Global political and economic events usually cause the structural changes of crude oil price behavior. If this problem concerning structural changes is not considered, the testing result would produce estimation bias which leads to incorrect explanation and inference. This paper investigates the long-run and short-run dynamic relationships between crude oil price and Crude oil volatility index (OVX) and between the price and stock market volatility index (VIX). The empirical result finds six structural change points by which further divide into seven sub-periods that relevant events occurred per sub-period. This study also finds that, because of the structural changes there exists the ARDL-ECM co-integrating relationship between oil price and VIX. If the seven sub-period data are inspected one by one, this relationships hold for the 2nd, 3rd, 4th, 6th and 7th sub-periods, while this relationships between oil price and OVX hold for the 2nd, 4th and 6th sub-periods. Furthermore, for the entire observation period, the two-way Granger causality holds for the price, but the price Granger causes VIX only. If the seven sub-period data are inspected again, VIX Granger causes the price only for the 1st and 2nd sub-period, but OVX Granger causes the price only for the 5th and 6th sub-periods. Finally, for the entire observation period, the price and OVX are affected by their 15 period lags each other, but VIX is affected by 8 period lag of the price. If the seven sub-period data are examined, VIX is affected by 2 period lag and 3 period lag of the price for the 1st and 2nd sub-period, respectively, while the price is affected by 2 period lag of OVX for both the 5th and 6th sub-periods. This finding implies that, the fear gauges provide the investors with the more market efficiency during the sub-periods than during the whole sampling period. Therefore, this paper suggests that, not only do the investors in the petroleum market take care of the impacts of crucial global political and economic events on the crude oil price, but they also consider the short-run dynamic relationships between the price and its volatility.