The price behavior characteristics of China and Europe carbon emission trading market based on the perspective of time scaling and expected returns.

China has the world's largest carbon market in terms of greenhouse gas emissions, but its system needs to be improved and enhanced. In comparison, the European carbon market stands as the most mature and well-developed carbon market globally. Carbon trading prices, serving as a barometer for the carbon market, are significantly influenced by investor behavior. Therefore, it is necessary to analyze the characteristics of carbon trading prices in both China and Europe, considering the impact of investor trading intervals and psychological expected returns. This study utilizes the Zipf method to characterize the dynamic behavior of carbon trading prices between China and Europe, conducting a comparative analysis. The results show distinctive asymmetry in the behavior of carbon trading prices in both markets. In the Chinese market, when τ < 277, the absolute deviation da(τ, ε) value gradually changes but consistently indicates a bullish trend. However, when τ ≥ 277, the da(τ, ε) value surges rapidly, reflecting a pronounced bullish sentiment among investors toward carbon trading prices in China. In the European market, within the sample period, regardless of variations in τ and ε, the da(τ, ε) value shows a linear upward trend, indicating a significant overall bullishness in prices. This suggests a higher probability of long-term bullishness in carbon trading prices. Investors' investment time scale (τ) and expected returns (ε) both influence the behavior of carbon trading prices in both China and Europe. Generally, a longer τ implies a higher probability of bullishness. As for ε, higher values lead to more extreme judgments on price movements, resulting in greater distortion in carbon trading prices. Short-term investors (τ<1 month) anticipate extreme fluctuations, exhibiting random behavior when ε < 0.15 and converging rapidly to extreme values of 1 or 0 when ε ≥ 0.15. Long-term investors (τ>quarter) are less biased, expressing a bullish outlook on both Chinese and European carbon prices. With increasing ε, the probability of bullishness either increases or decreases rapidly until reaching the saturation point. Once saturated, there is no further distortion in carbon price behavior. Furthermore, the Chinese carbon market displays a positive trend in carbon trading prices and a higher probability of long-term bullishness. For the European market, lower expected returns contribute to considerable carbon trading price fluctuations, exacerbating risk and uncertainty. The results of this study contribute to understanding the diverse trading behaviors in Chinese and European carbon markets and provide guidance for avoiding extreme volatility in carbon trading prices.