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

In this paper, we demonstrate that policy iteration, introduced in the context of HJB equations in [P. A. Forsyth and G. Labahn, J. Comput. Finance, 11 (2007), pp. 1--44], is an extremely simple generic algorithm for solving linear complementarity problems (LCPs) resulting from the finite difference and finite element approximation of American options. We show that, in general, $O(N)$ is an upper and a lower bound on the number of iterations needed to solve a discrete LCP of size $N$. If embedded in a class of standard discretizations with $M$ time steps, the overall complexity of American option pricing is indeed only $O(N(M+N))$, and, therefore, for $M\sim N$, it is 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.

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