateřina, 2024. "Option pricing under multifactor Black–Scholes model using orthogonal spline wavelets," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 309-340.

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