Modified B-Spline Collocation Approach for Pricing American Style Asian Options

Abstract

In this work, a stable numerical scheme based on modified bi-cubic B-spline collocation method is developed for the valuation of Asian options. The option prices are governed with Black–Scholes equation. We use the \(\theta \)-method for temporal discretization and the modified bi-cubic B-spline collocation approach in spatial direction. This approach provides an unconditionally stable scheme which has been proved by using Von-Neumann stability analysis. It should be noted that the case of American Asian option cannot be reduced to one dimension for the case of the fixed strike. So then, to solve the American style problem, we would have to keep track of both the stock price and its running average, which leads back to the two-dimensional formulation of the problem. The efficiency of the proposed scheme is tested by two test examples. High accuracy, simple implementation and low complexity for high-dimensional problems are the advantages of our approach. Results shown by this method are found to be in good agreement with some available benchmarks given in the literature.