Convergence of a finite element approximation to a degenerate parabolic variational inequality with non-smooth data arising from American option valuation

Abstract

This paper deals with the approximation of solutions to a degenerate variational inequality of parabolic type with a non-smooth final condition arising from American option pricing by the piecewise linear finite element method in space and an implicit time-stepping scheme. We show that the error of the approximation in a weighted Sobolev norm is of order 𝒪(h ^2/3+Δ t ^1/3) under some realistic regularity assumptions on the exact solution, where h and Δ t denote the mesh parameters in space and time, respectively. Numerical examples are presented to confirm our theoretical results.

Keywords:

• variational inequality
• complementarity problem
• finite element analysis
• error bounds
• American option pricing
• Black-Scholes equation
• non-smooth data
• degenerate parabolic partial differential equations

Acknowledgements

The work has been partially supported by the Discovery Grant no. DP0344330 from the Australian Research Council. We also wish to thank the anonymous referees for their constructive comments and suggestions towards the improvement of this paper.