<p>This research proposes a novel Extreme-BEKK-GARCH model to track the volatility transmission between the oil and U.S. stock market at different bivariate quantile levels using daily return data from August 1, 2014 to July 31, 2024. Unlike the traditional BEKK-GARCH model, our model allows the volatility parameters to vary across different bivariate quantile levels, enabling a more comprehensive capture of volatility transmission under various return scenarios. The empirical results indicate that when ignoring the quantile dependence (i.e., using the BEKK-GARCH model), we observe a large and significant volatility transmission from oil to stock, while negligible and insignificant reverse transmission. However, when considering the quantile dependence (i.e., applying the Extreme-BEKK-GARCH model), we detect significant volatility transmission from oil to stock but also find evidence of significant transmission from stock to oil at certain bivariate quantile levels, particularly when both markets are experiencing strong gains or declines. Our findings provide new insights into the existing literature and portfolios.</p>