2.1 Coupled Heston volatility model

The Heston model and its extended models are commonly used to describe the dynamic evolution characteristics of stock prices [21–24]. We extend the Heston model to describe the dynamic processes between N stock indices. The model is given by the following coupled stochastic differential equations:

(1)

Where x_i(t) and are the logarithmic price and realized volatility of the i -th stock price index at time t, is the mean reversion rate of the stock price index’s volatility , is the long-term mean of volatility , is the fluctuations of volatility, and dt is the time step. dZ_i and dY_i are correlated Wiener processes with the following statistical properties:

(2)

here, quantifies the correlation between the i-th stock market’s self-volatility and prices, and considers the correlated synchronization behaviors between the i-th and j-th stock markets stemming mainly from volatility.

2.2 Volatility synchronization indicator

Unlike correlation analysis, which only reveals the linear relationship between two variables, synchronization accounts for nonlinear dynamic phenomena such as market resonance and herd effects. Notably, Roll utilized the goodness-of-fit measure R² in the regression model to quantify the contribution of market factors to stock returns [25], and Morck transformed the goodness-of-fit R² accordingly to calculate the synchronicity of stock prices [10]. We will continue to refer to Morck’s research methodology [10]. By introducing the realized volatility estimated based on 5-minute high-frequency data and regressing the volatilities of two stock market indices, we obtain the fitted coefficients R², which are logarithmically deformed to derive the volatility synchronization indicator:

(3)

(4)

and () represent the realized volatilities of the two stock market indices. The correlation coefficient R_i,j is a statistic that measures the strength of linear correlation between two markets. The value of ranges from 0 to 1, and the closer it is to 1, the stronger the linear correlation between the two variables. Also, the larger the value of SYNCH_i,j, the stronger the volatility synchronization between the two stock markets. According to previous studies, the average test accuracies of the dynamic modeling methods using rolling time windows are all significantly higher than those of the static models [35,36].

2.3 Double coupled Heston volatility model and its dynamic synchronization

When considering only two markets or two stocks, based on the system of Eq (1), the double-coupled system can be transformed into the following coupled stochastic differential equations:

(5)

Where x_i(t) and are the logarithmic price and realized volatility of the i-th stock price index at time t, is the mean reversion rate of the stock price index’s volatility , is the long-term mean of volatility , is the fluctuations of volatility, representing the magnitude of the rise and fall in volatility, and dt is the time step. dZ_i and dY_i are correlated Wiener processes with the following statistical properties:

(6)

and quantify the correlation between the two stock markets’ self-volatility and prices, and considers the correlated synchronization behaviors between the two stock markets stemming mainly from volatility. These equations describe the coupling relationship between stock prices and volatility. Changes in volatility affect changes in asset prices, and vice versa.

By introducing the realized volatility estimated based on 5-minute high-frequency data and regressing the volatilities of two stock market indices, we obtain the fitted coefficients R², which are logarithmically deformed to derive the volatility synchronization indicator:

(7)

(8)

and represent the realized volatilities of the two stock market indices. According to previous studies, the average test accuracies of the dynamic modeling methods using rolling time windows are all significantly higher than those of the static models [35,36]. Therefore, based on the above volatility synchronization indicator, we introduce the rolling time window technique to establish a dynamic volatility synchronization indicator:

(9)

R_t denotes the coefficient of determination (R²) computed from the regression of realized volatilities of stock indices within a rolling time window.

2.4 Parameter estimation method

There are many methods for estimating model parameters, such as the likelihood function [37–39], Bayesian estimation [40,41], and so on. The maximum likelihood function estimation method is one of the traditional and commonly used methods [37–39]. In this paper, in order to discuss the problem of parameter estimation for the model, we first based on a sample set:

(10)

To discretize the stochastic model, the Eq (5) is differenced as follows, The parameters involved in the formula have been defined and specifically introduced in Sect 2.3 above.

(11)

Then the likelihood function can be expressed as:

(12)

The final set of unknown parameters can be denoted as . The values of these parameters are obtained by maximizing the log-likelihood function under the conditions specified above. Where,

The expression

signifies that our objective is to maximize the log-likelihood function in order to obtain the optimal estimates for the parameter ϕ. Meanwhile denotes the probability density function of a multivariate Gaussian distribution, Next, we can utilize this distribution for parameter estimation.

(13)

(14)

The above two formulas respectively represent the general form of the log-likelihood function under the multivariate Gaussian distribution and its specific expansion under this model. The first formula provides the general expression of the log-likelihood, while the second formula fully expands it based on the model parameters and variables, clarifying the contribution of each parameter to the likelihood function. By maximizing this log-likelihood function, the parameter ϕ can be effectively estimated. Estimated parameters (e.g., ) are inferred via maximum likelihood estimation unless otherwise stated; observable quantities (e.g., x_i(t) and ) are derived from market data. In addition, the parameters can also be obtained by calculating the minimum variance between the model and actual data through simplex computation.

3.1 Rolling time window forecasting method

This paper further analyzes the rationality and predictive ability of the proposed model. For comparison, we also discuss the ARCH model, the GARCH (1,1) model, and the SV model. Based on the machine learning methods, out-of-sample daily volatility forecasting using rolling time windows is conducted for the coupled Heston model proposed in this paper and the three comparative models mentioned above. Subsequently, Constructing a volatility synchronization index based on dual-market volatility prediction sequences. We apply a standard machine learning framework in which data are divided into training and prediction (or test) samples, and the model is trained to forecast volatility synchronization. Our data ranges from January 2, 2003, to December 29, 2023, with 5100 realized volatility series data.

By dividing the data samples (t = 1, 2,..., N = 5100) into two parts, the estimation sample and the prediction sample[36,42–44].The estimation sample contains data for H = 1000 trading days and the prediction sample includes data for M = 4100 trading days after the estimation sample (i.e., t = H + 1, H + 2, ..., H + M). The forecast horizon is one-day-ahead. The prediction method is constructed as follows:This study adopts the rolling window dynamic prediction method for volatility prediction, with its core logic being updating samples through “fixed window + periodic sliding” to generate predictions. Specifically, the method takes as input a time-series data matrix X (e.g., daily stock returns of enterprises), an initial window size , and total forecast days M, and outputs a predicted volatility sequence with length M; in operation, it first initializes an empty vector to store results, then conducts M rounds of cyclic prediction. This method features no data leakage (only using historical data before the forecast date), dynamic adaptability (sliding window incorporating the latest information), and stability with flexibility.

3.2 Forecast evaluation

With the predicted series of volatility synchronization values for the coupled Heston model proposed in this paper, along with the three comparative models described above, we can assess the deviation of these forecast values from the standard of real market volatility estimation, denoted as S_m (where S_m represents the RV data) [45,46]. Currently, academics are not clear on which loss function is the most reasonable standard for measuring predictive deviation [36,42–44]. Therefore, Hansen and Lunde suggest that as many different forms of loss functions as possible can be used as criteria for judging the accuracy of forecasting models [47]. Based on this, this paper adopts five widely used statistical loss functions to evaluate the prediction accuracy of various types of volatility models separately.

The five loss functions are labeled as Loss_i (i = 1, 2,..., 5), among which Loss₁ and Loss₂ are respectively called the Mean Squared Error (MSE) and the Mean Absolute Error (MAE), which are the two most commonly used forms of loss functions in such judgment. Loss₃, Loss₄, and Loss₅ are specifically the Mean Absolute Percentage Error (MAPE), Mean Squared Percentage Error (MSPE), and Root Mean Squared Error (RMSE). The specific definitions of each type of loss function are as follows [36,48]:

Where M is the length of the forecast set. However, relying simply on some loss functions as the evaluation criteria for comparing the prediction accuracy of the models, may lead to conclusions that are not robust. Therefore, it is necessary to further enhance the reliability of the results through some statistical tests. Hansen and Lunde proposed a so-called “Superior Predictive Ability (SPA) test" [47], which employs a bootstrap method for simulation [42,44]. The SPA test has superior model discrimination capability than the similarity Reality Check (RC) test proposed by White [49], and the conclusions drawn from the SPA test are more robust. In other words, compared with other tests based on a single sample, the test conclusions obtained from the SPA test are more reliable, and its findings can be generalized to other similar data samples [50,51].

4.1 Data

The overall volatility risk of a stock market is typically reflected by the volatility of that stock market index. In this paper, the daily realized volatility (RV) series data calculated from 5-minute high-frequency data of two stock market indices, the Shanghai Securities Composite Index (SSEC) and the Shenzhen Securities Component Index (SZI), are proposed to be selected as the research samples. Moreover, the daily closing price and daily realized volatility data samples of these two stock market indices are sourced from the CSMAR database, covering the period from January 2, 2003, to December 29, 2023, resulting in a total of 5,100 trading days of data. In this case, the calculation method for the daily Realized Volatility in the CSMAR database refers to the literature of Wei, Li, and Chen, which defines it as the sum of the squared logarithmic returns of every 5-minute trading data [45,46].

The descriptive statistics of the log prices, log returns, and realized volatilities of the SSEC and the SZI are given in Table 1. From Table 1, it can be observed that the realized volatility series of the market indices exhibit characteristics of sharp peaks and thick tails, which indicates that the volatility of both indices is quite drastic and far beyond the range assumed by the normal distribution (the Jarque-Bera statistic is significant in both cases). Meanwhile, the logarithmic price x_i, logarithmic return r_i, and realized volatility series data demonstrate very significant autocorrelation characteristics over a very long time horizon, suggesting persistent volatility with long-memory characteristics. Further, the ADF unit root test shows that the original hypothesis of the existence of unit root is significantly rejected for each series. Therefore, it can be considered that the individual series are smooth and thus can be further analyzed and modeled.

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Table 1. Descriptive statistics of log prices (), log returns (), and realized volatility () of stock market indices.

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4.2 Analysis of volatility synchronization

To illustrate the volatility synchronization model of the stock market, this paper calculates the volatility synchronization values between the SSEC and the SZI over the full sample period (Sect 4.1), using Eqs (7) and (8). The actual synchronization value is found to be SYNCH = 2.1223.

Additionally, to explore the volatility synchronization across stock markets at multiple scales, we conduct dynamic volatility synchronization simulations for the SSEC and the SZI at four scales (60, 252, 500, and 1,000 days) using the estimated parameters of the coupled Heston model. Fig 1 presents the results, showing significant synchronization between the two markets across all scales—with the highest synchronization observed at the 60-day (short-term) scale.

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Fig 1. The dynamic volatility synchronization of the SSEC and the SZI at multiple scales (60, 252, 500, and 1000 days).

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4.3 Comparison of in-sample fitting results

This paper selects the index price and volatility real data of the SSEC and the SZI to estimate the parameters of the model. We set the objective function to be the sum of the average mean squared errors between the coupled Heston model estimation sequence and the actual sequence. Then, we assign initial values to the parameters ϕ of the coupled Heston model. Subsequently, we iteratively use the Nelder-Mead method, a numerical optimization algorithm, to adjust the model parameters in order to minimize the value of the objective function. Finally, we identify the set of parameters ϕ that minimizes the objective function. Through the above steps, the estimated values of the parameters for the equation are obtained as follows: , , , , , , , , .

Then, we combined the serial data of parameter value estimation and the real data to calculate the probability density functions of prices and realized volatilities, as shown in Figs 2 and 3. The comparison between the real data (squares) and the model results (solid lines) for the SSEC and the SZI is presented in the figures. Fig 2a and 2b show the probability density functions of the index price and realized volatility for the SSEC, while Fig 3c and 3d represent those for the SZI, respectively. It is evident that the probability density functions of index prices and realized volatilities simulated by the coupled Heston model are in good agreement with the actual data results, indicating a good fit for the simulation outcomes. Meanwhile, based on the realized volatility series data obtained from the simulation, the simulated result volatility synchronization value is obtained as SYNCH^model = 2.1063, and the real synchronization value is SYNCH^data = 2.1223, which is a result that further demonstrates the reliability of the model.

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Fig 2. (a) and (b) are the probability density functions of the SSEC price and realized volatility.

The squares denoting the real data and the solid lines indicating the model results. The parameter fitting results are: , , , , , , , , .

https://doi.org/10.1371/journal.pone.0334853.g002

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Fig 3. (c) and (d) are the probability density functions of the SZI price and realized volatility.

The squares denoting the real data and the solid lines indicating the model results. The parameter values are consistent with those in Fig 2.

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4.4 Comparison of out-sample dynamic predictions

Fig 4a shows the dynamic volatility synchronization prediction results of the coupled Heston stock price dynamics model mentioned in the paper for the prediction sample interval t = 501, 502,..., 5100 (represented by the solid line), while the dynamic volatility synchronization results computed based on the real data are represented by the red dashed line. Similarly, Fig 4b shows the dynamic volatility synchronization prediction results of the stock market for the predicted sample interval t = 1001, 1002,..., 5100. As can be seen from the comparison of the figures, the prediction results of the proposed model in this paper align well with the actual results under different prediction sample intervals.

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Fig 4. The out-of-sample rolling forecasting results of the coupled Heston model with different estimation sample intervals (H = 500 and 1000).

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To explore the out-of-sample volatility synchronization predictive performance of the models proposed in this paper, we conducted a comparison of their out-of-sample prediction capabilities. Table 2 gives the results of the volatility synchronization value prediction test of various volatility models under five loss functions, where the definitions of the five loss functions Loss_i are shown in Sect 3.2. The results from the table show that:

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Table 2. The loss function values for the synchronization prediction of various models.

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(i) The coupled Heston model attains the lowest loss across the evaluated loss functions and ranks first. This indicates that the model better predicts the dynamic volatility synchronization of the stock market, demonstrating superior synchronization value forecasting performance than other models.

(ii) As far as the comparison with other ARCH, GARCH, and SV models is concerned, the GARCH model is ranked second in terms of the loss functions MSE, MAPE, MSPE, and RMSE and has relatively good predictive performance, whereas the SV model achieves the lowest prediction accuracies across all loss functions.

However, further SPA testing of this prediction is necessary to obtain more robust and broadly applicable conclusions. The SPA test is a statistical method for comparing the predictive performance of multiple models, which evaluates whether a base model significantly outperforms a set of alternative models under a specified loss function.

Table 3 shows the SPA test results obtained after 10,000 bootstrap simulation processes. Column 1 of Table 3 represents the five loss functions Loss_i, while column 2 lists the name of the model selected as the base model ( respectively denote the coupled Heston model, ARCH model, GARCH model, and SV model).The numbers in the table represent the p-values of the SPA test. Specifically, under a certain loss function Loss_i judgment criterion, if the SPA test p-value of the base model is larger (closer to 1) relative to the other models, it indicates that the base model is the best-performing forecast model. Conversely, if the p-value is smaller, it suggests that the base model exhibits inferior predictive performance relative to the comparison models. The results from the table show that:

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Table 3. The SPA test values for predicting volatility synchronization for various models.

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(i) Using the SSEC and the SZI (representing two major stock markets in China) as examples, the SPA test results for the five loss functions all indicate that when the coupled Heston model proposed in this paper is used as the base model, the SPA test p-value is approximately equal to 1 relative to the other models. This suggests that the coupled Heston model, which considers that the correlated synchronization behavior between stock markets primarily originates from volatility, can better predict the dynamic volatility synchronization among the stock markets.

(ii) The SV model is the worst-performing forecasting model under almost all loss function criteria. Overall, the four models’ ability to predict dynamic volatility synchronization across stock markets ranks in the following order (from best to worst): the coupled Heston model outperforms the GARCH model, the GARCH model outperforms the ARCH model, and the ARCH model outperforms the SV model.

(iii) The conclusions drawn from the SPA test results in Table 3 are generally consistent with those obtained in Table 2, and thus these conclusions have considerable universality for the prediction of volatility synchronization among stock markets.

5.1 Synchronous correlation mechanism

In the previous sections, we conducted in-sample and out-of-sample empirical comparative analyses of the proposed method’s dynamic forecasting performance, using the Shanghai Composite Index and Shenzhen Component Index as examples, based on the coupled Heston model and its dynamic synchronization framework. The results demonstrate that the in-sample estimation of the proposed model aligns closely with market behavior and exhibits superior predictive performance, thereby addressing the research question of how to dynamically measure and forecast cross-market volatility synchronization. To explore the dynamical mechanisms underlying volatility synchronization, this paper will further investigate the dynamical mechanisms of volatility synchronization across markets by combining dynamic simulations and multivariate empirical mechanism analysis.

Upon scrutiny of the current model, it becomes evident that each synchronization metric derived from real data is intricately linked to a distinct set of estimated parameters within the framework of our proposed coupled model. The determination of each synchronization metric is contingent upon a spectrum of estimated parameters, including but not limited to , , and . This realization paves the way for a detailed comparative analysis between the synchronization metrics and the respective parameters of the coupled model.

Our investigation will focus on identifying any discernible patterns that may exist between the oscillations in synchronization metrics and the alterations in these parameters. The ultimate goal is to unravel the dynamical mechanisms at play, providing a deeper understanding of the system’s behavior. Of particular interest is the pronounced connection between the synchronization metrics and the parameter , as demonstrated by the following Eq (15). This relationship will be the focus of our examination, as we seek to elucidate the patterns of their fluctuations and impact on the dynamic mechanisms, thereby guiding our understanding of the synchronization process.

(15)

In this study, our analysis is based on comprehensive datasets comprising 4,101 synchronization metrics, as previously obtained, along with their corresponding optimal parameter set: , θ₁, , , , θ₂, , , and . A rolling window with an average span of 1,000 data points has been employed to smooth out short-term fluctuations and highlight longer-term trends. To ensure the date’s authenticity and reliability for our analysis, stringent filtering and normalization processes have been applied.

Through comparative analysis, we have identified a notable correlation between the synchronization metrics and the parameter . As illustrated in Fig 5, there is a striking similarity in the frequency characteristics between the synchronization metrics and the as proposed by our model. This is particularly evident in the synchronicity of peak occurrences, suggesting a dynamic interplay that warrants further investigation. Our findings underscore the significance of in understanding the underlying mechanisms that drive synchronization within the system.

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Fig 5. Comparison of synchronization metrics with frequencies.

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The synchronicity in the occurrence of peaks within two sets of data may indicate an underlying shared dynamical mechanism or interplay between them. Such synchronization holds substantial significance within dynamical systems, as it may unveil the intrinsic coupling behaviors within the system. This suggests an interdependent, mutually influential, and regulatory relationship between the synchronization metrics and the parameter . By delving into these studies, we can achieve a more profound comprehension of the internal generative mechanisms of synchronization metrics and how they respond to changes in external conditions. This understanding allows us to anticipate how synchronization metrics will evolve over time and adapt to various stimuli, providing a foundation for more effective management and decision-making in complex dynamic environments.

5.2 Multivariate analysis of synchronous dynamics mechanism

Table 4 provides descriptive statistical data for ten variables, including the mean, standard deviation (SD), first quartile (p25), median (p50, which is also the second quartile), and third quartile (p75). Based on this table, the following conclusions can be drawn: The data in the second row, namely the volatility synchronization metric, has a median (p50) of 0.906, which is close to the mean of 0.903, indicating that the volatility synchronization metric is generally high and the distribution is relatively symmetrical. Additionally, the first quartile (p25) and the third quartile (p75) are 0.860 and 0.957, respectively, showing the range of the middle of the synchronization metric data. The synchronization metric exhibits a certain standard deviation, suggesting that synchronization varies under different parameter conditions.

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Table 4. Descriptive statistics of the main variables.

https://doi.org/10.1371/journal.pone.0334853.t004

The data from rows 3 to 11 represent the main parameters of the coupling model, and the majority of them are very stable with minimal standard deviation, implying that these parameters exhibit consistent behavior within the studied system. The distribution of all data is close to a normal distribution, as the median is near the mean, and the interquartile range is relatively narrow. These parameters are used to describe the dynamical model, where the and σ parameters may be related to the system’s response characteristics. The stability of the θ parameters may indicate that their role in the model is fixed or unaffected by external conditions, while the ρ parameters may be associated with the strength of synchronization.

Based on the information mentioned above, we conducted a baseline regression analysis between the parameters in the coupling model and the synchronization metrics to further understand the impact of these model parameters on synchronization. The results are shown in Table 5, from which we can observe the following: and represent the mean reversion speeds of the volatility of stock price indices for two different stock markets. Their coefficients are −0.413 and −0.438, respectively, and are significant at the 0.05 significance level. This indicates that when the mean reversion speed of the volatility of the stock price indices in the stock markets increases, the synchronization between the two markets decreases. θ₁ and θ₂ indicate the long-term mean volatility of stock price indices for the two stock markets. The coefficient for θ₁ is −36.106, significant at the 0.1 significance level, while the coefficient for θ₂ is −11.265, which is not significant. This may imply that θ₁ has a negative impact on synchronization, but the effect of θ₂ is not significant.

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Table 5. Baseline regression results of the synchronization metric with the model parameters.

https://doi.org/10.1371/journal.pone.0334853.t005

and denote the fluctuations in volatility of two stock markets. The coefficient for is -3.327, significant at the 0.1 significance level, while the coefficient for is −3.096, which is not significant. This suggests that has a negative effect on synchronization, but the impact of is not significant. and reflect the correlation between the volatility and prices within each of the two stock markets. Their coefficients are 3.071 and 3.171, respectively, and are also significant at the 0.1 significance level. This indicates that when the correlation between volatility and prices within the stock markets is stronger, synchronization also increases. manifests the primary source of the correlated synchronous behavior between the two stock markets. Its coefficient is 3.167, significant at the 0.1 significance level. This suggests that the correlated synchronous behavior between the two stock markets has a positive effect on synchronization.

5.3 Identification of significant extreme events based on volatility synchronization

To further identify and detect significant risk events through extreme volatility synchronization, Fig 6 presents the identification chart of significant risk events in volatility synchronization on a 60-day scale. We can observe that after major risk events such as the “money crunch” in 2013, the stock market crash in 2015, the Trump administration’s announcement of continued tariff hikes on China in 2019, and the “COVID-19” pandemic in 2020, the peaks of dynamic volatility synchronization between the two stock markets, namely the Shanghai Composite Index and the Shenzhen Component Index, have identified these significant risk events.

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Fig 6. Volatility synchronization significant risk event identification chart.

The reference lines for the gray area are the mean ± 1 standard deviation.

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Meanwhile, the measured volatility synchronization values when these identified significant risk events occurred are all above the sum of the mean and the standard deviation. This indicates that the two markets experienced high-intensity volatility synchronization simultaneously, revealing the linkage effect of volatility synchronization triggered by risk synchronization, where the risk occurring in one market spreads to the other market.

In addition, after the implementation of the QDII (Qualified Domestic Institutional Investor) system in 2007, it can be found that the volatility synchronization between the two markets has significantly increased. This conclusion is consistent with the statement that measures such as the liberalization of QFII (Qualified Foreign Institutional Investors) and QDII quotas have increased the connectivity between China and international capital markets and enhanced the linkage of fluctuations in China’s stock market.

Therefore, the dynamic volatility synchronization index constructed based on the realized volatility in this paper can effectively measure the volatility synchronization between stock markets, and significant risk events can be further identified through the peaks of volatility synchronization.