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 journalArticlepeer-review

2 Citations (Scopus)


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 languageEnglish
Pages (from-to)690-704
Number of pages15
JournalMathematics and Computers in Simulation
Publication statusPublished - 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)


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

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)
  • Numerical Analysis
  • Modelling and Simulation
  • Applied Mathematics


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