Generalized Control Variate Methods for Pricing Asian Options

The conventional control variate method proposed by Kemna and Vorst (1990) to evaluate Asian options under the Black-Scholes model can be interpreted as a particular selection of linear martingale controls. We generalize the constant control parameter into a control process to gain more reduction on variance. By means of an option price approximation, we construct a martingale control variate method, which outperforms the conventional control variate method. It is straightforward to extend such linear control to a nonlinear 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 option under stochastic volatility models. We demonstrate multiple choices of controls and test them under MC/QMC (Monte Carlo/ Quasi Monte Carlo)simulations. QMC methods work significantly well after adding a control, the variance reduction ratios increase to 260 times for randomized QMC compared with 60 times for MC simulations with a control.