Risk estimation via weighted regression

In this paper we propose a method based on weighted regression for the estimation of risk in nested Monte Carlo simulation. The mean squared error (MSE) of a standard nested simulation converges at the rate k^-2/3, where k is the computational budget. Similar to the regression method proposed in Broadie, Du, and Moallemi (2011b), the MSE of the proposed weighted regression method converges at the rate k^-1 until reaching an asymptotic bias level, which depends on the size of the regression error. However, the weighted approach further reduces MSE by emphasizing scenarios that are more important to the calculation of the risk measure. We find a globally optimal weighting strategy for general risk measures in an idealized setting. For applications, we propose and test a practically implementable two-pass method, where the first pass uses an unweighted regression and the second pass uses weights based on the first pass.