From a practical perspective, it is crucial to hedge the crude oil price risk in periods of dramatic price change. In this study, we directly investigate the performance of crude oil hedge portfolios in the five periods in which the largest oil price shocks in history occurred. We use stochastic volatility (SV), GARCH, and the diagonal BEKK model to estimate the minimum variance hedge ratio of hedge portfolios. Our empirical results provide evidence that hedging strategies based on the SV model are able to outperform the GARCH and BEKK models in terms of variance reduction. Our results are also consistently valid for various hedge horizons. Interestingly, although it is important to estimate variance and covariance accurately when constructing minimum variance portfolios, we find that reducing the mean squared and mean absolute errors does not guarantee superior hedge performance. (C) 2019 Elsevier B.V. All rights reserved.