Pricing continuous Asian options: a comparison of Monte Carlo and Laplace transform inversion methods

Abstract

In this paper, we investigate two numerical methods for pricing Asian options: Laplace transform inversion and Monte Carlo simulation. In attempting to numerically invert the Laplace transform of the Asian call option that has been derived previously in the literature, we point out some of the potential difficulties inherent in this approach. We investigate the effectiveness of two easy-to-implement algorithms, which not only provide a cross-check for accuracy, but also demonstrate superior precision to two alternatives proposed in the literature for the Asian pricing problem. We then extend the theory of Laplace transforms for this problem by deriving the double Laplace transform of the continuous arithmetic Asian option in both its strike and maturity. We contrast the numerical inversion approach with Monte Carlo simulation, one of the most widely used techniques, especially by practitioners, for the valuation of derivative securities. For the Asian option pricing problem, we show that this approach will be effective for cases when numerical inversion is likely to be problematic. We then investigate ways to improve the precision of the simulation estimates through the judicious use of control variates. In particular, in the problem of correcting the discretization bias inherent in simulation when pricing continuous-time contracts, we find that the use of suitably biased control variates can be beneficial. This approach is also compared with the use of Richardson extrapolation.