Black-Box CoVaR and Its Gradient Estimation

CoVaR is an important measure of financial systemic risk. Because of the complex structures and dependence of portfolios, estimating the CoVaR of financial systems as well as its gradient information could be very challenging, especially in a black-box setting, where only noisy estimates are available. For CoVaR and its gradient estimation, we propose a stochastic approximation algorithm that is applicable in various complicated cases, including black-box and streaming data scenarios. We establish the strong consistency and rate of convergence of the algorithm under appropriate technical conditions. In particular, the convergence rates of our proposed CoVaR and its gradient estimators are of orders [Formula: see text] and [Formula: see text], respectively, where n is the number of algorithm iterations. Simulation experiments are also performed to support our theoretical findings. History: Accepted by Bruno Tuffin, Area Editor for Simulation. Funding: The work of H. Cao was supported by the National Natural Science Foundation of China (NSFC) [Grant 723B2006]. The work of J.-Q. Hu was supported by the NSFC [Grants 72350710219, 72033003, 72342006, and 72293565]. The work of J. Hu was supported by the National Science Foundation [Grant CMMI-2027527]. 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.2024.0833 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0833 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .