The arbitrage strategy in the crude oil futures market of shanghai international energy exchange

Abstract

This research conducts an empirical study on arbitrage opportunities in the crude oil futures market of Shanghai International Energy Exchange in the period of China’s economic change, 2020–2022. We use the daily closing price data of crude oil futures sc2303 and sc2212 to test whether there is a statistical arbitrage opportunity in China’s crude oil futures market. Taking the most commonly used 12 + 6 rolling window mode in statistical arbitrage, we select several one-year periods that pass the cointegration test during the formation period and apply the optimal opening, closing, and stop loss thresholds based on the highest yield during the formation period to the trading period. Finally, we draw the following conclusions: (i) the pair trading strategy based on the cointegration model is profitable in China’s crude oil futures market; (ii) the 12 + 6 window model can be applied to the pair trading strategy based on China’s crude oil futures. Our research proves the effectiveness of pair trading strategy in China’s crude oil futures market for institutional and individual investors.

Introduction

As an important energy source for human survival, the price of crude oil has been constantly fluctuating. Since the COVID-19 epidemic swept the world in 2020, its price fluctuation has intensified, especially the price of international crude oil futures. The continued spread of the COVID-19 pandemic around the world and the Russian–Ukrainian war are undoubtedly the main reasons for the recent sharp fluctuations in China’s crude oil futures prices (Yousef and Shehadeh, 2020; Yu et al. 2020; Farhad et al. 2021;). The Russian –Ukraine war has affection on the global commodity and stock markets seriously,especially in financial markets. During these time, oil is the safe haven for investors ( Belucio et al. 2022; Diaconaşu et al. 2022).The global economy may even suffer multiple shocks, because of repeated outbreaks (Zaremba et al. 2020; Zhang et al. 2020; Chinazzi et al. 2020; Barbier and Burgess, 2020). Under the continuous influence of the COVID-19 pandemic and the Russian–Ukrainian war, crude oil futures prices may continue to fluctuate, bringing opportunities for statistical arbitrage.

With the volatility of global financial markets during the epidemic and the continued decline in investor risk appetite, more and more institutional and individual investors are looking for an investment strategy that can avoid risk and obtain stable returns. The severe volatility of crude oil futures and its derivatives prices has attracted widespread attention from institutional and individual investors (Dutta et al. 2020; Adekoya et al., 2021). Under the influence of COVID-19 and the Russian–Ukrainian war, the trading volume and open interest of China’s crude oil futures not only did not decrease, but instead achieved stable growth (Corbet et al. 2020; Fu et al. 2020; Mensi et al. 2020; Wen et al. 2021; Hu et al. 2022; Wang et al. 2022a; Zhao et al. 2022; Zheng et al. 2022a). Data show from January to February 2022 that the trading volume of medium-quality sour crude oil futures at the Shanghai International Energy Exchange was 6,780,600 lots, or an increase of 654,100 lots compared to the same period in 2021 and a year-on-year increase of 10.68%. At the same time, on May 20, 2022, the single-day open interest of SC2707, the main force of domestic crude oil futures, increased by 0.714%. In a market environment where crude oil futures prices fluctuate violently, a slight misjudgment by both the seller and the buyer can result in huge losses (Wang et al. 2021a; Wang et al. 2022b; Long et al. 2022; Yin et al. 2022a; Zheng et al. 2022c). Therefore, both institutional and individual investors are reluctant to go all out when the market is unstable. Although the price of crude oil futures fluctuates violently, the same characteristics of the subject matter make the price of crude oil futures fluctuate for the same reason, and the occasional price deviation between crude oil futures will be corrected in a timely manner. Based on this, this study aims to explore the neutral investment strategies of individual and institutional investors based on crude oil futures.

Paired trading is a commonly used method in statistical arbitrage strategies. After 1978, it has been often used by investment banks and hedge funds and is a market-neutral strategy with weaker risk appetite (Gatev et al. 2006). It is based on the high correlation between two assets or two stocks during market movements (Gloukhov et al. 2014; Miao, 2014). When this correlation is broken and a deviation occurs, this deviation from historical experience will return to its original equilibrium, and so one can sell overvalued assets and buy undervalued assets to try and gain excess profits. Pair trading relies on the deviation of the relative price of certain securities in the short term. Since the relative price will eventually return to a reasonable range in the long run, investors will open positions when the price deviates, hoping to close the position and make a profit when it returns to a reasonable range (see Table 1).

Table 1.

Summary of the literature review

Author (s)	Model	Period	Sample	Key-related findings
Quan Gu, Xinghui Lei (2018)	Statistical arbitrage method: crush arbitrage	January 4, 2013, to December 2016	Soybean, soybean meal and soybean oil futures rapeseed, rapeseed meal and rapeseed oil futures	Compare with soybean rapeseed crushing arbitrage strategy portfolio returns better
Hoffman(2021)	Partial co-integration	January 1990 to November 2020	Johannesburg Stock Exchange	Partial co-integration trading strategy made higher returns during bear cycles compared with bull cycles
Li Chen, Guang Zhang(2022)	Parametric pairs trading model	January 1, 2015, to December 5, 2021	Energy-related securities, including futures, stocks, and ETFs traded in the USA	The strategy performed well before COVID-19 but yielded poor results in the pandemic era
Nicolas Huck (2009)	Paired trading	1999 to 2006	S&P 100 index stocks	Paired trading captures positive excess returns
Hanxiong Zhang,Andrew Urquhart (2019)	Pairs trading	January 1996 to July 2017	Mainland China and Hong Kong on highly liquid large‐cap and midcap stocks	If investors can trade across Mainland China and Hong Kong, pairs trading is profitable after adjusting for risk and transaction costs, where the annualized abnormal return is 9%

Quite a few scholars have studied the profitability of pair trading based on different markets. Based on the US stock market, Gatev et al. (2006) selected 40 years of stock data to study pair trading strategies. Their findings showed that pair trading strategies generate an average annual return of 11%, are low risk relative to just buying stocks, and have limited exposure to known stock risk factors. Since this original paper, the related literature has expanded significantly. Do and Faff (2010) conducted a further study using the pair trading model constructed by Gatev et al. and the same research subjects, extending the analysis interval to 2009, and came to similar conclusions as Gatev et al. Bowen and Hutchinson (2016) proved that the strategy of pair trading got the maximum monthly income in October 1987, and the first five pairs and the first twenty pairs of pair trading returned 46% and 36%, respectively, between 2007 and 2008. They also show that unconditional returns of the strategy do not relate to recognized systematic risk factors. Zhang and Urquhart (2019) switched the research subjects to stocks in the China market and the Hong Kong market, applied the pair trading strategy to both markets, and found that the strategy produced statistically and economically significant net monthly excess returns and net abnormal returns.

Based on previous research, this study applies pair trading strategy to China's crude oil futures market and makes the following contributions. First, it proves the effectiveness of the pair trading strategy in China’s crude oil futures market. Second, this paper verifies the effectiveness of the cointegration model based on a 12 + 6 rolling window. The original and pioneering studies of pair trading strategies began from Gatev et al. (2006), whose GGR model has been adopted by most scholars studying pair trading strategies, especially the dynamic rolling window mode of 12 + 6 (that is, the formation period is 12 months and the trading period is 6 months). The 12 + 6 rolling window mode has not been verified in China’s crude oil futures market, and the trading time of China’s crude oil futures is usually more than one year or more, which can be used to test the validity of the 12+6 window mode. Therefore, this paper determines the rolling window is 12+6. Based on the 12+6 rolling window, we apply the daily closing price data of crude oil futures sc2303 and sc2212 from March 2, 2020, to April 29, 2022. During the 12-month formation period, the paired combinations are screened and a pairing trading strategy based on the cointegration model is constructed. Finally, through the verification of the six month trading period of the screening matching, the earnings of the matching trading strategy in the China crude oil futures market of the Shanghai International Energy Exchange were obtained. Verify the effectiveness of China's crude oil futures market, and make a qualitative analysis of all investors' expectations of the market, so that investors can better grasp the market and improve the effectiveness of the market. The guidance of futures on spot prices should be achieved, so as to make the economic operation more stable.

Scholars in previous literature have used pair trading strategies to find arbitrage space in the fund market, stock market, stock index futures market, and even the Shanghai 50ETF market. However, few studies have studied the effectiveness of pair trading strategies in China’s crude oil futures market. Considering this situation, this paper takes crude oil futures whose prices have fluctuated violently in the financial market recently as the research object and explores the effectiveness of the pair trading strategy in this market. The purpose of this article is to provide a practical approach to individual and institutional investors in China’s crude oil futures market. More and more investors use the method of pair trading to compare transactions, which could eliminate arbitrage opportunities and improve market efficiency. It is thus beneficial for market makers to provide more stable and reasonable bilateral quotations. Furthermore, this paper has a certain reference value for regulatory authorities to improve supervision efficiency and maintain market order. Finally, with its large number of participants, the improvement in market efficiency, and the improvement in regulatory laws and regulations, China’s crude oil futures trading products can be built into Brent crude oil futures.

The remainder of this paper runs as follows. Section 2 provides the literature review. Section 3 introduces the construction of a crude oil futures pair trading model. Section 4 presents the empirical analysis. Section 5 concludes and discusses the policy implications.

Literature review

Pair trading is divided into three steps: the first is the construction of the pairing pool; the second is to screen out the appropriate paired combination from the pairing pool; the third is to develop appropriate trading strategies.

On the study of pairing pool selection, Zhang (2012) selects six industries with high homogeneity to build a matching pool, including real estate, steel, coal, electricity, banking, and automobile. The author chooses homogeneous industries, because of the high degree of similarity in internal operations, products, or services in homogeneous industries, and hence, the price sequence of all enterprises in the industry is more similar. Therefore, it is more conducive to the screening of the stock pair with long-term cointegration.

Hu (2013) selects commodity index futures and Shanghai commodity ETF as specific research objects. Allowing short selling on ETF Shanghai commodity stocks, the article proves the effectiveness of the two-span arbitrage strategy in China’s futures market. Most domestic scholars choose two specific research objects in the study of pair trading strategies. According to the brokerage research report, Huang (2015) takes the real estate stocks with high homogeneity in the current industry classification as the research object. She selects 12 large estate stocks to build a matching pool. Based on the assumption that the spread follows the O-U process, it is then proved that an arbitrage strategy based on this process has the advantages of low cost, high income, and low risk compared with the traditional arbitrage strategy based on the cointegration theory.

By using the strong correlation between some A stocks and US stocks, statistical arbitrage is carried out. Gatev (2006) studies the performance of pair trading strategies in the US stock market, using all stocks in the market as paired pools, and then, the minimum distance method defined by them is used to screen paired combinations. Perlin (2009) selects 100 stocks with the highest liquidity in Brazil between 2000 and 2006 to construct the matching pool according to the principle of liquidity, which finally proves the effectiveness of the pair trading strategy based on the stocks with better liquidity.

Do and Faff (2010) expand the research scope of Gatev et al. (2006) by studying the performance of a pair trading strategy in the US market from 1962 to 2009, selecting all the stocks in the US market to construct the matching pool. Since stocks in the same index generally have many common properties, Bolgun (2010) finds a good matching effect when Turkey ISE-300 index component stocks are chosen as an alternative stock pool. In contrast with the study by Gatev et al. David A. Bowen (2014) selects 767 stocks included in the FTSE full stock index to build a matching pool to remove illiquid stocks and finds that the performance of pair trading in the UK market does not decline over time.

As far as the method of selecting paired combinations in the pairing pool is concerned, scholars have developed many different methods in academic or practical terms, including the minimum distance method, cointegration, the random price difference method, and others. In recent years, with the development of science and technology and the extension of theory, some algorithms in machine learning have also been applied to the screening of paired combinations. Han and Chen (2007) select 50 component stocks of the Shanghai Stock Exchange Index to build a matching pool. In the screening of stock pairs, they draw on the idea of cointegration and employ a stepwise regression method to determine the portfolio and pricing subspace.

Cui et al. (2011) select the Shanghai Stock Exchange 50 index component shares to construct the matching pool. They use the minimum distance method as the technical means of empirical analysis to select the paired combination part. Considering the cost, there is still a lot of profit from pairing and the profit is less affected by market risk.

Wang (2013) conducts a pair study of stocks according to the gap between different stock prices. Via a thorough analysis of the final results, he notes that the method could obtain approximately 1% of the proceeds, and the final data will not fluctuate too much. In addition, the trading method will not be affected by the market environment, so that the safety factor is high. Hu (2016) combines cointegration with the minimum distance method and explores in depth the matching behavior in stocks. Through the analysis of the final conclusion, it is found that the effect of the combination of the two methods is much higher than that of any method. In addition, this method is largely affected by the parameters involved. The trading time should not be too short, and the threshold must fluctuate within the specified range.

Whistler (2004) adopts the correlation coefficient method to screen the paired combination based on whether the correlation coefficient is close to positive or negative. The statistical data such as price difference or price ratio and cumulative probability are used to determine the opening or closing of the position. Nath (2003) conducts an in-depth study of the use of pair trading schemes in British bonds. He mainly used the minimum distance method instead of the cointegration method when screening stocks, and explored the opening and closing time in the paper, and finally verified and evaluated the final conclusion through the omega function. Elliott et al. (2005) points out that the price difference can be used to verify the post-pair situation. He arranges the gap between stock prices and constructs a linear space model based on this. After that, if the real spread deviates from the theoretical spread calculated by the model beyond a certain range, then a trading opportunity arises.

A relatively perfect theory of distance methods was first proposed by Gatev et al. (2006), which can be called the GGR method. They consider n stocks with data for the first 12 months selected as a training set. The sum of the euclidean distance squared (SSD) of n*(n-1)/2 possible combinations is then calculated in the training set. In the process, the combinations are sorted by SSD from small to large with only the first 20 combinations. Binh (2006) uses the random spread model in the selection of stock pairs to carry out further research. Researchers such as Chang (2009), Perlin (2009), and Faff (2010) have explored pair trading with minimal distance. Through a lot of research and a series of improvement work, Bertram (2010) obtains the optimal solution of price difference from the O-U random process.

For the design of a pair trading model, Agarwal (2004) studies the trading signal and sets it according to the degree of the price difference in the paired stocks’ deviation from the historical average level, and the result is that the research object has higher annualized returns. But its shortcomings are also more obvious, that is, the spread range is not static, but changes over time. Gatev et al. (2006) carries out a thorough exploration of statistical arbitrage. The concrete approach first takes the processing of the stock price and then selects the optimal stock pair through the model. On this basis, the sum of squares of the price difference between the two stocks is obtained, and the stock portfolio with the least sum of squares is kept. If the price difference is 2 times higher than the standard deviation, you can start to prepare to establish a position. Through the above conclusions, this method can bring more profits.

Many researchers have perfected GGR strategies. For example, Papadakis and Wysocki (2007) reformulates the standard of building a position and finds that the absolute value of the price difference being greater than the pre-set threshold is the new standard of building a position. Engelberg (2009) establishes a ream-skimming closing strategy that closes the position when the profit is greater. Mai and Wang (2014) combine the GGR method with the Herlemont method to construct the FTBD strategy to set up the trading position. The above three schemes are used to pair stocks listed on the exchanges in Shanghai, Shenzhen, and Hong Kong. As fully reflected in the final results, the profits obtained through the FTBD scheme are the largest, and the profits obtained in the stock pairings of the Shenzhen Stock Exchange are much higher than those of the Hong Kong Stock Exchange. In a better economic development market, the effect of matching exchanges is not obvious. Because China has developed its financial sector in just a short period of time, the market has not yet formed a sound system, and relevant policies are not perfect, thus limiting the role of pair trading to a large extent there.

Ouyang and Li (2015) study the pair trading strategy in depth, hoping to find an optimal threshold. They apply it to Chinese A + H stocks on the basis of this threshold. Their final exploration results fully show that when the parameters no longer change, the profits obtained through pair trading can be greater. If the relevant parameter values are not stable, then the profits will decrease under the influence of the threshold. Multiple trading activities mean greater costs, which have a great negative impact on profitability. Hu et al. (2017) combine a pair trading strategy with reinforcement learning to build a new trading model with dynamic optimization parameters and apply it to China’s bond market. Compared with the traditional pair trading strategy with fixed parameters, their findings offer a better return. Fu et al. (2019) use the O-U process modeling to design specific transaction models when studying dynamic management of fund assets. Wang et al. (2019) focus on checking the SC crude oil futures intertemporal arbitrage strategy. This paper takes the heteroscedasticity of the spread series into account. We compare EARCH(1,1) with GARCH(1,1) and choose the latter to fit the conditional heteroskedasticity of spread series. Robert B, Pavel T (2020) took the 10 US commodity futures, and by capturing the dynamics of the futures volatility terms structure with three factors, the paper shows that in most markets the slope factor is strongly negative in certain periods and at best only weakly negative in other periods. High inventory levels are found to correspond to flatter volatility term structures in seven futures.

As mentioned above, pair trading is divided into three main steps. For the selection of the pairing pool, we use fundamental analysis, or select all securities contained in a securities index, or select related indicators such as liquidity indicators to build the pairing pool. However, from the literature, the selection methods of the pairing pool are much the same and have not been studied. For the screening of paired combinations, the methods are divided into the minimum distance method and cointegration method. The minimum distance method and cointegration method are adopted by most scholars, and their effectiveness is widely proved. The cointegration method is an innovation of the minimum distance method which avoids the disadvantages of a similar trend of paired combination selected by the distance method and cointegration method, such as fewer number of transactions caused by infrequent price fluctuation, long-term position holding, etc (Feng et al. 2021; Wang et al. 2021b; Peng et al. 2022; Wang et al. 2022b; Yang et al. 2022; Yin et al. 2022b; Zheng et al. 2022b, c). Therefore, the screening of paired combination has improved and is more innovative.

For the design of a pair trading model, the key lies in seeking the best transaction threshold and trading opportunity. On the basis of determining the threshold, the random price difference method makes use of its good predictive property and takes the initiative to choose the best time to build and close positions, thus maximizing arbitrage profit. By the stochastic control method, the HJB equation is established and the optimal threshold is obtained. However, the shortcomings of these two methods are obvious—that is, the assumptions based on them are harsh and different from reality, and so it is difficult to apply them directly to actual transactions. Therefore, the innovation of this step mainly lies in seeking a more approximate process for price difference or the solution of optimal threshold. In general, a pair trading strategy is a very mature strategy system, and the research covered by the relevant literature has been extremely rich and comprehensive.

There are several problems in the existing literature on pair trading. First, when selecting the research target, most of them choose stocks, stock indices, ETFs, and futures, and no literature studies the effectiveness of a pair trading strategy in the crude oil futures market. Based on this, this research studies the effectiveness of a pair trading strategy in the crude oil futures market in order to further improve the relevant literature on pair trading. Second, when selecting the optimal transaction threshold, the traditional research sets the fixed transaction threshold based on experience. In recent years, some scholars have used the random control method to model asset pricing dynamically and obtain the optimal transaction threshold. The disadvantage is that the assumption of price difference is harsh, different from reality, and not universal. Therefore, it is difficult to apply it to practical trading to guide statistical arbitrage. Therefore, this study adopts the dynamic trading threshold. During the formation of the pairing combination, it traverses different positions and stop loss combinations, selects the optimal threshold, and applies it to the trading period, in order to explore whether the dynamic rolling arbitrage strategy of 12+6 in the crude oil futures market has a basis.

Construction of a crude oil futures pair trading model

As mentioned above, the main steps of pair trading are divided into three steps. For the selection of pairing pool, we can use fundamental analysis or select all securities contained in a securities index or select related indicators such as liquidity indicators to build a pairing pool. For the screening of paired combinations, the methods are divided into minimum distance method and cointegration method. For the design of a pair trading model, the key lies in seeking the best transaction threshold and trading opportunity.

Trading pair screening

The screening of transaction pairs is during the formation period of pair trading. Pair trading is divided into the formation period and trading period. The formation period refers to the stage of screening paired combination. First, the original futures price sequence is logarithmic in the formation period. Second, the cointegration theory is used to find the paired combination, which passes the cointegration test.

Correlation analysis

The pair trading should first screen out the futures pairs with the long-term equilibrium relationship of the price trend in the formation period—that is, the price time series of the paired futures should have a high correlation. The famous statistician Pearson has proposed correlation coefficients to measure the close relationship between variables. This study uses the correlation coefficient to measure the correlation degree between the time series of future price. Assuming that the price variables of the two futures are Z_t and Y_t ,respectively, the correlation between the two futures is expressed by $\gamma$ with the following calculation method:

$$\gamma =\frac{\sum _{i=1}^n(Z_i-\overline{Z})(Y_i-\overline{Y})}{\sqrt{\sum _{i=1}^n{(Z_i-\overline{Z})}^2}\sqrt{\sum _{i=1}^n{(Y_i-\overline{Y})}^2}}$$ 1

The correlation coefficient $\gamma$ between price time series Z_t and price time series Y_t can be calculated by formula (1). The value is between 0 and 1, and the closer the absolute value $\gamma$ is to 1, the higher is the correlation between variables.

Cointegration test

Vidyamurthy pioneered the introduction of the cointegration method into the pair trading strategy to screen initially qualified pair combinations. After the correlation analysis, the cointegration test is conducted on the matching combination with the correlation coefficient above the set critical value, and the eligible paired combination is finally selected. The steps of the cointegration test are as follows.

The first step is to test the single integer number of time series $\left\{Y_{1t}\right\},\left\{Y_{2t}\right\},...\left\{Y_{\mathit{kt}}\right\}$. If only two variable sequences are included, then the single integer number of two variables should be the same.

The second step is to use the OLS method for the covariant regression and the OLS method for the regression Eq. (2) (also known as the cointegration regression equation).

$$Y_{1t}=\alpha +{\beta }_2{Y}_{2t}+...+{\beta }_k{Y}_{\mathit{kt}}+{\mu }_t$$ 2

The residual sequence (3) is now obtained.

$$e_t=Y_{1t}-\left(\widehat{\alpha }+\widehat{{\beta }_2}{Y}_{2t}+...+\widehat{{\beta }_k}{Y}_{\mathit{kt}}\right)$$ 3

The third step is to test the smoothness of e_t If it is smooth, then $\left\{Y_{1t}\right\},\left\{Y_{2t}\right\},...\left\{Y_{\mathit{kt}}\right\}$. pass the cointegration test, and vice versa.

Trading strategy

Setup of trading signal

The basic idea of a pair trading strategy is to use the mean recovery characteristic of a price difference to capture the short-term deviation of such price difference by setting the threshold of trading operation such as opening, closing, and stop loss, so as to obtain the arbitrage income. This study determines the trading signal by the deviation of the logarithmic price difference between the two futures contracts from its long-term mean. Most studies use this method when determining trading signals (see, for example, Gatev et al. 2006; Perlin 2009; Do and Faff 2010; David A. Bowen and Mark C. Hutchinson 2014; Hanxiong Zhang and Andrew Urquhart 2019). Most studies use traditional trading signals, such as the multiple value of the fixed standard deviation of the valence difference sequence. This multiple is often determined by empirical values, usually when the price difference sequence exceeds 0.75 times the standard deviation.

According Vidyamurthy (2004), when the residual sequence is normal and the trigger condition is 0.75 times the optimal value of the transaction signal, the earnings can be maximized. In practice, the residual sequence does not conform to the normal distribution, and the financial time series has the characteristics of high peak and thick tail distribution. It would thus be inappropriate to use a zero-fold optimal transaction signal for normal distribution. There is no guarantee that the earnings will be maximized. Scholars choose these parameters according to experience, and most of them take stocks as the research target.

This study chooses crude oil futures as the research object. Therefore, the parameters selected according to experience may not be applicable to crude oil futures. The paper adopt dynamic opening and stop loss threshold and use the fluctuation information reflected in the formation period of crude oil futures contract pairing. This study calculates the optimal open position threshold and stop loss threshold to guide the trading of the futures. The method of traversal is used to calculate the optimal parameters by traversing different open positions and stop loss threshold combinations during the formation period. The opening line of position is set between 1.1 and 1.5, the closing line of position is set between 2.1 and 2.5, and the stop loss of position is set between is 0.3 and 0.5. The step size is 0.2. Traversing the operation process is implemented with Python. This article also makes an innovation in trading signals—that is, when the spread suddenly expands from the closing line to above the opening line, the trading strategy will not be to open an position. This is because the “leapfrogging” or sudden change in the spread is likely to be caused by non-systematic risk and the change is inertial. Pair trading is a neutral strategy, which minimizes transaction risk and is more appropriate with pair trading strategies.

Long-short position ratio

In pair trading there are two commonly used long-short positions: coefficient neutral strategy and capital neutral strategy. A capital neutral strategy means that the initial capital can be 0 without considering the transaction cost. The funds obtained from short selling futures are used to buy a short, and the market risk is small during the holding period. At the same time, the income obtained by closing positions is relatively small. A coefficient neutral strategy refers to the coefficient obtained by using cointegration pairing to construct a linear model as the ratio of a long-short position, and the coefficient neutral strategy is used to construct the ratio of a long-short position strictly according to the regression coefficient. This study uses the cointegration method to construct the matching transaction. Considering that the margin should be paid when the future is bought, the ideal situation of initial fund 0 cannot exist, and so the coefficient neutral strategy is adopted to build the long-short position ratio according to the regression coefficient:

$$\frac{\text{share,A}\ast \text{price,A}}{\text{share,B}\ast \text{price,B}}=\text{beta}$$ 4

Share A and share B are the position quantity of the paired futures contract in Eqs. (4). Price A and price B are the price when the paired futures contract reaches the opening condition. Beta is the coefficient of the model independent variable of the paired combination through a cointegration test in the formation period.

Transaction costs

The cost in the course of trading is mainly composed of several parts: transaction handling fee is 2 yuan per contract, exercise right handling fee is 2 yuan per contract, exercise right pre-hedging fee is 2 yuan per contract, and trading margin (futures seller). However, futures trading has capital requirements, which are specified as follows: the balance of funds available in the margin account for five consecutive trading days before applying for the opening of the transaction code or trading authority is not less than RMB 100,000 yuan, and so the initial fund of this study is 100,000 yuan. Selling crude oil futures requires a certain margin, and this study also takes the margin into account. The margin calculation formula is the larger value of the following two:

1. Futures contract settlement price * underlying futures contract trading unit + underlying futures contract trading margin—futures contract value *0.5(equal value and real value futures contract has a value of 0).

2. Futures contract settlement price * underlying futures contract trading unit + underlying futures contract trading margin *0.5

In this study, the margin is sufficient—that is, the margin required to sell crude oil futures is always sufficient. In practice, when the position pair expands exponentially, the initial capital also expands accordingly.

Evaluation indicators for trading models

This research selects the monthly average rate of return and the Sharp ratio to measure the return and risk of the strategy. Pair trading is a neutral trading strategy. Investors who use pair trading often have a low appetite of risk—that is, risk is an important consideration when they choose their portfolio. The Sharp ratio is a classical index to consider benefit and risk as a whole. Therefore, the monthly average yield and Sharp ratio are selected as the model evaluation index.

1. Monthly rate of return

Because the paper set up a window mode of 12 + 6, the rate of return calculated by the trading period is the monthly yield rate. After the pair trading is completed throughout the trading period, the monthly yield of the strategy is calculated. Assuming that the total amount after the completion of the trading period is Q₁, the initial capital is C, and the formula(5) for calculating the monthly rate of return is:

$$r=\frac{Q_1-C}{Q_1}$$ 5

• (2)Sharp ratio

This paper uses the Sharp ratio to measure the risk factors of a pair trading strategy. This indicator is considered to be a standardized indicator of performance evaluation in the fund industry, and it considers both the return of the portfolio and the risk of the portfolio. For rational investors in the securities market, when considering investment objects, they will use the Sharpe ratio as a measure, that is, to choose the investment object with the highest return expectation under the same risk level or the investment object with the lowest risk under the same return expectation. Therefore, it is reasonable to believe that when investors choose investment portfolios, they will require that the expected return of the investment object is greater than the return of risk-free assets. The Sharp ratio is designed to calculate the excess return per unit risk of the portfolio. The larger the value of the Sharpe ratio, the higher the return on the selected investment object.

The Sharp ratio(6) is calculated as follows:

$$\text{Sharp,Ratio}=\frac{E\left(R_p\right)-R_f\ast \frac{t}{250}}{{\sigma }_p}$$ 6

Here, $E\left(R_p\right)$ is the expected rate of return of the venture portfolio. This study uses the cumulative rate of return of the pair trading portfolio, where t is the trading days of the formation period or trading period and $R_f$ is the risk-free rate of return. In this paper, the risk-free rate is 3.75, and ${\sigma }_p$ is the standard deviation of the return rate. Herein, the standard deviation of the rate of return on previous trades in a pair trading portfolio is used.

Empirical analysis

Data

When selecting data in this paper, the paper first excludes crude oil futures contracts with a duration of less than one and a half years, because when designing the pair trading strategy, the paper selects the 12+6 window mode most used by scholars in the past. The period can characterize the numerical characteristics of the price difference of the future pairing portfolio, and in the 6-month trading period, the price difference will not deviate significantly from this range. Based on this, the paper eliminates a series of crude oil futures contracts with a duration of less than one and a half years, such as sc2003 and sc2109.

Second, according to the principle of liquidity, 30% of the crude oil futures contracts after the open interest are excluded, because the pairing trading strategy can only describe the range of the spread well when the liquidity is strong. When the liquidity is poor, the price often cannot reflect the real price. The market situation is highly contingent.

Third, excluding the abnormal point of crude oil futures price, the paper stipulate that when the crude oil price changes by more than 50% at some time, it is considered that its price is exhibiting an excessive response to market information, and so it is kicked out. The selected data come from the official website of Shanghai International Energy Exchange, and its annual data are updated to April 29, 2022.

This study therefore uses the crude oil futures corresponding to sc2303 and sc2212 when establishing a two-way position and selects the daily closing price data of two crude oil futures contracts for empirical research. Due to some missing data, only data with the same time in the two futures contracts are retained.

Matching combination screening

Before the correlation analysis, the crude oil futures sc2303 and sc2112 are first divided into 8 pairs of different time periods according to the 12-month formation period, and the last pair is from October 2020 to October 2021. The annual data package of Shanghai International Energy Center is only updated to April 29, 2022 for the time being. In the data processing process, the paper eliminates 30% of the crude oil futures contract data after liquidity and only selects the data of the two crude oil futures contracts with the best liquidity. Table 2 shows different formation and trading periods of the two futures contracts.

Table 2.

Different formation and trading periods

SC2303 and SC2212
Formative period	Trading period
2020.3–2021.3	2021.3–2021.9
2020.4–2021.4	2021.4–2021.10
2020.5–2021.5	2021.5–2021.11
2020.6–2021.6	2021.6–2022.12
2020.7–2021.7	2021.7–2022.1
2020.8–2021.8	2021.8–2022.2
2020.9–2021.9	2021.9–2022.3
2020.10–2021.10	2021.10–2022.4

Correlation analysis

First of all, in pair trading, crude oil futures contracts whose price trends have a long-term equilibrium relationship should be screened out—that is, the price time series of paired futures should have high correlation. This study chooses the Pearson correlation coefficient to test the correlation between contracts. The paired combinations selected in this paper are sc2303 and sc2212, which are divided into 8 time periods as shown in Table 2. Calculating by Formula (1), correlation tests are carried out for different formation periods, and the time period with the correlation coefficient higher than 0.9 is selected for the next cointegration test. Table 3 shows the correlation coefficient of the price of the paired combination formation period in different time periods. As shown in Table 3, the price correlation coefficients in the formation period of the 8 time periods are all higher than 0.9, which can be tested in the next step.

Table 3.

Correlation coefficients of different formation periods

Ranking	Formative period	Correlation coefficient
1	2020.3–2021.3	0.999
2	2020.4–2021.4	0.998
3	2020.5–2021.5	0.998
4	2020.6–2021.6	0.997
5	2020.7–2021.7	0.997
6	2020.8–2021.8	0.996
7	2020.9–2021.9	0.995
8	2020.10–2021.10	0.995

It can be seen from Table 3 that the correlation between the prices in the formation period of 8 different time periods is very high, all above 0.99, and so the 8 groups of paired combinations all pass the correlation test. At the same time, Table 3 sorts different time periods according to the size of the correlation coefficient. Therefore, the paired combinations of 8 different time periods pass the correlation analysis.

Stationarity and cointegration tests

After screening out the matching combination of 8 pairs of futures contracts with correlation coefficient above 0.9, it is necessary to carry out the cointegration test. After that, the paper selects the paired combination that passes the cointegration test during the formation period of 12 months. Taking the first pairing combination as an example, the paper selects the daily closing price data with the formation period from March 2, 2020 to March 2, 2021. The Augmented Dickey–Fuller test (ADF test) results of the first group of original sequences and the first-order difference sequences are displayed in Table 3.

It can be seen from Table 4 that the logarithmization of the two futures contracts is a first-order single integer, and so they are tested by the EG two-step method. First, the regression results are sorted as shown in Table 5. According to the regression results, the residual expression is obtained:

$$u_i=lnSC2303-0.9405lnSC2112-0.3697$$

Table 4.

Test results of original sequence and first-order differential ADF

Variable (after taking logarithm)	ADF value	1% critical mass	5% critical mass	10% critical mass	P value	Stability
Log-SC2303	− 2.163	− 3.46	− 2.87	− 2.57	0.220	Not smooth
Log-SC2212	− 2.240	− 3.46	− 2.87	− 2.57	0.192	Not smooth
First-order difference of SC2303	− 13.554	− 3.46	− 2.87	− 2.57	0.000	Stable
First-order difference ofc344	− 18.040	− 3.46	− 2.87	− 2.57	0.000	Stable

Table 5.

EG first-step results of the two-step approach

Variable	Coefficient	Standard deviation	T value	P value
const	0.3697	0.091	4.057	0
Log-SC2212	0.9405	0.015	61.825	0

Next, EG is the second step of the two-step method, and the residual sequence is ADF tested in Table 6.

Table 6.

EG results of the second step of the two-step approach

Variable	ADF value	1% critical mass	5% critical mass	10% critical mass	P value	Stability
$u_i$	− 4.423	− 2.58	− 1.94	− 1.62	0.000	Stable

Table 6 shows that the ADF value of the residual sequence is less than the critical value at the level of 1%. It is considered that the residual sequence satisfies the stationary test and the two variables have a long-term equilibrium and stability relationship, which adheres with the precondition of paired transaction.

Setting up the trading strategy

After correlation analysis and cointegration test, the price difference sequence is:

$$\text{Spread}=lnSC2303-0.9405lnSC2112$$

The paper then standardizes the spread and make trading decisions according to the transaction signal. Before this, the optimal parameters are obtained by traversing different combinations of values of open and close positions in Python, taking the yield rate in the formation period as the criterion. Different from the dynamic open position and stop loss threshold in stock pair trading, this study only needs to determine one pair in the formation period. Because all futures contract subjects in the pairing pool are crude oil, the reason for fluctuation is the same. Table 7 shows the statistics of the top five in terms of yield rate in the first pair according to traversal results.

Table 7.

Searching for optimal parameters

Opening threshold	Clearance threshold	Stop loss threshold	Forming period rate of return
1.1	0.5	2.1	30.2%
1.1	0.5	2.3	26.4%
1.1	0.5	2.5	29.7%
1.1	0.3	2.1	28.6%
1.1	0.3	2.3	29.4%

The specific trading strategy is as follows. When the spread scale (standardized price difference) is larger than 1.1, the paper shorts the price difference, selling SC2303 and buying SC2212 according to the calculated proportion in a neutral strategy; when the spread scale is smaller than − 1.1, the paper longs the price difference, buying SC2303 and selling SC2212; when the absolute value of the spread scale is larger than 2.1, the paper reverses the operation and stops the loss in time; when the absolute value of the spread scale is less than 0.5, the paper closes the position and obtains earnings from the arbitrage.

After obtaining the optimal parameters according to the traversal method, the paper draws the price difference sequences between the formation period and the trading period of the first group by Python. According to Fig. 2, the trading model constructed according to the high frequency data of the formation period performs well. In Figs. 1 and 2, the blue solid line is the closing line, the green dashed line is the building line, and the red dotted line is the stop line Figs. 3 and 4.

Fig. 2.

Standardized spread sequence of au2006c344 and au2006c340 in the trading periods

Fig. 1.

Standardized spread sequence of au2006c344 au2006c340 in the formation period

Fig. 3.

Standardized spread sequence of au2012c432 au2012c440 in the formation period

Fig. 4.

Standardized spread sequence of au2012c432 and au2012c440 in the trading period

Empirical results of pair trading

According to the pair trading strategy, the paper back-tests the data of the formation period and trading period of SC2303 and SC2212. Taking the first group of paired combinations as an example, 168 transaction signals are captured in the formation period, and all 168 transactions have achieved positive returns. On the premise of considering the transaction cost and the transaction margin, the yield obtained by the transaction is 65.76%, and the Sharp ratio is 5.96. When the spread deviates from the mean value, it can return to the mean value at a fast rate. The model captures two transactions in the trading period data, and both transactions have positive returns. Considering the transaction cost and the trading margin, the yield or monthly gain rate in one month obtained by the two exchanges is 5.24, and the Sharp ratio is 0.65.

Table 8 shows the statistical information on the formation and trading periods of 8 pairs of paired combinations screened. Table 8 shows that, of the 8 pairs, all pass the cointegration test, which is conducted by the ADF test in the Python arch package. The average monthly rate of return for the 8 pairs screened is 37.13%, and the Sharp ratio is 3.72. For each pair of the 8 pairs, they at least once open a position. Of the 8 pairs, one pair generates negative returns during the trading period, and 7 pairs produce positive returns. Table 8 shows the statistical results.

Table 8.

Rate of return and Sharp ratio of portfolio

SC2302 and SC2212
Pair group	Forming period rate of return	Sharp ratio of formation	Trading rate	Sharp trading rate
***1	65.76%	5.96	5.24%	0.65
**2	48.26%	7.84	2.68%	4.32
***3	29.71%	10.09	− 3.49%	− 7.11
**4	69.26%	21.65	7.20%	8.33
**5	25.32%	2.02	9.62%	5.77
**6	62.08%	20.01	20.87%	5.33
*7	52.86%	9.21	17.26%	9.45
**8	41.37%	8.72	24.57%	2.09

***Indicates stability at the 1% level,** indicates stability at the 5% level,* indicates stability at the 10% level

Conclusion and policy implications

Scholars in previous literature have used pair trading strategies to find arbitrage space in the fund market, stock market, stock index futures market, and even the Shanghai 50ETF market. However, few studies have studied the effectiveness of pair trading strategies in China’s crude oil futures market. Considering this situation, this paper takes crude oil futures whose prices have fluctuated violently in the financial market recently as the research object and explores the effectiveness of the pair trading strategy in this market. The paper selects the futures pool according to the liquidity principle and select 8 pairs of future contracts with a correlation coefficient greater than 0.9 through correlation analysis in period of Chinese economic change 2020–2022. The paper then empirically analyzes the 8 pairs that meet the pair trading conditions as empirical objects. Finally, the average monthly return of the 8-pair portfolio far exceeds the average monthly return of the stock market, reaching as high as 10.51%, with a Sharpe ratio of 3.61. This paper provides the first evidence of the profitability of pair trading strategies in China’s crude oil futures market.

Most pair trading with futures adopts the rolling window model of 12 + 6—that is, 12 months to characterize the price range of the research object, and 6 months to implement the trade. The duration of crude oil futures contracts is usually greater than two years, and so the 12+6 rolling window model can be applied. On this basis, this paper selects the paired combination with a cointegration relationship, uses its statistical information to build a pairing trading model with a 12-month formation period, applies it to the trading period, and achieves better returns. It is proved that the 12+6 rolling window model is effective in China’s crude oil futures market.

We traverse the formation period of transaction signal setting to obtain the optimal parameters and then apply it to the transaction period. Because the reasons for the changes of crude oil futures in different periods are different, the optimal threshold needs to be determined for the specific formation period. Overall, with full consideration of trading margin, the pair trading strategy achieves an average yield of 49.32% in the formation period, a Sharp ratio of 10.68, a monthly average yield of 10.51% in the trading period, and a Sharp ratio of 3.61. The decline of the Sharp ratio is because the model is built on the formation period and is expected.

This study overall confirms the effectiveness of pair trading and the 12 + 6 window model in China’s crude oil futures market. China’s crude oil futures contracts have a natural high correlation, because of the same underlying objects. At the same time, its T+0 trading mechanism makes it possible to quickly open or close positions when the price deviates and returns, compared with the mechanism that can only be operated again on the next day in several stock markets, including A shares. Combined with the strategic asset allocation of crude oil futures and its derivatives under the epidemic situation, this study provides operational guidance for institutional and individual investors. It confirms that pair trading strategies also have arbitrage space in the crude oil futures market. These studies combined indirectly prove that the financial market in China is not efficient. Quantitative trading strategies represented by pair trading have matured in recent years and are expected to become the mainstream risk-neutral trading strategies in the future against the backdrop of large fluctuations in the global financial market. Pair trading not only brings higher investment returns to investors in practice, but also improves the value discovery function and operation efficiency of the market through a large number of arbitrage behaviors in the market.

In this paper, we suppose that the initial fund is 100,000 yuan. Actually, in the world of arbitrage one needs to borrow the money from a bank. There exists an interest cost. In our study, we do not consider the interest cost. Interest will be a limitation and future recommendation.

Declarations

Conflict of interest

The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.