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

Abstract

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.

Appendices

Appendix

Why do stochastic differential equations fail to adequately model CEA prices?

Let $i=1, 2,\cdots , 122$ denote the sequence from December 21, 2023 to June 26, 2024, and let $s_1,s_2, \cdots ,s_{122}$ represent the CEA prices, as shown in Table 1. Assume that the CEA prices obey a stochastic differential equation

$$\begin{aligned} \textrm{d}S_t =\mu S_{t}\textrm{d}{t} + \sigma S_{t}\textrm{d}W_{t}. \end{aligned}$$

(20)

Here, $S_t$ represents the CEA price, $\mu$ and $\sigma$ are the unknown parameters to be determined, and $W_t$ is a Wiener process. For each index i with $i=2,3,\cdots ,122$, we solve the updated stochastic differential equation

$$\begin{aligned} \textrm{d}S_t =\mu S_{t}\textrm{d}{t} + \sigma S_{t}\textrm{d}W_{t}, \quad S_{i-1} =s_{i-1}, \end{aligned}$$

and we find that $S_i$ is a lognormal random variable which follows $\ln {S_i}\sim N(\ln {s_{i-1}}+\mu -\frac{1}{2}\sigma ^2,\sigma ^2)$. Therefore, we can obtain the probability distribution of $S_i$ as below

$$\begin{aligned} \Phi _i(x)= \frac{1}{\sqrt{2\pi }\sigma } \int _{0}^{x}\frac{1}{y}\exp \left( -\frac{(\ln {y}-(\ln {s_{i-1}+\mu -\frac{1}{2}\sigma ^2 }))^2 }{2\sigma ^2}\right) \textrm{d}y \end{aligned}$$

in which $\Phi _{i}(S_i)$ is a uniform random variable ${\mathcal {U}}(0,1)$. Subsequently, we replace $S_i$ with the actual observed data $s_i$, and write

$$\begin{aligned} \varepsilon _{i}(\mu ,\sigma )=\Phi _i(s_i). \end{aligned}$$

(21)

Here, $\varepsilon _{i}(\mu ,\sigma )$ is referred to as the i-th residual of the equation (20). Next, we solve the following system of equations

$$\begin{aligned} \frac{1}{122-1}\sum _{i=2}^{122}\varepsilon _i^p(\mu ,\sigma )=\frac{1}{1+p},\quad \text {for }p=1,2 \end{aligned}$$

to obtain the estimated parameters, where the left-hand side represents the p-th sample moment of the residuals and the right-hand side represents the corresponding p-th population moment of the distribution ${\mathcal {U}}(0,1).$

Fig. 6

Residuals plot of equation (22)

Therefore, we can obtain the moment estimation $(\mu ^*,\sigma ^*)=(0.0016,0.0171),$ and the estimated stochastic differential equation is as follows

$$\begin{aligned} \textrm{d}S_t =0.0016S_{t}\textrm{d}{t} + 0.0171S_{t}\textrm{d}W_{t}. \end{aligned}$$

(22)

After substituting $\mu ^*$ and $\sigma ^*$ into equation (21), we obtain a series of 121 residuals $\varepsilon _2, \varepsilon _3 \cdots ,\varepsilon _{122}$ for the stochastic model (22), as illustrated in Fig. 6. When we divide these residuals into two parts, i.e.,

$$\begin{aligned} (\varepsilon _{2},\varepsilon _{3},\cdots ,\varepsilon _{69})\ and\ (\varepsilon _{70},\varepsilon _{71},\cdots ,\varepsilon _{122}), \end{aligned}$$

the Ansari-Bradley test indicates that the two parts from the residuals mentioned above do not come from the same population, with a p-value of 0.0417 obtained using the function “ansaribradley” in MATLAB. Therefore, the residuals $\varepsilon _{2},\varepsilon _{3},\cdots ,\varepsilon _{122}$ are neither white noise in the framework of probability theory, nor do they follow a uniform probability distribution ${\mathcal {U}}(0,1)$. Consequently, it is inappropriate to apply a stochastic differential equation to model the CEA spot prices.