In this paper, we investigate the multifractal characteristics of the cross-correlations between price and volume in China's first crude oil futures market (INE), and compared the INE market with Brent and WTI market. Multifractal detrended cross-correlation analysis (MF-DCCA) was applied to comprehensively research the multifractal characteristics of the three different crude oil futures markets based on the data of the past five years. The results show that the cross-correlations between price change and volume change have significant multifractal characteristics in three markets, and the degree of multifractality is stronger in the WTI market for the full sample. Due to the INE market appeared late, the sample size of INE is not consistent with that of the other two markets. In order to solve this influence, we further compare the multifractal characteristics of the three crude oil futures markets in the same period (March 26, 2018 to November 20, 2020) and compares the results with those of the full sample period. The empirical results show that the degree of multifractality in both Brent and WTI markets is increased. To further verify whether this result is due to the emergence of the INE crude oil market, the entire sample of the Brent and WTI markets are divided into two periods according to the launch time of INE market. The result shows that the degree of multifractality and the risk of these two markets are all stronger in the last two years. Thus, there are reasons to believe that the emergence of INE market has a certain impact on the traditional Brent and WTI market, and a greater impact on Brent market. However, MFDCCA only consider the overall situation and cannot describe the multifractal cross-correlation between the upward and downward trends of the market. Thus, it probably won't accurately reflect the nonlinear cross-correlation. Furthermore, we study the characteristics of the asymmetric multifractal cross-correlations in different trends (upwards or downwards) by using the multifractal asymmetric detrended cross-correlation analysis (MF-ADCCA) method. Empirical results indicate that the cross-correlation between price and volume in three markets shows obvious asymmetry. Specifically, when the price changes series is on an upward trend, or when the volume changes series is on a downward trend, the degree of asymmetry in three markets is stronger, and the risk situation of the market is more complex. Therefore, for a portfolio of assets with different markets, investors should fully observe and understand different market trends and make reasonable investment based on their own risk preference.