On the Use of Policy Iteration as an Easy Way of Pricing American Options

TL;DR

This paper shows that policy iteration is a simple, efficient, and robust method for pricing American options, with complexity comparable to European options, and provides insights into its numerical properties.

Contribution

It demonstrates that policy iteration is an effective, generic algorithm for solving the linear complementarity problems in American option pricing, with complexity analysis and robustness discussion.

Findings

Policy iteration solves American option LCPs efficiently.

Complexity is O(N(M+N)), similar to European options.

Policy iteration is robust and numerically stable.

Abstract

In this paper, we demonstrate that policy iteration, introduced in the context of HJB equations in [Forsyth & Labahn, 2007], is an extremely simple generic algorithm for solving linear complementarity problems resulting from the finite difference and finite element approximation of American options. We show that, in general, O(N) is an upper and lower bound on the number of iterations needed to solve a discrete LCP of size N. If embedded in a class of standard discretisations with M time steps, the overall complexity of American option pricing is indeed only O(N(M+N)), and, therefore, for M N, identical to the pricing of European options, which is O(MN). We also discuss the numerical properties and robustness with respect to model parameters in relation to penalty and projected relaxation methods.