Non-asymptotic convergence rates for the plug-in estimation of risk measures

Let $\rho$ 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 $\mu$ of $X$ is unknown, or the numerical computation of $\rho(\mu)$ is not possible. In both cases, either relying on historical data or using a Monte-Carlo approach, one can resort to an i.i.d.\ sample of $\mu$ to approximate $\rho(\mu)$ by the finite sample estimator $\rho(\mu_N)$ (where $\mu_N$ denotes the empirical measure of $\mu$). In this article we investigate convergence rates of $\rho(\mu_N)$ to $\rho(\mu)$. We provide non-asymptotic 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 neither on the dimension of the underlying assets, nor on the number of options available for trading.