Estimation VaR and CVaR by Gradient and Langevin MC Methods

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.