Efficient Nested Simulation for Estimating the Variance of a Conditional Expectation

In a two-level nested simulation, an outer level of simulation samples scenarios, while the inner level uses simulation to estimate a conditional expectation given the scenario. Applications include financial risk management, assessing the effects of simulation input uncertainty, and computing the expected value of gathering more information in decision theory. We show that an ANOVA-like estimator of the variance of the conditional expectation is unbiased under mild conditions, and we discuss the optimal number of inner-level samples to minimize this estimator's variance given a fixed computational budget. We show that as the computational budget increases, the optimal number of inner-level samples remains bounded. This finding contrasts with previous work on two-level simulation problems in which the inner- and outer-level sample sizes must both grow without bound for the estimation error to approach zero. The finding implies that the variance of a conditional expectation can be estimated to arbitrarily high precision by a simulation experiment with a fixed inner-level computational effort per scenario, which we call a one-and-a-half-level simulation. Because the optimal number of inner-level samples is often quite small, a one-and-a-half-level simulation can avoid the heavy computational burden typically associated with two-level simulation.