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.

中文翻译：

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Black–Scholes 方程的最优非均匀有限差分网格

摘要 在本文中，我们提出了 Black–Scholes (BS) 方程的最优非均匀有限差分网格。有限差分法主要用于使用均匀网格，并且在 BS 方程下为多个选项定价需要相当长的时间。维度越高，问题就越严重。在我们提出的方法中，我们通过基于包含该点的网格上的数值解重复去除具有最小误差的网格点，从均匀网格中获得最佳非均匀网格。我们用一维、二维和三维 BS 方程进行了几次数值测试。对现金或无现金和股票挂钩证券 (ELS) 期权进行了计算测试。