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

• 成果类型: Article
• 署名作者: Bartl, Daniel; Tangpi, Ludovic
• 署名单位: University of Vienna; Princeton University
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2022.1333
• 发表日期: 2023
• 页码: 2129-2155
• 关键词: weak continuity; invariant; functionals; Robustness

摘要

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 independent and identically distributed sample of mu to approximate rho(mu) by the finite sample estimator rho(mu N) (mu N denotes the empirical measure of mu). In this article, we investigate convergence rates of rho(mu N) to rho(mu). 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.