A Finite Element Approach to the Numerical Solutions of Leland's Mode

TL;DR

This paper applies finite element methods to solve Leland's option pricing model with transaction costs, demonstrating stability and accuracy comparable to existing finite difference approaches.

Contribution

It introduces a finite element approach with specific spatial and temporal schemes for Leland's model, including mesh-size stability analysis.

Findings

Finite element solutions align well with finite difference results.

Mesh-size ratios influence stability of the numerical method.

The approach provides a viable alternative to finite difference methods.

Abstract

In this paper, finite element method is applied to Leland's model for numerical simulation of option pricing with transaction costs. Spatial finite element models based on P1 and/or P2 elements are formulated in combination with a Crank-Nicolson-type temporal scheme. The temporal scheme is implemented using the Rannacher approach. Examples with several sets of parameter values are presented and compared with finite difference results in the literature. Spatial-temporal mesh-size ratios are observed for controlling the stability of our method. Our results compare favorably with the finite difference results in the literature for the model.