Since the Industrial Revolution, the significant increase in carbon dioxide emissions has caused the earth’s ecological environment to deteriorate sharply. Furthermore, it has been associated with more frequent natural disasters globally, such as the rare low-temperature freezing in Texas; extreme heat and out of control fires in North America; and the drought followed by a desert locust plague in Africa. To mitigate climate change, rapid measures are needed to reduce greenhouse gas emissions (Zhou, 2021). To this end, more than 100 countries signed the “Kyoto Protocol” in Japan in 1997 (Weng and Xu, 2018), which put forward specific carbon dioxide emission reduction requirements for developed countries, and established international emission trading, joint implementation, and clean development mechanisms to promote the achievement of emission reduction targets. In 2005, with the official enforcement of the “Kyoto Protocol”, carbon emission rights became publicly traded commodities, and carbon trading markets were established and have developed rapidly since then (Wang et al., 2019). Carbon emission trading helps to distribute the pressure of energy saving and emission reduction to enterprises, and serves as an important means to achieve carbon emission reduction targets by using market-oriented mechanism. The European Union-Emissions Trading System (EU-ETS) is by far the world’s largest carbon market and a leader in carbon trading.

At the December 2020 Climate Ambition Summit, China pledged to strive for “carbon peak” by 2030 and “carbon neutrality” by 2060 (Ruan and Tu, 2021). Subsequently, at the Central Economic Work Conference in March 2021, it was listed as one of the eight major tasks to be done. In this context, improving the carbon emission trading market may be the fastest and most effective way to reduce emissions and meet international climate commitments. The outline of China’s 12th Five-Year Plan in 2011 clearly proposed to “establish and improve the statistical accounting system for greenhouse gas emissions and gradually establish the carbon emission trading market”. This was the first time that the Chinese government proposed the establishment of a domestic carbon trading market in an official document. Compared the international carbon markets, China’s carbon emission trading started late and it remains relatively behind in its development. Clearly, China’s carbon trading market can and needs to be further improved.

Due to the scarcity of carbon emission quota, carbon emission rights have the value attribute and become a new type of asset (Zhang, 2011). This makes it possible to undertake carbon trading and establish a carbon market. Diversified trading products and flexible implementation methods can greatly enhance a market’s vitality, and ensure the sustainable and stable development of the carbon trading market. As an important trading product in the financial market, options play an important role in risk management, asset allocation, and price discovery. Option contract holders can choose to execute or not execute the option contracts according to the changes in market price to minimize the risk loss. Clearly, for a better carbon trading market, carbon option products are indispensable. Carbon options refer to the right of both parties to sell or purchase the asset at a specified price within a fixed time with carbon emission right as the underlying asset. Enterprises with heavy carbon emissions can obtain more choices through carbon option trading, thus reducing operating costs and increasing profits. Meanwhile, investors can use carbon options to make more reasonable hedging decisions and reduce the risk of carbon trading. Because there is no dividend in a carbon asset, carbon option is studied as a European option in this study.

Scholars have devoted substantial attention to asset return distribution and European option pricing model theory. In 1973, the Black-Scholes (B-S) Model, the first complete option pricing model created by Black and Scholes was publicly used; since then, it has been widely used in European option pricing. However, the B-S model cannot capture phenomena such as skew, thick tails, and volatility aggregation in financial time series. Considering these problems, Benoit (1963) was the first to propose that stock returns follow a stable distribution. The property of this distribution was first deduced by Lévy (Schinckus, 2013); the distribution exhibits sharp peaks and thick tails at the mean value (Geng and Fei, 2016).

However, in reality, the tail of the return on assets is thicker than the normal distribution and thinner than the stable distribution. Therefore, the Tempered Stable (TS) distribution was introduced. Rosiński (2007), Küchler and Tappe (2013) and Sztonyk (2010) studied the properties of this distribution. Barndorff-Nielsen and Shephard (2001a, b) and Carr et al. (2003) used the process of TS in financial mathematics to simulate the stochastic volatility of financial markets. The TS distribution is an infinitely divisible distribution family, which combines a stable distribution and Gaussian trend (Rosiński, 2007), and approximates a Brownian motion in a long time (e.g., CGMY model) (Carr et al., 2002).

Thereafter, the TS distribution gradually expanded into a kind of distribution which aroused the interest of various researchers. For instance, the TS process is known as the truncated Lévy flight model in physics (Schinckus, 2013), which has been widely used in financial modeling (Bianchi et al., 2010). The bilateral gamma distribution is a special case of the TS distribution. Küchler and Tappe (2008a; 2009) studied the application of the bilateral gamma process in financial mathematics and explored the properties of the bilateral gamma distribution through financial data. The authors found that the combined model was closer to the truth than other models. Lei et al. (2021) found that the bilateral gamma distribution could better fit the distribution of return on assets and proposed a new risk measurement method. Bellini and Mercuri (2014) proposed the conditional bilateral gamma model for option pricing, demonstrating that the model results are more accurate.

Most carbon option scholars focus on the role of carbon option and the significance of introducing it. However, to our knowledge, few studies explore the pricing of carbon options. Hong (2016) introduced the role of carbon option in improving the carbon market mechanism, and pointed out the necessity and significance of studying carbon option. Xu et al. (2016) showed that carbon option trading can hedge future risks, reduce spot price level and price volatility, and simultaneously help achieve emission reduction targets. Yu et al. (2020) studied the carbon option pricing model based on investors’ attitude towards risk; however, their study was limited to only considering investors’ preference for carbon option pricing. Liu and Huang (2021) constructed the fractional Brownian motion model optimized by the generalized autoregressive conditional heteroscedastic (GARCH) model for carbon option pricing. While the prediction result of this model has the smallest error versus the actual value when compared with the benchmark model, the prediction error of asset volatility in the early stage is large because the model is greatly affected by time, which cannot reflect the real volatility. Liu et al. (2022) selected two companies in the power and steel industries to improve the B-S model for carbon option pricing by considering multiple industry-related influencing factors; yet, their model suffers from certain limitations.

To overcome these challenges, this study introduces the bilateral gamma distribution into the carbon option pricing model. Focusing on the carbon trading products in the EU market, the European Union Allowance (EUA) futures is taken as the underlying asset of carbon options in the study. Descriptive statistical analysis of the return series of the EUA futures daily closing price reveals that the series is more consistent with the relevant characteristics of the gamma distribution. Therefore, the bilateral gamma distribution is used to fit the carbon quota yield series, and then the volatility of the carbon price is calculated. Finally, the pricing formula is used to price the carbon emission option. In addition, this study uses the bilateral gamma distribution to get the joint distribution of the continuous rise and fall rate of return, in which the scale parameter is a variable that changes with the trading volume. This is used to make a probability deduction for the future price and obtain the conditional probability of continuous rise to fall.

The bilateral gamma distribution can be expressed as the convolution of two gamma distributions, which basically has the same analytical tractability as the gamma distribution and is more flexible for the modeling of nearly symmetrical data sets. The concept of gamma distribution is given by Definition 2.1 (Lei et al., 2018).

Definition 2.1.

If the density function of the random variable X is:

We denote by X\sim \text{Γ}(\alpha ,\beta ) as a gamma distribution, where α and β are the shape and scale parameters, \alpha >0, \beta >0, $\Gamma(\alpha)=\int_{0}^{\infty} x^{\alpha-1} \mathrm{e}^{-x} \mathrm{~d} x(\alpha>0)$ is called the Gamma function.

Then, the bilateral gamma distribution can be expressed as:

where {\alpha }^{+}>0,{\beta }^{+}>0,{\alpha }^{-}>0,{\beta }^{-}>0. \ast is the convolution operation.

Definition 2.2 (Küchler and Tappe, 2008a).

The bilateral gamma distribution is defined as the difference of two independent gamma distributions:

Z\sim \text{Γ}\left({\alpha }^{+},{\beta }^{+};{\alpha }^{-},{\beta }^{-}\right), if Z=X-Y with X\sim \text{Γ}\left({\alpha }^{+},{\beta }^{+}\right), Y\sim \text{Γ}\left({\alpha }^{-},{\beta }^{-}\right).

According to Equation (1) and Definition 2.2, the characteristic function of the bilateral gamma distribution can be obtained as follows:

Proposition 1(Eliazar and Klafter, 2003):

Let vbe a measure defined on {ℝ}^d\backslash \left\{0\right\}. If then v is called a Lévy measure. Among them, v is the measure; {ℝ}^d\backslash \left\{0\right\} is the set of d real numbers without 0; y is a random variable; \left|·\right| is the absolute value operation; |y|²^1 is the minimum operation of |y|² and 1. The formula less than infinity indicates that the formula has a specific value, the same below.

Lévy-Khintchine theorem (Eliazar and Klafter, 2003):

Let \mu \in M({ℝ}^d) be infinitely divisible measure. If there are vectors b\in {ℝ}^d, symmetric positive definite matrices {\rm A}\in {ℝ}^{d\times d}, and Lévy measures v on {ℝ}^d\backslash \left\{0\right\}, then:

This is also called the Lévy-Khintchine formula. {{\rm I}}_{\left\{x>0\right\}}and are indicative functions; i is an imaginary number; w is the parameter; x,y are random variables, \langle ·,·\rangleis a vector.

According to the characteristic function, the bilateral gamma distribution is stable under convolution and infinitely divisible distribution (Küchler and Tappe, 2008b). Therefore, the Lévy density can be obtained as follows:

According to Proposition 1 and Equation (6), F(dx) can be seen as a Lévy measure. Based on the Lévy-Khintchine theorem, the characteristic function of bilateral gamma distribution under this measure can be written as:

The accumulative generating function is the logarithm of the moment generating function; that is, \psi (t)=\ln \text{E}\left({\text{e}}^{tx}\right). E represents the expected function, so E_X represents the expected function for X. The cumulant generation function of the bilateral gamma distribution can be obtained from Equation (7):

Therefore, the n-order generator function of random variable Z in Definition 2.2. is as follows:

If Z\sim \text{Γ}\left({\alpha }^{+},{\beta }^{+};{\alpha }^{-},{\beta }^{-}\right), according to Equation (10), we can obtain the following properties:

Mean:

Variance:

Skewness:

Kurtosis:

The carbon emission option mentioned in this study is an option contract with the carbon emission quota as the underlying asset, which gives the option holder the right to purchase carbon emission rights at a certain price at a specific time. The trading direction of carbon options depends on the buyer’s judgment on the price trend of carbon emission rights. Option buyers can lock in the return level by purchasing call or put options. In addition, the combination of call and put options with different maturity and strike price can be traded to lock in profits and avoid risks. The following parameters will be involved in the option pricing process:

Underlying asset price (S_t): It refers to the price of the assets sold or purchased at t (0<t<T). Here, this is the price of EUA futures.

Strike price (K): The contractual price at which an option holder purchases the underlying asset at expiration.

Expiry date (T): It refers to the holding period of the option holder for the option contract, which is the time for the execution of the contract agreed by both parties. On that date, the contract holder has the right to decide whether to exercise the option contract based on the price of the carbon emission quota in the carbon market and option strike price.

Risk-free interest rate (r): The interest rate at which funds is invested in an investment object without any risk. Here, the risk-free interest rate is the yield on one-year UK government bonds.

In this pricing model, the following prerequisites must be met:

(1) The price of options depends only on the price of the underlying asset and risk-free interest rate;

(2) There are no taxes or transaction costs in the market, and no dividends are paid during the transaction;

(3) The price of the underlying asset satisfies the bilateral gamma distribution.

To accurately predict the future price rise and fall probability of EUA futures, determining the main factors affecting its price change is crucial. Wang et al. (2020) used the CSI 300 stock index futures to test the causal relationship between yield and trading volume. The authors found that yield is the Granger cause of trading volume. Ren and Li (2017) showed that the yield is positively correlated with trading volume. Thus, the trading volume affects the price change. Therefore, the regression model is used to study the relationship between EUA futures yield series and trading volume. The estimated test P-values of the model coefficients are all less than 0.001, indicating that the model coefficients are significant. The Pearson correlation coefficient is 0.633, indicating strong correlation. Therefore, EUA futures yield and trading volume have a linear relationship. Next, the correlation analysis of the succession of rising and falling returns shows that they have some correlation.

Next, the influence of trading volume on price is analyzed. If the continuous rise yield sequence {X} or continuous fall yield sequence {Y} of EUA futures price obey the gamma distribution, then their density function is:

Let \ln \left(X\right)=\ln \left(\beta \right)+\ln \left(\frac{1}{\beta }X\right), where \ln \left(\frac{1}{\beta }X\right) is the standard logarithmic gamma distribution with the following density function:

Assuming that the scale parameter \betais affected by trading volume v, as the logarithmic return rate presents a linear relationship with trading volume, then:

Then, \beta =\exp (\theta +\delta v).

The data used in Section 3 is again used to perform the regression analysis on the rising and falling yield series with their corresponding trading volume. The coefficient at the time of continuous growth is θ₁=0.009 and ξ₁=0.008. P< 0.0001 indicates that the coefficient is significant at the 95% confidence level. Then, the scale parameter corresponding to the continuous rise yield X can be represented as:

The coefficient of continuous decline is θ₂=-0.009, ξ₂=-0.008. The P-values are 0.0000 and 0.0112, respectively, indicating that the coefficients are significant. Thus, the scale parameter corresponding to the continuous fall yield Yis:

The relationship between volume and return rate can be determined through this analysis. In addition, the continuous rise and fall yield are correlated. Then, the final probability inference can be made by calculating their joint distribution.

Assuming that the continuous rise yield Xand continuous fall yield Ymeet X\sim \text{Γ}\left({\gamma }_1,{\beta }_1\right) and Y\sim \text{Γ}\left({\gamma }_2,{\beta }_2\right), respectively, the following equation can be obtained:

Since Xand Yare related, if Z_i\sim \text{Γ}\left({\alpha }_i,1\right), i=1,2; Z\sim \Gamma \left(\alpha ,1\right),\alpha =Cov\left(X,Y\right); Z₁, Z₂ and Z are independent, let {\beta }_{1}X=Z_1+Z, {\beta }_{2}Y=Z_2+Z, yielding:

Therefore, according to the theoretical knowledge described in Section 2, the joint distribution of continuous rise and fall returns can be expressed as follows:

The values of parameters β₁ and β₂ in Equation (23) under different trading volumes are obtained by Equations (20) and (21), respectively.

Taking the carbon emission quota yield series as an example, X\sim \text{Γ}(1.69,0.01), Y\sim \text{Γ}(1.21,0.01),and Cov(X,Y)=0.973 can be obtained based on the results in Table 2. Then, Equation (20) can be used to obtain the following:

Under the condition of continuous rise yield R₁≤a, the conditional probability of continuous fall yield R₀≤b can be expressed as follows:

The price fluctuation of yield mainly includes four situations: falling after continuous rise, rising after continuous rise, rising after continuous fall, and falling after continuous fall. Here, Equation (24) is used to calculate the conditional probability of price decline under different trading volumes and different values of continuous rise yield. The results are shown in Table 3. The conditional probability of other situations can be obtained similarly.

Clearly, the trading volume, and yields of rise and fall affect future price changes. With the same continuous rise yield rate, an increase in trading volume further increases the probability of falling after a continuous rise yield. With the same trading volume, the greater the rising rate of return, the greater the probability that the continuous fall yield will fall by a larger value; that is, if the yield drops after the continuous rise, it is very likely to fall continuously; when the continuous rise yield is small, the falling yield is more likely to fall less. Clearly, the conditional probability of price changing from continuous rise to decline is also related to the value of historical continuous rise yield: a small (big) rise followed by a small (big) fall. This is the inevitable trend of market price change due to people’s speculation.

Reasonable carbon option pricing is conducive for promoting the development and improvement of the carbon trading market. Carbon option serves the functions of hedging and price discovery. Its trading is similar to the margin system, and can improve the liquidity of the carbon market and reduce the operating costs for enterprises. Improving the carbon trading market and enriching carbon financial derivatives in the market can further promote the innovative and green development of enterprises with heavy carbon emissions and make it cheaper and easier to undertake emission reductions. This study fits the EUA futures yield series using a bilateral gamma distribution. The parameters pass the K-S goodness of fit test and exhibit a good fit. The fit parameters are then combined with the B-S pricing formula to optimize the option pricing model. This model is then used to study carbon options pricing. The pricing results show that the option price shows the same trend and volatility as the corresponding underlying asset price. The call (put) price is negatively (positively) correlated with the strike price. The proposed pricing model can serve as a reference for the pricing of carbon financial derivatives; however, some limitations remain in this research. Only a fixed value can be obtained while predicting the volatility. A time- varying analysis can be conducted to predict the sequence of future volatility changes with time.

Predicting the conditional probability of price rise and fall has important value for investors to make investment decisions. However, many influencing factors have complex relationships with the rise and fall in carbon emission quota prices. This study finds a correlation between continuous rise and fall yield. This influencing factor must not be ignored when predicting future price rise and fall. Next, trading volume is an important indicator reflecting market supply and demand. This study shows that it is linearly correlated with market price changes, and affects the rise and fall in prices. Trading volume can promote the yield: A larger trading volume will lead to a larger rise or fall. Overall, this study not only considers the relationship between continuous rise and fall returns, but also considers the influence of trading volume on price. Combined with these two influencing factors, this study deduces the formula for the conditional probability of price rise and fall by using the bilateral gamma distribution, and predicts the future rise and fall in carbon quota prices. Importantly, this study's probability deduction result is more reliable. This method can also be used for probability inference in other fields.