Nonasymptotic Convergence Rates for the Plug-in Estimation of Risk Measures

Abstract

Let ρ be a general law-invariant convex risk measure, for instance, the average value at risk, and let X be a financial loss, that is, a real random variable. In practice, either the true distribution μ of X is unknown, or the numerical computation of [Formula: see text] is not possible. In both cases, either relying on historical data or using a Monte Carlo approach, one can resort to an independent and identically distributed sample of μ to approximate [Formula: see text] by the finite sample estimator [Formula: see text] (μN denotes the empirical measure of μ). In this article, we investigate convergence rates of [Formula: see text] to [Formula: see text]. We provide nonasymptotic convergence rates for both the deviation probability and the expectation of the estimation error. The sharpness of these convergence rates is analyzed. Our framework further allows for hedging, and the convergence rates we obtain depend on neither the dimension of the underlying assets nor the number of options available for trading. Funding: Daniel Bartl is grateful for financial support through the Vienna Science and Technology Fund [Grant MA16-021] and the Austrian Science Fund [Grants ESP-31 and P34743]. Ludovic Tangpi is supported by the National Science Foundation [Grant DMS-2005832] and CAREER award [Grant DMS-2143861].