China’s carbon emission allowance prices forecasting and option designing in uncertain environment

Carbon emissions trading is pivotal for advancing China’s low-carbon goals. As the primary tradable asset in the carbon market, carbon emission allowances inevitably experience price fluctuations. However, numerous empirical studies show that the frequency of real-world data is highly unstable, which results in the failure of probabilistic modeling. Therefore, this paper aims to model the dynamics of carbon emission allowance prices in China using four mainstream uncertain differential equations. The optimal model is chosen through rolling window cross-validation using the criterion of minimizing average testing errors. Parameters of the optimal model are determined by moment estimation based on residuals, and the model’s effectiveness is also assessed through uncertain two-sided hypothesis testing. Additionally, we forecast carbon emission allowance prices and their 95% confidence intervals for the next 14 business days. To manage trading risks, we propose a customized carbon option contract for pricing European carbon options and conduct sensitivity analysis on key parameters. Finally, we present a paradox of stochastic differential equations for modeling carbon emission allowance prices.