Standard Galerkin formulation with high order Lagrange finite elements for option markets pricing

Abstract

A semi-discrete Galerkin formulation (combined with high order Lagrangian finite elements) is employed for the approximate solution of the parabolic partial differential equation (widely known as Black–Scholes equation), which governs the evolution of the non-arbitrage (equilibrium) value of an option contract written on a singe underlying security. The Crank–Nicolson method is employed for the discretization in the time domain. Extensive numerical experimentation with American call and put stock options (where the stock may pay discrete cash dividends) and comparison with existing analytical, as well as, with approximate solutions, confirms the efficiency and accuracy of the proposed formulation. Moreover, it is verified that the p-extension (increasing the order of the polynomial interpolants, on a relatively coarse finite element mesh) is much more efficient (in terms of both accuracy and CPU time) than the h-extension (reducing the element sizes, with fixed low polynomial order). The work may be extended to more complicated option pricing models (e.g., multi-asset options or options on assets with stochastic volatilities).