Partial differential equations sometimes have critical points where the solution or some of its derivatives are discontinuous. The simplest example is a discontinuity in the initial condition. It is well known that those decrease the accuracy of finite difference methods. A common remedy is to stretch the grid, such that many more grid points are present near the critical points, and fewer where the solution is deemed smooth. An alternative solution is to insert points such that the discontinuities fall in the middle of two grid points. This paper compares the accuracy of both approaches in the context of the pricing of financial derivative contracts in the Black-Scholes model and proposes a new fast and simple stretching function.

翻译：部分差异方程式有时有关键点,其解决办法或其某些衍生物不连续。最简单的例子就是初始条件的不连续性。众所周知,这些方法降低了有限差异方法的准确性。一个共同的补救办法是拉长网格,使更多的网格点靠近关键点,而认为解决办法平稳的网格点较少。另一个解决办法是插入点,使不连续性在两个网格点中间下降。本文件比较了两种方法在黑雪球模型中金融衍生品合同定价方面的准确性,并提出了一个新的快速和简单拉伸功能。