Non-asymptotic rates for the estimation of risk measures
Daniel Bartl and
Papers from arXiv.org
Consider the problem of computing the riskiness $\rho(F(S))$ of a financial position $F$ written on the underlying $S$ with respect to a general law invariant risk measure $\rho$; for instance, $\rho$ can be the average value at risk. In practice the true distribution of $S$ is typically unknown and one needs to resort to historical data for the computation. In this article we investigate rates of convergence of $\rho(F(S_N))$ to $\rho(F(S))$, where $S_N$ is distributed as the empirical measure of $S$ with $N$ observations. We provide (sharp) non-asymptotic rates for both the deviation probability and the expectation of the estimation error. Our framework further allows for hedging, and the convergence rates we obtain depend neither on the dimension of the underlying stocks nor on the number of options available for trading.
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