Weighted Monte Carlo: Large Sample Properties and Applications
We analyze Monte Carlo estimators defined as weighted averages of independent replications, in which the weights are chosen to constrain the weighted averages of auxiliary control variables. Because the number of constraints in typically much smaller than the number of replications, there may be many feasible solutions; we consider weights minimizing a convex objective subject to the constraints. These are maximally uniform feasible weights. Estimators of this form arise (sometimes implicitly) in several settings, including at least two in finance: calibrating a model to market data (as in Avellaneda et al.) and solving the optimal stopping problem embedded in pricing American options. We distinguish two cases (unbiased vs. biased) depending on whether the control averages are constrained to their population means or to some other values.We show that in the unbiased case all convex objective functions within a large class produce estimators that ate very close in a strong sense. In contrast, in the biased case the choice of objective function does matter. This is joint work with Bin Yu.