Large Sample Properties of Weighted Monte Carlo Estimators

A general approach to improving simulation accuracy uses information about auxiliary control variables with known expected values to improve the estimation of unknown quantities. We analyze weighted Monte Carlo estimators that implement this idea by applying weights to independent replications. The weights are chosen to constrain the weighted averages of the control variables. We distinguish two cases (unbiased and biased), depending on whether the weighted averages of the controls are constrained to equal their expected values or some other values. In both cases, the number of constraints is usually smaller than the number of replications, so there may be many feasible weights. We select maximally uniform weights by minimizing a separable convex function of the weights subject to the control variable constraints. Estimators of this form arise (sometimes implicitly) in several settings, including at least two in finance: calibrating a model to market data (as in the work of Avellaneda et al. 2001) and calculating conditional expectations to price American options. We analyze properties of these estimators as the number of replications increases. We show that in the unbiased case, weighted Monte Carlo reduces asymptotic variance, and that all convex objective functions within a large class produce estimators that are very close to each other in a strong sense. In contrast, in the biased case the choice of objective function does matter. We show explicitly how the choice of objective determines the limit to which the estimator converges.