In this paper, the authors investigate two numerical methods for pricing Asian options: Laplace transform inversion and Monte Carlo simulation. In attempting to numerically invert the Laplace transform of an Asian call option that has been derived previously in the literature, some of the potential difficulties inherent in this approach are indicated. The effectiveness of two easy-to-implement algorithms are investigated. These algorithms provide a cross-check for accuracy and also demonstrate superior precision to two alternatives proposed in the literature for the Asian pricing problem. The theory of Laplace transforms for this problem is extended by deriving the double Laplace transform of the continuous arithmetic Asian option in both its strike and maturity. The authors then contrast this 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, it is shown that this approach will be effective for cases when numerical inversion is likely to be problematic. Ways to improve the precision of the simulation estimates through the judicious use of control variates are considered. In particular, in the problem of correcting the discretization bias inherent in simulation when pricing continuous-time contracts, it is found that the use of suitably biased control variates can be beneficial. This approach is also compared with the use of Richardson extrapolation.