COVID-19 was first reported in China and quickly spread throughout the world. Weak investor confidence in government efforts to control the pandemic seriously affected global financial markets. This study investigated chaos in China’s futures market during COVID-19, focusing on the degree of chaos at different periods during the pandemic. We constructed a phase diagram to observe the attractor trajectory of index futures (IFs). During the COVID-19 outbreak, overall chaos in China’s futures market was increasing, and there was a clear correlation between market volatility and the macroenvironment (mainly government regulation). The Hurst index, calculated by rescaled range (R/S) analysis, was 0.46. The price and return of IFs showed long-term correlation and fractal characteristics; the relevant dimensions of the futures market were 2.17. Overall, under the influence of an emergency (COVID-19), chaos in China’s financial market intensified, creating a need for timely government intervention and macrocontrol of the market. This study’s findings can help improve the government’s understanding of the phenomenon of financial chaos caused by emergencies. This study also provides theoretical guidance for controlling financial chaos and maintaining healthy economic development when faced with similar events in the future.

Chaos in financial systems arises from instability in their internal operations. External manifestations of such chaos include sharp, abnormal fluctuations and financial crises. Moreover, inefficient market-resource allocation can greatly affect the economy and society. Therefore, evidence of chaos can have significant effects on regulators and on short-term trading strategies.

Although China’s financial market has developed rapidly, it has an incomplete and unbalanced structure, and it has gradually shown the characteristics of chaos. Studies of the Shanghai Stock Exchange have found that China’s stock market has significant chaotic features, with a chaotic attractor dimension of 2–3 [

COVID-19 quickly spread throughout China in late 2019 and propagated globally. Since then, numerous studies have investigated the various effects of COVID-19 [

At the beginning of the COVID-19 outbreak, investors in China were initially optimistic, but after Wuhan was locked down on January 23, 2020, their confidence was shaken. In early February, China’s financial market saw an unprecedented plunge. However, from February 4 to February 21, 2020, as the government took effective COVID-19 control measures, investor confidence rebounded, and the market improved somewhat. Yet, in March 2020, COVID-19 began to spread out of control overseas, the WHO declared it a pandemic [

This study focused on chaos in China’s futures market after the COVID-19 outbreak. It mainly contributes to the literature on the effect of emergencies (pandemics) on economic chaos in China. Taking China’s futures market as the research object, we selected price series of index futures (IFs) before and after the main development period of the pandemic in China (November 1, 2019, to May 31, 2020) to establish a time series. The next section introduces the methods. Then, Section 3 describes the data set and the empirical results; we also use the concept of chaos to examine the effect of “big events” during COVID-19 on futures market chaos in China. Section 4 summarizes the paper and discusses the limitations.

This study used big data to simulate chaos in China’s futures market during COVID-19. The methods included a phase diagram, maximum Lyapunov exponent, correlation dimension, and Hurst exponent.

We reconstructed the phase space of IF exponential time series, calculated the most prominent Lyapunov exponent of the series, and used visualization techniques to visualize chaotic attractors. Meanwhile, the fractal dimension and Hurst index were used to judge the chaotic characteristics of the futures market.

Time-series phase diagrams can be used to analyze changes in a nonlinear system with a time state; this can reflect the system attractor’s spatial structure. A chaotic attractor of the high-dimensional region is first proposed in the phase space to restore the chaotic system. The principle here is that the evolution of any component in a system depends on the interaction between other members. Information about these related components will be retained during the development of any element. We can extract and restore the original law of the system from the time-series data of any component, which is a trajectory in high-dimensional space. The revolution reflects the regularity of the chaotic system. The evolution of a chaotic system will eventually fall into a specific course, which is the chaotic attractor.

We used the C–C algorithm to calculate time-lag parameter t and embed dimension m of the IF time series [

Suppose there is a chaotic time series,

…

where N is an integer multiple of t. The statistics are defined, and the block average strategy is adopted:

where

If the time series is independent and identically distributed, then for a fixed-delay parameter t and embedding dimension m, N →∞. For any r,

This measures the maximum deviation of

Based on the above principles, we find that the first local minimum occurs when the delay parameter is 4. Accordingly, we selected 4 as the best delay parameter.

In practical application, the Lyapunov exponent is an important index for quantitatively measuring a system’s dynamics. Its geometric meaning is to quantify the exponential divergence of the initial closed-loop orbit and the amount of chaos in the nonlinear system. It represents the adjacent orbits of the system in the phase space—that is, the average exponential rate of convergence or divergence. (The Lyapunov exponent is greater than 0. This means that no matter how small the initial trajectory distance in the system’s phase space is, it will increase exponentially with the evolution of time and is ultimately unpredictable. It is a chaotic phenomenon.)

Assuming there is a time series, the time-delay parameter is t, the embedding dimension is m, and the phase space of the reconstructed time series is

For each

For each

where q is the number of nonzeros, and the regression line is made using the least-squares method. The slope of the line is the largest Lyapunov exponent. In 1983, Gribo proved that the time series with the largest Lyapunov exponent greater than 0 is chaotic. Therefore, we do not need to calculate all of the Lyapunov exponents; we only need to figure out whether the most prominent Lyapunov exponent is greater than 0.

The small data volume method is relatively reliable for calculating the most prominent Lyapunov exponent of a time series [

The existence of strange attractors is an essential feature of chaotic dynamic systems. The divergence of the attractor in space is an irregular movement in a limited area, and we used the fractal dimension to describe the singularity of its activity. For the phase space, we define the distance between these points as the maximum component difference between the two points:

If the distance is less than the positive number of the field’s radius, it is called the associated point. Assuming there is such a point in the phase space, calculate the correlated topics’ logarithm. The proportion of all possible species pairings becomes the correlation integral; namely,

Select an appropriate choice so that there is a specific interval, where D is the correlation dimension. If the correlation dimension calculated by the system’s attractor is a fractal dimension, the system attractor appears in a chaotic state when it moves. The fractal dimension of the phase space depends on the complexity of the system. All variables in the phase space that are higher than the next integer of the fractal size are the dynamic system model’s minimum dynamic variables.

The Hurst index is a parameter for judging whether a time series is dependent on time. In this study, we used rescaled range (R/S) to calculate the Hurst index of the time series and calculate the fractal dimension of the string through the Hurst index.

The main steps of the R/S method are as follows: Return the logarithm of the sequence length divided into subsequences, and the interval length of each subset is

Choose a subset arbitrarily and calculate the cumulative and standard deviations of the firsts points one by one:

Calculate the fluctuation range R of subset a and the standard deviation S of each subset a pair sequence:

The standard deviation S is used for each subset to standardize the fluctuation range R and obtain the R/S rescaling range. Multiple experiments can be performed by selecting different

Calculate the Hurst index as follows:

where the K is a constant.

The data for this study came from the China Financial Futures Exchange (

After the trend-elimination processing of the sequence in

We reconstructed the phase space of the financial time series and constructed the phase diagram using the visualization method. The phase diagram was used to determine the chaotic characteristics of IF time series. First, it was necessary to confirm the phase-space reconstruction to determine the time-delay parameter t and the embedding dimension m. The best embedding time t we chose in Section 2.1 was 4, and the best embedding dimension m was 4.

As shown in

For a chaotic economic system, its Lyapunov exponent is the system’s fractal dimension, and g is the range:

Then, the vector is

This is the system’s chaotic degree vector, where λ is the largest Lyapunov exponent of the system.

We split the data to examine chaos in the futures market during different periods. Experiments showed that the overall maximum Lyapunov exponent λ from November 2019 to May 2020 finally converged to 0.775. The system’s fractal dimension d was 2.173, and the logarithmic return range g was 0.1217.

Sample interval | λ | d | g |
---|---|---|---|

2019.11–2020.02 | 0.7517 | 1.955 | 0.121764 |

2020.01–2020.03 | 0.2333 | 2.092 | 0.121764 |

2020.02–2020.04 | 0.4133 | 2.102 | 0.121764 |

2020.03–2020.05 | 0.411 | 1.731 | 0.061671 |

2020.04–2020.06 | 0.6894 | 1.736 | 0.037273 |

2020.05–2020.07 | 0.633 | 1.408 | 0.059755 |

2019.05–2020.05 | 0.775 | 2.173 | 0.121764 |

From

1. Chaos existed in China’s futures market.

2. After the outbreak, the degree of chaos in the futures market showed an upward trend over time, and the overall degree of chaos was higher than the local ones.

3. There were large fluctuations in China’s futures market and the macroenvironment (mainly government intervention: when the government exhibited better control of the pandemic, investors had more confidence; otherwise, they were less willing to invest).

Time | Peak | Index | Event |
---|---|---|---|

January 2020 | Low | 3961.8 | COVID-19 outbreak in China |

February 2020 | Peak | 4192.6 | Government responded quickly to control the epidemic |

March 2020 | Low | 3477.0 | Linkage effect caused by the crash of US stocks |

April 2020 | Peak | 3911.8 | Pandemic in China was largely contained |

As shown in

The sudden global public health event of COVID-19 at the end of 2019 increased nonlinearity in global economic systems and produced chaos. The differing strategies of various countries in response to the pandemic affected the global economy in a nonlinear manner. China’s financial system became unstable, and in February and March 2020, its futures market experienced two sharp declines, which had negative effects on economic development and social stability.

This study used the C–C method to reconstruct the phase space of the time series. Moreover, we observed the chaotic attractor trajectory of China’s futures market in the phase diagram. The maximum Lyapunov exponent of the system was greater than 0 and therefore had a chaos effect. The overall level of chaos in China’s futures market increased during the pandemic; this phenomenon was related to intervention by the macroenvironment.

During the pandemic, the fractional dimension of the attractor of China’s futures market was 2.173. The fractal dimension also revealed that, for financial markets, we need to consider many uncertain factors, such as human psychological factors, decision-making, and information/information asymmetry. The market and its movement are still dominated by factors that are relatively difficult to measure, and the macroenvironment (COVID-19 in this case) can magnify these uncertainties. The fractal dimension is the smallest variable of the system-measurement factor. This means that during a pandemic, the futures market needs at least three dynamic variables, which we believe should be investor expectations, price estimates, and market liquidity.

The Hurst index of China’s futures market was 0.46. This means that the market had fractal characteristics—namely, self-similarity and long-term correlation. Self-similarity refers to recursion and pattern similarity while long-term correlation means there is no significant correlation between past and future. This feature causes the market to have certain risks.

External performance refers to the continuous rise and fall of futures prices. Volatility clustering makes the large-scale fluctuation of endogenous variables concentrate on the period while small-scale undulation occurs during a short period. Although chaos and randomness increased in China’s financial trading market after COVID-19, investors could still profit from it. Their ability to realize profits was dependent on investors’ understanding of stock-market characteristics.

This study has some limitations. First, we did not consider how the linkage effect between global stock markets affected chaos in China’s futures market during COVID-19. Second, because of the intensification of chaos in the futures market caused by an emergency, we did not conduct a deeper exploration of the subsequent development of the market. The next step will be to dig deeper into how financial market chaos affects subsequent market development and find ways to control such chaos.