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Abstract:

We analyze a class of Monte Carlo estimators that are stochastically weighted averages of independent replications. The weights are chosen to constrain the weighted averages of auxiliary control variables. Because the number of constraints is typically much smaller than the number of replications, there may be many feasible solutions. We select weights that minimize a separable 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 work of Avellaneda et al.) and calculating conditional expectations in order to price 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. In the first case, the weights are intended to reduce variance whereas in the second case their purpose is to correct errors in a simulated model. We show that in the unbiased case 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 function determines the limit to which the estimator converges. 1

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