A Multilevel Approach to Control Variates

Control variates are a popular technique for reducing the variance of Monte Carlo estimates. Recent literature has enlarged the set of potentially useful control variates. Still, finding an control variate that efficiently reduces estimation error can be a challenging task for which the theoretical literature provides little guidance. In this note we show by theory and example how to construct an efficient control variate when the underlying simulation is based on a discrete approximation that converges to a limiting model. To illustrate the technique, we price Asian put options in the Black-Scholes-Merton framework and show the control variate we prescribe is competitive with other commonly used control variates and dominates them when small discretization errors are required. Finally, we explore the applicability of these ideas under randomized quasi-Monte Carlo (low-discrepancy) sampling. The results from our example suggest that our method provides even greater error reduction in this setting than in that of simple random sampling.