Nested Simulation for Value-at-Risk with Precision Tolerance

This paper investigates nested simulation for estimating Value-at-Risk (VaR), a widely adopted risk measure in practice. We formulate a mathematical framework tailored to a practical setting where a risk estimate is measured up to a certain precision level, and any risk estimate within certain range specified by the precision tolerance level is deemed acceptable. Within this framework, we propose a rounded estimator of VaR that explicitly accounts for the pre-specified precision tolerance level, and demonstrate that its error can decay exponentially fast as the sampling budget increases. An important implication of our theoretical result is that a finite inner sample size may suffice for nested simulation in our setting, leading to a budget allocation rule that deviates substantially from the standard nested simulation procedure. Numerical examples confirm the theoretical findings, showcasing the consistent performance of the proposed rounded estimator.