Inserting or Stretching Points in Finite Difference Discretizations

TL;DR

This paper compares grid stretching and point insertion techniques for finite difference methods in PDEs with discontinuities, proposing a new stretching function to improve accuracy in financial derivative pricing.

Contribution

It introduces a new fast and simple grid stretching function and evaluates the effectiveness of point insertion versus stretching in PDE discretizations.

Findings

Point insertion improves accuracy near discontinuities.

The new stretching function enhances computational efficiency.

Both methods reduce errors in financial PDE solutions.

Abstract

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.