Estimation VaR and CVaR by Gradient and Langevin MC Methods

Abstract

Value at Risk (VaR) is a key risk management metric used to quantify potential losses in financial 
portfolios over a specified time horizon under normal market conditions. Traditional methods for 
estimating VaR, such as historical simulation and parametric approaches, can be limited by their 
reliance on strong assumptions about the underlying return distributions and their inability to accurately 
capture the behavior of extreme market events. In this paper, we introduce the Gradient Monte Carlo 
(GMC) and Langevin MC methods as robust alternatives for VaR estimation. By leveraging the 
principles of MC simulation combined with gradient-based optimization techniques, GMC efficiently 
approximates the distribution of portfolio returns, enabling accurate and reliable estimation of VaR 
while LMC provides a rigorous theoretical framework for the application of GMC and LMC in risk 
assessment and demonstrate their advantages over conventional methods through extensive empirical 
validation. Our results indicate that GMC and LMC not only enhance computational efficiency but also 
improve the accuracy of VaR and CVaR estimates, particularly in tail regions where conventional 
approaches may fail.