Sequential Nested Simulation for Estimating Expected Shortfall

Expected shortfall (ES) is a widely used tail risk measure in the financial industry. Estimating the ES of a financial portfolio usually requires nested simulation, which is computationally burdensome. In a standard nested simulation procedure, one first simulates a set of plausible evolution of underlying risk factors, or the scenarios. Then, conditional on each outer scenarios, inner simulations are run to evaluate the financial positions in that scenario. In this work, we propose a sequential nested simulation procedure that dynamically allocates a fixed simulation budget to accurately estimate the expected shortfall. The goal is to gradually concentrate the simulation budget on the tail scenarios with the largest losses, as these scenarios are most relevant in ES estimation. Our numerical experiments show that, with the same simulation budget, the proposed procedure significantly improves the estimation accuracy of ES compared to a standard nested simulation procedure.