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This paper applies the extended MFDFA method to analyze the volatility characteristics, multifractal features and asymmetry in EU carbon emission permits trading market(EUA) and the main Chinese carbon emission permits market including HBEA, SZA and GDEA. We divided the EUA market into three stages according to the phased construction plan of European commission. The empirical analysis shows that the fluctuation in the second and third stage of EUA market presents a long-term memory feature. However, there is obvious anti-persistent characteristics in the volatility of three main Chinese carbon emission permits trading markets and the first stage in EUA market. The multifractal degree in three main Chinese carbon emission permits trading markets is less than that in the first stage of EUA market, but far greater than the second and third stages. We also verified that the HBEA market has the weakest volatility and the least multifractal degree among three Chinese markets, and whose validity is second to the third stage of EUA market. In addition, the fluctuation of the EU and Chinese carbon emission permits trading markets are asymmetric. The EUA market has stronger reaction to good news, while Chinese carbon emission permits trading market still displays the leverage effect which verifies the volatility is more sensitive to the bad news in the second and third stages. Therefore, the results powerfully demonstrate that the Chinese carbon emission permits trading market is less mature and highly dynamic, which is similar to the level of first stage of EUA market. The volatility in Chinese markets is strong, and exists much noise in the system. So, the efficiency in Chinese markets is weaker than EU. Chinese markets are still far from maturity and efficiency. • The market price volatility does not possess the significant long-term memory characteristics, it shows up the anti-persistent features. • With the development of the EUA market, the multifractal characteristics become weaker and weaker, and the reaction of the market price fluctuation to information become weaker as well. • The market asymmetry reflects the reaction of the price volatility to the positive and negative information. [ABSTRACT FROM AUTHOR]