This article proposes the Range Value at Risk under model uncertainty, denoted by RG-VaR, a risk measure that explicitly incorporates both best-case and worst-case scenarios within sublinear expectation framework. We prove that RG-VaR satisfies monotonicity, translation invariance, positive homogeneity, and comonotonic additivity, and we establish its exact relationship to the G-Value at Risk (G-VaR) and G-Expected Shortfall (G-ES). For G-normally distributed risks, we derive closed-form expressions. A counter example demonstrates that G-ES, in either its best-case or worst-case version, can fail to be subadditive. Nevertheless, we show that G-VaR, G-ES, and RG-VaR are subadditive when risks are independent and G-normal. Sensitivity analysis reveals that significance levels are the primary drivers of the risk measures, while volatility uncertainty affects them asymmetrically: upper volatility perturbations dominate worst-case risk, whereas lower volatility changes mainly influence best-case risk. We further propose hypothesis tests for risk measures under model uncertainty and validate <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$G$</EquationSource> <EquationSource Format="MATHML"><math> <mi>G</mi> </math></EquationSource> </InlineEquation>-ES and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textrm{R}G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>R</mtext> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation>-VaR through numerical simulations and an empirical study on the S&P 500 index. The results show that, under the proposed backtesting procedure, worst-case <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textrm{R}G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>R</mtext> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation>-VaR performs best and remains stable across different significance levels. By incorporating model uncertainty, the proposed framework provides both worst-case and best-case range-based risk assessments, offering conservative and optimistic benchmarks for tail risk in complex market environments.

Range value at risk under model uncertainty

期刊论文
发表日期：

2026年6月19日

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Jie Peng,
Panyu Wu,
Lingqi Meng

摘要

This article proposes the Range Value at Risk under model uncertainty, denoted by RG-VaR, a risk measure that explicitly incorporates both best-case and worst-case scenarios within sublinear expectation framework. We prove that RG-VaR satisfies monotonicity, translation invariance, positive homogeneity, and comonotonic additivity, and we establish its exact relationship to the G-Value at Risk (G-VaR) and G-Expected Shortfall (G-ES). For G-normally distributed risks, we derive closed-form expressions. A counter example demonstrates that G-ES, in either its best-case or worst-case version, can fail to be subadditive. Nevertheless, we show that G-VaR, G-ES, and RG-VaR are subadditive when risks are independent and G-normal. Sensitivity analysis reveals that significance levels are the primary drivers of the risk measures, while volatility uncertainty affects them asymmetrically: upper volatility perturbations dominate worst-case risk, whereas lower volatility changes mainly influence best-case risk. We further propose hypothesis tests for risk measures under model uncertainty and validate $G$ $G$ -ES and $\textrm{R}G$ $R G$ -VaR through numerical simulations and an empirical study on the S&P 500 index. The results show that, under the proposed backtesting procedure, worst-case $\textrm{R}G$ $R G$ -VaR performs best and remains stable across different significance levels. By incorporating model uncertainty, the proposed framework provides both worst-case and best-case range-based risk assessments, offering conservative and optimistic benchmarks for tail risk in complex market environments.