Variance reduction for risk measures with importance sampling in nested simulation

Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are two standard risk measures that are widely adopted in both financial and insurance industries. Simulation-based approaches including nested simulation and least-squares Monte Carlo are effective strategies to yield reliable estimates of these risk measures, but there remain open questions on how importance sampling can be incorporated to improve estimation efficiency. In this paper, we extend the scope of importance sampling from simple Monte Carlo to nested simulation settings and its adaptations for American-type options; we also establish the asymptotic consistency of importance sampling. Numerical results consistent with our theoretical analysis are provided to verify its effectiveness.