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

Daniel Bartl () and Ludovic Tangpi ()
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Daniel Bartl: Department of Mathematics, Vienna University, 1010 Vienna, Austria
Ludovic Tangpi: Operations Research and Financial Engineering, Bendheim Center for Finance, Princeton University, Princeton, New Jersey 08544

Mathematics of Operations Research, 2023, vol. 48, issue 4, 2129-2155

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 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.

Keywords: Primary: 91B82; 91B30; 91B16; decision analysis; risk; statistics; estimation; decision analysis; approximations (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:48:y:2023:i:4:p:2129-2155

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