**3. Results**

For each individual time series of the absolute normalized returns, we created a cumulative distribution function and investigated how fast its tail decays. In order to quantify the tail behavior, we fit the empirical histograms with selected models that are of the highest significance in this context: the power-law function, the stretched exponential function, and the *q*-Gaussian function. Figure 3 shows sample return distributions with the three best-fitted models of interest. We refer the reader to the specific subsections for a discussion on the asset statistical properties; here, we consider only the fits. In each panel, it is evident that the power-law function (dashed line) is able to reproduce the empirical histograms in their far-tail region while it fails to describe the central part of the distributions completely. The stretched exponential and *q*-Gaussian functions perform much better in the central parts, while only the latter works well in the tails. However, as the *q*-Gaussian and power-law functions converge to the same behavior in the tail regions and as both the parameters *α* and *q* are related with each other via a relation,

$$q = \frac{3+\mathfrak{a}}{1+\mathfrak{a}'} \tag{8}$$

henceforth, we omit the *q*-Gaussian fit parameter *q* and explicitly give the fitted values of *α* and *β* only. For simplicity, we also omit the acronym "CFD" and use the asset names only, but one has to realize that the CFD contracts and the assets they refer to are not the same financial entities and that the statistical properties of the former may not necessarily reflect the properties of the latter.

**Figure 3.** The least-square best fits of the power-law function (red dashed), the stretched exponential (green dotted), and the *q*-Gaussian function (blue dash-dotted). Sample cumulative distribution functions of the returns for the EUR/USD exchange rate (**top**) and the S&P500 index CFDs (**bottom**) are shown with different sampling intervals Δ*t* from 1 s to 1 h.

#### *3.1. Stock Market Indices*

Let us start with the cumulative return distributions of the stock market index CFDs representing six principal indices, NASDAQ100, DJIA, S&P500, DAX30, FTSE100, and CAC40, and the five time scales, 1 s, 10 s, 1 min, 10 min, and 1 h. Figure 4 shows that the return distribution for the three U.S. indices (NASDAQ100, S&P500, and DJIA) does not show an inverse cubic decay. Dow Jones is the closest, but this may be due to the fact that this index has the least number of aggregated stocks (30 vs. 100 and 500). As the time scale Δ*t* increases, we observe a gradual decrease in the thickness of the distribution tails, but this decrease is not so large that a convergence to the normal distribution could firmly be involved. The best power-law fits for Δ*t* = 1 s are *α* ≈ 3.9 (S&P500), 3.8 (NASDAQ100), and 3.6 (DJIA). For a complete record of the fitted power-law and stretched exponential function parameters, see Table 1. For longer time scales, the tails appear to be significantly thinner only for NASDAQ100, and for Δ*t* = 1 h, they reach *α* ≈ 4.6. For DJIA and S&P500, we do not observe any convergence to the normal distribution, and therefore, we assume that there is no such convergence for the scales up to 1 h. A strong discrepancy between the inverse cubic and the empirical distribution is also visible in the case of DAX30. For the returns with Δ*t* = 1 s, we obtain the power law with *α* ≈ 3.5, and for the higher scales, we have a trace of *α* → 4; however, this is by no means a monotonous increase.



The return distribution for the FTSE100 and CAC40 indices are different. Especially in the case of the latter, we observe an approximate inverse cubic decay *α* ≈ 3 for Δ*t* = 1 s. It is also clearer than in the previous cases that the tails become much thinner with increasing scale, and for Δ*t* = 1 h, we see *α* → 5. In the case of FTSE100, we do not deal with a homogeneous distribution but, rather, with two or more different distributions imposed. This is visible especially for the shortest time scale, where *α* < 3. As the scale increases, we see a behavior similar to that of CAC40, although it is even more pronounced due to the thicker tail at 1 s.

These results can be compared to those obtained for the high-frequency data from 1998–1999, which included both DJIA and DAX30 [24]. The distributions for the shortest scale analyzed (Δ*t* = 5 min) displayed tails close to those of the inverse cubic ones (even more in the case of DJIA than DAX30), but a crossover was visible for the scales Δ*t* > 2 h for DJIA and Δ*t* > 30 min for DAX30. Due to the limited maximum scale considered in the present study, we cannot conclude what the DJIA distributions for the 2 h scale look like, but it seems that, for the shorter scales, these distributions are slightly thinner than before. With regards to the results for the S&P500 and DAX30 data from the years 2004–2006 [32], the tail slope decrease was power-law starting from *α* ≈ 4 for Δ*t* = 1 min to *α* ≈ 6 for Δ*t* = 1 h for the American index and from *α* ≈ 3.5 to *α* ≈ 5 for the German index, respectively. These results differ from what we obtained here for the years 2017–2020. It seems that the tendency of the inverse cubic scaling regime to shift towards shorter time scales has at least stopped. The distribution tails also scale worse now than before. However, one has to notice that the years 2004–2006 were characterized by much lower volatility than the years 2017–2020, with a lack of comparably significant, dramatic events, which can have some impact on the results.

**Figure 4.** Cumulative distribution functions of the CFD returns for stock market indices NQ100 (NASDAQ), DJIA (New York SE), S&P500 (both New York SE and NASDAQ), DAX30 (Deutsche Börse), FTSE100 (London SE), and CAC40 (Euronext). Different sampling intervals (time scales) are shown from 1 s to 1 h. The inverse cubic scaling *α* = 3 (dashed line) and the stretched exponential with *β* = 0.5 (dotted line) are shown in each panel to serve as a guide.

#### *3.2. Individual Stocks*

The return distributions for all individual stocks collected from four mature markets: the U.S., German, British, and French ones are shown in Figure 5. For the shortest time scale analyzed, three markets display approximate inverse cubic scaling of their tails: *α* ≈ 3.2 (U.K. and France) and *α* ≈ 3.3 (Germany), while the U.S. market shows a larger exponent: *α* ≈ 3.6 (see Table 2). With increasing Δ*t*, the distribution tail becomes thinner, and already for Δ*t* = 10 s, the exponent reaches ≈3.5 (the European markets) and ≈4.0 (the U.S. market). This seems to be the quickest departure from the *α* ≈ 3 behavior observed so far for individual stocks. The scaling index increases gradually up to Δ*t* = 10 min, but for 1 h, this picture is altered and only the U.S. stocks show a further increase (*α* ≈ 5.0), while the exponent either stops—Germany—or even decreases—the U.K. and France (for these two longest scales, the stretched exponential function fits the empirical distribution better). This makes the situation less clear, but such a non-monotonous behavior was also

observed for some scales in [16] despite a much larger set of stocks considered there (1000). In that study (the years 1994–1995), *α* ≈ 5.0 was observed for the returns sampled every 50–70 trading days. Later studies reported that the scaling regime with *α* ≈ 3 already broke at Δ*t* = 2 h for 30 DJIA stocks and at Δ*t* = 5 min for 30 DAX stocks [24] (1998–1999) and then that *α* ≈ 3 was valid up to Δ*t* = 1 min and *α* ≈ 5 was reached for Δ*t* = 2 h [32] (1000 U.S. stocks, 1998–1999). In fact, even though in Table 2 we do not observe a convincing convergence of the empirical distributions for the European stocks towards the normal distribution, our results show that contemporary stocks experienced an accelerated time flow compared with the past. Of course, since we analyzed the CFD contracts instead of the stock share spot quotations as in [16,24,32], we have to be careful in drawing decisive conclusions from the comparison between these two asset types.

**Figure 5.** Cumulative distribution functions of the CFD returns for the stock shares representing different markets: the U.S. market (USA), the German market (GER), the U.K. market (UK), and the French market (FR). In each case, the aggregated distributions for 30 stocks with the largest capitalization are shown. Different time scales are shown from 1 s to 1 h. The inverse cubic scaling *α* = 3 (dashed line) and the stretched exponential with *β* = 0.5 (dotted line) are shown in each panel to serve as a guide.

The faster convergence of aggregated returns nowadays, compared with a more or less distant past, is among others a consequence of a decreasing autocorrelation time [16,24,32]. On the other hand, somehow, an opposite process is the increase in the cross-correlation magnitude among different stocks, which leads to stronger violation of the CLT assumption about random variable independence and thickening of distribution tails for the stock market indices, which can cause a later crossover to the CLT regime. For the shortest scales available for analysis of the order of an inter-trade interval, the cross-correlations are relatively weak due to strong noise and a longer time needed for information to spread over

the market. However, by increasing Δ*t*, we also increase the cross-correlation magnitude, which can eventually reach a saturation level with a magnitude dependent on the stocks considered (the same industrial sector vs. different sectors, whether the stocks are included in the same index, etc.) [34,41,44,124]. If we review the available results on this problem, we can see that, in 1971, the saturation of the cross-correlation coefficient for the stocks of the largest capitalization was reached at Δ*t* ≈ 1 day [41], while it was 1/2 h for 1998–1999 for the largest companies and a few hours for the medium-sized companies [44]. In order to learn how much time is needed for the cross-correlation magnitude to saturate nowadays, we calculated the Pearson cross-correlation coefficients *Cij*(Δ*t*) (*i*, *j* = 1, ... , 30) for all pairs of stocks within each of the markets studied here. Figure 6 shows the results for the mean coefficient *Cij* together with a mean length of the zero-return sequences in the analyzed time series and the largest eigenvalue of the 30 × 30 correlation matrix **C**(Δ*t*) in which the elements are *Cij* for a given Δ*t* and a given market. We also added two sets of U.S. stocks that represent medium-sized and small capitalization stocks. For all sets of stocks, a trace of saturation is observed already at the time scales of a few minutes, which is much less than the numbers presented above that from earlier works. This validates our statement that the market time "felt" by the assets accelerates. There is also a clear dependence of the mean cross-correlation coefficient on stock capitalization: the larger the capitalization, the stronger the correlation Figure 6.

**Table 2.** Estimated tail exponent *α* for the aggregated distributions of the CFD returns for 30 U.S. stocks with the largest, medium, and small capitalizations and 30 stocks representing selected European markets.


It is well-known that the cross-correlations are not stationary and that they strongly fluctuate across time [34,125,126]. Figure 7 displays the evolution of *Cij*(*t*) calculated in 30-day windows over the years 2018–2020. Two time scales are considered: 1 s and 1 h. *Cij*(*t*) fluctuates with a larger amplitude for Δ*t* = 1 h than for Δ*t* = 1 s. One of the periods associated with the largest values of *Cij*(*t*) is 9–27 March 2020 (the COVID-19 pandemic outburst in the U.S.), when the markets underwent strong turbulence [39]. As the cross-correlations were particularly strong during that period, we suspect that it could contribute substantially to the tail shape of the stock return distributions.

To verify this hypothesis, we removed this period from the time series and constructed artificial stock indices by aggregating the returns for all stocks belonging to the same set. Figure 8 shows both the complete distributions and the resultant no-COVID ones. After removing the COVID-19 outburst period, the distribution tails became substantially thinner, which is particularly evident for Δ*t* = 1 h (see Table 3). This supports our hypothesis that strong cross-correlations among the stocks can prevent stock indices from showing CLT convergence for short time scales. In this case, the stretched exponential function fits the empirical distribution better than the power-law function (Table 3). The numbers in this table illustrate how the stock-stock correlation strength can influence the stock index returns. While the stretching parameter *β* is comparable for each group of the U.S. stocks

at Δ*t* = 1 s, it becomes significantly different at Δ*t* = 1 h, where the medium and small companies have thinner tails than the large companies. This is because the former are less cross-correlated than the latter and the distributions can more easily converge towards a Gaussian in this case, even though the medium and small companies should experience a slower time flow than the large ones, which acts towards tail thickening. From this example, we can see that both effects compete against each other and that the actual tail behavior depends on the interplay of both factors.

**Figure 6.** (**Top**) The mean Pearson cross-correlation coefficient *Cij*(Δ*t*) for the CFD returns as a function of time scale Δ*t* for 30 companies, with the largest capitalization representing four stock markets, French (FR), German (GER), British (UK), and American (US), and for 30 companies with medium and small capitalization from the American market. Averaging was carried out over all pairs *i*, *j* with *i* > *j* and *i*, *j* = 1, ... , 30. (**Middle**) The same was performed as above, but here, the zero returns were filtered out before calculating the correlation coefficients. (**Bottom**) The largest eigenvalue of the correlation matrix **<sup>C</sup>**(*s*) constructed from the Pearson cross-correlation coefficients *Cij*(*s*) for the same sets of stock share CFDs.

**Figure 7.** Evolution of the mean Pearson cross-correlation coefficient *Cij*(Δ*t*)(*t*) for the CFD returns of 30 companies, with the largest capitalization representing four stock markets, French (FR), German (GER), British (UK), and American (US), and for 30 companies with medium and small capitalization from the American market. The coefficient was calculated in a moving window of length of 30 days, and averaging was carried out over all pairs *i*, *j* with *i* > *j* and *i*, *j* = 1, ... , 30. Two time scales with Δ*t* = 1 s and Δ*t* = 1 h are shown in each case.



**Figure 8.** Cumulative distribution functions of the returns of an artificial index constructed as a sum of the stock share CFD quotes *I*(*t*) = ∑*i Pi*(*t*) for the 30 largest companies representing four stock markets, the U.S. market (USA), the German market (GER), the U.K. market (UK), and the French market (FR). Two time scales with Δ*t* = 1 s and Δ*t* = 1 h are shown and denoted by solid lines. In addition, the analogous distributions constructed from the CFD return time series after removing the COVID-19 outburst period corresponding to the strongest cross-correlations among the stock shares (9–27 March 2020) are denoted by dashed lines. The inverse cubic scaling *α* = 3 (dashed line) and the stretched exponential with *β* = 0.5 (dotted line) are shown in each panel to serve as a guide to the eye.
