Generalized control variate methods for pricing Asian options

The conventional control variate method proposed by Kemna and Vorst for evaluating Asian options using the Black-Scholes model utilizes a constant control parameter. We generalize this method, applying it to a stochastic control process through the martingale representation of the conventional control. This generalized control variate has zero variance in the optimal case, whereas the conventional control can only reduce its variance by a finite factor. By means of option price approximations, the generalized control is reduced to a linear martingale control. It is straightforward to extend this martingale control to a non-linear situation such as the American Asian option problem. From the variance analysis of martingales, the performance of control variate methods depends on the distance between the approximate martingale and the optimal martingale. This measure becomes helpful for the design of control variate methods for complex problems such as Asian options using stochastic volatility models. We demonstrate multiple choices of controls and test them with Monte Carlo and quasi-Monte Carlo simulations. Quasi-Monte Carlo methods perform significantly better after adding a control when using the two-step control variate method, while the variance reduction ratios increase to 320 times for randomized quasi-Monte Carlo methods, compared with 60 times for Monte Carlo simulations with a control.