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This paper investigates the impact of structural breaks on volatility spillovers between Asian stock markets (China, Hong Kong, India, Indonesia, Japan, Korea, Singapore, and Taiwan) and the oil futures market. To this end, we apply the bivariate DCC-GARCH model to weekly spot indices during the period 1998-2015. The results reveal significant volatility transmission for the pairs between the Asian stock and oil futures markets. Moreover, we find a significant variability in the time-varying conditional correlations between the considered markets during both bullish and bearish markets, particularly from early 2007 to the summer of 2008. Using the modified ICSS algorithm, we find several sudden changes in these markets with a common break date centred on September 15, 2008. This date corresponds to the collapse of Lehman Brothers which is considered as our breakpoint to define the global financial crisis. Also, we analyse the optimal portfolio weights and time-varying hedge ratios based on the estimates of the multivariate DCC-GARCH model. The results emphasize the importance of overweighting optimal portfolios between Asian stock and the oil futures markets.