Pricing continuously sampled Asian options with perturbation method

Abstract

This article explores the price of continuously sampled Asian options. For geometric Asian options, we present pricing formulas for both backward-starting and forward-starting cases. For arithmetic Asian options, we demonstrate that the governing partial differential equation (PDE) cannot be transformed into a heat equation with constant coefficients; therefore, these options do not have a closed-form solution of the Black-Scholes type, that is, the solution is not given in terms of the cumulative normal distribution function. We then solve the PDE with a perturbation method and obtain an analytical solution in a series form. Numerical results show that as compared with Zhang's (2001) highly accurate numerical results, the series converges very quickly and gives a good approximate value that is more accurate than any other approximate method in the literature, at least for the options tested in this article. Graphical results determine that the solution converges globally very quickly especially near the origin, which is the area in which most of the traded Asian options fall. © 2003 Wiley Periodicals, Inc.