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

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 ρ(µ) 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 ρ(µ) by the finite sample estimator ρ(µ_N) (µ_N denotes the empirical measure of µ). In this article, we investigate convergence rates of ρ(µ_N) to ρ(µ). 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.