Metamodeling techniques have been applied to approximate portfolio loss as a function of financial risk factors, thus producing point estimates of various measures of portfolio risk based on Monte Carlo samples. Rather than point estimates, this paper focuses on the construction of confidence intervals (CIs) for a widely used risk measure, the so-called conditional value-at-risk (CVaR), when the least-squares method (LSM) is employed as a metamodel in the point estimation. To do so, we first develop lower and upper bounds of CVaR and construct CIs for these bounds. Then, the lower end of the CI for the lower bound and the upper end of the CI for the upper bound together form a CI of CVaR with justifiable statistical guarantee, which accounts for both the metamodel error and the noises of Monte Carlo samples. The proposed CI procedure reuses the samples simulated for LSM point estimation, thus requiring no additional simulation budget. We demonstrate via numerical examples that the proposed procedure may lead to a CI with the desired coverage probability and a much smaller width than that of an existing CI in the literature. History: Accepted by Bruno Tuffin, Area Editor for Simulation. Funding: This research was supported by the National Natural Science Foundation of China (NNSFC) [Grants 72101260 and 72471232], the Research Grants Council of Hong Kong (RGC-HK) [General Research Fund Project 11508620], InnoHK Initiative, the Government of the HKSAR, and Laboratory for AI-Powered Financial Technologies, and NNSFC/RGC-HK Joint Research Scheme [Project N_CityU 105/21]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0394 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0394 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

The Greeks are derivatives of option price with respect to market parameters and play an important role in financial risk management. Among various simulation methods for estimating the Greeks, the pathwise method typically has a low variance. However, when the option payoff is discontinuous, the pathwise method is not applicable and the Greek involves a conditional expectation taken over a hypersurface that is a probability-zero set. In this paper, we propose an importance sampling (IS) method to estimate this conditional expectation. More specifically, IS is applied in a way that all simulated observations fall into a set constructed by thickening the hypersurface. Allowing the thickness of the set to go to zero then leads to a new representation of the Greek as an ordinary expectation, thus leading to an unbiased estimator. The resulting estimator makes use of the pathwise derivatives and can be viewed as an extension of the pathwise method to cases with discontinuous payoffs. Numerical results show that the proposed IS method works well.