Volatility Modeling and Tail Risk Estimation of Financial Assets: Evidence from Gold, Oil, Bitcoin, and Stocks for Selected Markets

Abstract

As investment portfolios become increasingly diversified and financial asset risks grow more complex, accurately forecasting the risk of multiple asset classes through mathematical modeling and identifying their heterogeneity has emerged as a critical topic in financial research. This study examines the volatility and tail risk of gold, crude oil, Bitcoin, and selected stock markets. Methodologically, we propose two improved Value at Risk (VaR) forecasting models that combine the autoregressive (AR) model, Exponential Generalized Autoregressive Conditional Heteroskedasticity (EGARCH) model, Extreme Value Theory (EVT), skewed heavy-tailed distributions, and a rolling window estimation approach. The model’s performance is evaluated using the Kupiec test and the Christoffersen test, both of which indicate that traditional VaR models have become inadequate under current complex risk conditions. The proposed models demonstrate superior accuracy in predicting VaR and are applicable to a wide range of financial assets. Empirical results reveal that Bitcoin and the Chinese stock market exhibit no leverage effect, indicating distinct risk profiles. Among the assets analyzed, Bitcoin and crude oil are associated with the highest levels of risk, gold with the lowest, and stock markets occupy an intermediate position. The findings offer practical implications for asset allocation and policy design.