1. Introduction
A proper risk assessment is one of the key prerequisites of any prospective financial investment. Even for an asset of moderate volatility, underestimating the probability of occurrence of an event of a given magnitude can lead to severe outcomes. Among the methods of dealing with risk assessment is the determination of a correct probability distribution function for asset price fluctuations in order to construct an adequate model of that asset’s price dynamics. This issue has been of central interest since the early years of econometrics. It was Bachelier who proposed a model of the stock option price dynamics based on an uncorrelated random walk with a Gaussian distribution of fluctuations [
1]. Later, it was found that the Gaussian noise hypothesis was only a poor approximation of the empirical data, which shows non-vanishing higher moments of the fluctuation distributions, i.e., skewness and positive excess kurtosis. Based on an observation of the cotton price dynamics, Mandelbrot proposed to model the logarithmic price increments (returns) with a process of Lévy flights, which is described by a heavy-tailed probability distribution function that is stable [
2,
3]. These distributions are defined by their characteristic function as they do not have a closed analytic form. However, their tails decrease as a power law in the limit of large
x:
where
.
According to Mandelbrot [
4], such a process can account for the absence of a convergence of the aggregated return distribution to the normal distribution as expected by the central limit theorem (CLT). The heavy tails are thus viewed as a natural limit of the aggregated independent or weakly dependent factors provided they are described by the stable distributions. However, this hypothesis has a weak point because the empirical data cannot exhibit the infinite variance required to maintain the distribution stability under aggregation. After the pioneering work of Mandelbrot, many researchers investigated financial time series in order to verify his outcomes. For example, Fama reported that the daily returns of stocks are better approximated by the infinite-variance distribution than the normal distribution or a mixture of the normal distributions [
5]. The Lévy stability of the return pdfs in their central parts was also confirmed, among others, by Blume (
) [
6], Teichmoeller (
) [
7], and Blattberg and Gonedes (
) [
8]. Some reports pointed out that, although central parts of the return distribution can be approximated by the stable distributions, the same cannot be said about the distant parts of their tails, which decay faster than expected. Officer found that the tails of the daily and monthly return distributions are no doubt thicker than Gaussian but at the same time thinner than Lévy-stable [
9]. Barnea et al. observed that the daily return distributions for some stocks are well approximated by stable distributions, while for other stocks, they are not [
10]. Much later, Young and Graff reported that the real-estate annual return distribution can be fitted by a stable function using
[
11].
Along with the research on empirical data, much effort was devoted to developing models that could mimic the market dynamics. Among such models, the subordinated stochastic processes do not require an assumption of the Lévy-stable character of the underlying dynamics and assume that the price movement is a Brownian motion that takes place in time, which itself is a stochastic process with positive increments and finite variance (e.g., a lognormal process) [
12]. In practice, the subordinating process is assumed to be volume or transaction number. As an alternative, Engle proposed that the distribution tails are heavy because of the heteroskedasticity of the return-generating process, in which large returns are caused by a locally large variance of the process [
13]. Mantegna and Stanley found a dual structure of the stock index return distribution (S&P500 index during the years 1984–1989), with its central part being in agreement with a Lévy-stable distribution and with exponentially decaying distant tails [
14]:
While considering the aggregated returns at different time horizons, they did not find any trace of a convergence to the normal distribution. Based on these findings, they proposed a new model for the price return dynamics: a truncated Lévy flight process. They also showed that the heteroskedastic model (GARCH) does not fit the data well [
14]. This type of distribution (
1.6–1.7) was also reported from an analysis of the same S&P500 index recorded over a longer interval (1986–2000). In contrast, the aggregated returns showed a crossover to a CLT regime around a time scale of 20 days [
15].
Plerou et al. and Gopikrishnan et al. presented two parallel, comprehensive studies of the stock market high-frequency data representing stock price returns for 1000 American companies and S&P500 index returns [
16,
17], in which they observed the cumulative distribution function tails obtained from aggregated returns over a substantial spectrum of time scales from 5 min (stocks) and 1 min (index) to 4 years. They found that the return distributions have power law tails, with the exponent
depending on a stock. However, despite the fact that they did not fit the Lévy-stable domain (
), these distributions were invariant under a change in the time scale up to
days. Only for the sampling intervals longer than 16 days, a slow transition to a normal distribution was observed [
16]. An analogous invariance of the return distribution shape with the power exponent
under the time-scale change was observed for the S&P500 index, but in that case, the crossover occurred earlier at
days. Only for the time scales longer than 4 days, a slow convergence to a Gaussian distribution was seen. A similar behavior was found in the indices from other stock markets (Nikkei & Hang-Seng) [
17]. This surprising behavior of the stock markets led the authors to formulate the so-called “inverse cubic law”—a conjecture that the power-law tails of the return distributions with the scaling exponent
are a universal property of all stock markets at short and medium time scales [
18]. Indeed, similar statistical characteristics were found by other researchers in data collected from other stock markets [
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35], Forex [
36], commodity markets [
36,
37], and the cryptocurrency market [
36,
38,
39,
40].
The only possible explanation of this result is that the analyzed data violated the assumptions of the central limit theorem, i.e., the returns were significantly correlated. Indeed, the cross-correlations among the stock returns representing different companies are an obvious characteristics of all stock markets [
34,
41,
42,
43,
44,
45]. It was shown that the inter-stock cross-correlation strength has a strong impact on the index return distributions and can even modify their tail behavior, leading to a kind of alternation between different power-law regimes: stable and unstable [
43]. On the other hand, the cross-correlations between different stock markets can also induce a significant regime change [
21,
22]. The existence of autocorrelation in returns is a more delicate issue: while the returns reveal some short-term memory lasting for a few minutes, the existence of long-term memory is doubtful [
16,
17,
24,
46,
47], even though there were reports stating that the returns can show some autocorrelation or persistence over long terms [
48,
49,
50,
51,
52]. On the other hand, there is consensus over a fact that the long-term autocorrelation is present in absolute returns (volatility) and in some more fundamental observables such as fluctuations in stock market orders, transaction size, and market liquidity [
53,
54,
55]. The existence of a return autocorrelation can be considered an important factor that can destroy market efficiency [
56,
57]. These ubiquitous manifestations of the inverse cubic scaling in the financial data encouraged Gabaix et al. to propose a model that was able to account for this phenomenon [
58]. According to this model, the inverse-cubic return fluctuations were a result of two processes: the volume fluctuation that forms a probability distribution function with the tail index 1.5 and a specific square-root form of the price impact function, which together produce a tail index equal to 3 [
58]. However, Farmer and Lillo pointed out that the price impact function is specific to individual markets and even to individual stocks; thus, it cannot produce any universal behavior. Also, the dependence on transactions is slower than the square root and the volumes are not power-law distributed, so they cannot lead to a power-law behavior of the returns with
[
59]. The price changes are driven by more factors than simply volume and transaction number fluctuations—it can be the order book structure, for example [
60,
61]. Moreover, there is plenty of published evidence that various financial assets either do not have the power-law distribution tails [
29,
62,
63,
64,
65,
66,
67] or their scaling exponent
differs from 3 even for the short time scales [
36,
68,
69,
70,
71,
72,
73]. Given these results together, the inverse cubic scaling cannot be considered a universal property of financial returns and, thus, cannot be called “a law”. However, it manifests itself sufficiently often to allow us to view it as one of the possible reference models describing the empirical return distribution tails (there is a plethora of volatility models, which takes into account various factors; a review of such models can be found in Poon and Granger [
74]).
The power law tails of the return distributions, which are among the financial stylized facts, can be reproduced with a broad range of the scaling exponent by means of various models based on stochastic processes [
63,
75,
76,
77,
78,
79,
80,
81,
82,
83], including multiplicative processes [
84,
85,
86], the minority game and other agent-related dynamics [
87,
88,
89,
90,
91], as well as spin dynamics [
23,
92].
Apart from power-law functions, the tail behavior of the return distributions can also be approximated by exponential functions and stretched exponential functions [
93]. The latter are defined by the following expression:
Such a functional form allows for the stretched exponents to locally resemble the power laws. There were many published studies in which the return distributions were approximated successfully by the exponents, and some researchers advocate using these functions instead of the power laws [
23,
29,
31,
63,
64,
65,
66,
67,
69,
80,
94,
95]. Another type of exponential function that is sometimes considered in the context of financial data is the Laplace distribution function
. This function can also demonstrate heavy tails. It was observed that some empirical return distributions can be approximated by this function [
62,
96].
The functions that have been discussed so far do not exhaust the possible models that can be used to approximate the empirical return distributions. In a financial context, a particularly important class is the
q-Gaussian functions. They were derived as a part of the formalism of nonextensive statistical mechanics based on the Tsallis nonadditive entropy [
97]:
where
is some probability distribution and
is a positive constant. Under certain conditions, this entropy is maximized by a family of
q-Gaussian distributions given by
where
provided that
and that
and
are
q-mean and
q-variance, respectively. The
q-Gaussians generalize both the normal distribution (
) and the Lévy distributions (
). Their attractiveness comes from the fact that, for the correlated random variables, the
q-Gaussians become stable distributions. Moreover, their tail behavior can also resemble the power laws [
98]. As the price returns are correlated, one can expect that these functions can describe the statistical properties of returns. Indeed, there is a growing evidence that the
q-Gaussian distributions can approximate the empirical return distributions [
30,
32,
66,
99,
100,
101].
The
q-Gaussians are among the functions borrowed from the nonextensive statistical mechanics that were exploited in this context. Another example is the
q-exponent given by Equation (
6), which was also reported to fit the empirical returns from a stock market [
102]. Finally, some researchers consider the normal-inverse Gaussian function to be a prospective model that can successfully be fitted to the data [
71].
This short review of the return distribution modeling approaches shows that there is a cornucopia of the reported results that were even contradictory sometimes. The only firm observation that is shared by all the studies is that the return distributions reveal heavy tails, at least at short time scales. On medium and long time scales, the situation depends strongly on a data set, a market, and a financial instrument. Drożdż et al. attempted to resolve this problem by noticing that the most well-known results regarding the return distributions, i.e., Mandelbrot’s Lévy stability (
) [
4]; Mantegna and Stanley’s truncated Lévy flights [
14]; Plerou and Gopikrishnan’s unstable power-law tails (
), which are persistent under aggregation of the returns until the time scales of days or even a month [
16,
17]; and their own results with the
regime already breaking at the time scale of hours [
24], were based on the data covering different epochs: 1816–1958 (Mandelbrot), 1984–1989 (Mantegna), 1926–1995 (Plerou and Gopikrishnan), and 1998–1999 (Drożdż). One can follow the whole historical process of the financial market development, introduction of new financial instruments, technological innovations, transition from the classic “floor-based” markets to the digital markets, computing power increase, telecommunication revolution, etc. from past to present. This inevitably leads to the constantly increasing number of investors, transactions, and pieces of information that arrive at the market. These are accompanied by the increasing amount of money and information processing speed, which, if taken together, result in an overall acceleration of the market time flow. Any unit of time nowadays corresponds to a much longer interval in the past. From this perspective, the market properties once observed, say, at a daily scale, now can be observed at scales of hours, minutes, or even seconds. This may be the very reason why Mandelbrot observed the Lévy-stable distributions that are hardly seen today and why Plerou and Gopikrishnan reported the crossover to the CLT-related convergence of distribution tails at the time scale of many days, while today, such a behavior is observed within hours or minutes. This hypothesis formulated by Drożdż et al. was later supported by other analyses as well [
24,
30,
32,
69,
72].
However, based on data covering a given time interval, one can observe an analogous phenomenon by considering, e.g., the stocks representing companies with different capitalization. Since there is statistically a relation between the capitalization of a company and the number of transactions involving its stock shares, the highly capitalized stocks “feel” that time flows faster than their lower-capitalized counterparts. In consequence, the properties of the corresponding return distributions substantially differ between both groups, with the former displaying thicker tails than the latter [
24,
34,
35,
100,
103]. Qualitatively similar observation can be made by comparing the distributions for the data from the markets of different developmental stage, e.g., the mature markets and the emerging markets. The former are characterized by higher liquidity and a higher transaction number than the latter; therefore, generally, the situation is parallel to the previous cases. Studies of the data from the emerging markets report thick tails with small scaling exponents more frequently than the mature markets [
25,
26,
28,
52,
66,
94,
104,
105,
106,
107,
108,
109,
110].
Another issue related to return distributions is their asymmetry between positive and negative parts. It was investigated in various works as it is also an important factor in investment risk assessments (the gain–loss asymmetry). Typically, this property was tested by means of the third moment (skewness) of return distributions, in which a negative value means a higher probability of a significant gain with respect to a significant loss while a positive value means the opposite. The negative skewness is associated, thus, with a positive tail of the distribution being heavier than the negative tail. There are mixed outcomes of the empirical data reported in the literature, including indications of either positive, negative, or neutral skewness as well as the scaling exponent difference between the left and the right tails (in the case of power-law tails) dependent on the analyzed time intervals, markets, and securities (e.g., References [
14,
16,
17,
20,
28,
31,
36,
62,
71,
94,
111,
112,
113,
114,
115,
116,
117,
118,
119]). However, even though a difference between the positive and negative tails exists in the data, it has a much weaker impact on the distribution shape and the related investment risk than the heavy tails. Therefore, in many studies reported in the literature, only absolute returns are considered, neglecting their actual signs (e.g., References [
16,
17,
24,
30,
32,
39]). As our study is focused on an investigation of the tail exponent stability with respect to the time scale
and, based on literature and our previous experience, we expect larger effects due to the time-scale change than due to the left–right tail asymmetry, we neglect the return sign and consider both tails together by analyzing the absolute return values. In fact, our major new finding is that, in recent years, the market’s “internal” time stopped accelerating with respect to our ordinary “clock” time. Other factors also affect the convergence of return distributions to the Gaussian with increasing
, especially those that cause extreme volatility and strong cross-correlations between assets such as COVID-19. We discuss the interplay of these two factors in the following sections.
The remainder of our paper is organized as follows: in
Section 2, we present the data sets that were analyzed; in
Section 3, we discuss the results; and in
Section 4, we collect the main conclusions of our study.
4. Summary
In this study, we analyzed high-frequency quotations of the CFD contracts associated with the stock market indices, the stocks themselves, and the selected commodities as well as with the most frequently traded currency exchange rates and the cryptocurrency prices. All of the data sets covered the years 2017–2020 except for the stock share CFDs, which covered the years 2018–2020. We analyzed the returns at a few different time scales from 1 s to 1 h and constructed the return distributions in order to investigate their tails. Our principal objective was to compare the tail behavior of the distributions derived from contemporary data with the behavior of the distribution tails in the past for the same assets. We applied the power-law function and the stretched exponential one to model the empirical distributions. A hypothesis that we planned to verify was the one formulated in [
24,
32,
34], which states that, together with the acceleration of the information flow and processing across the financial markets, we can observe a significant change in the statistical properties of the returns at a particular time scale related to an effective acceleration of the market time with all of the possible consequences of this fact.
The results are mixed. On the one hand, the stock market indices (DJIA, DAX30, and S&P500, for which the present results can be compared directly with earlier works) do not show any further signatures of the time acceleration compared with the data from 1998–1999 and 2004–2006. It seems that the acceleration that was reported in [
24,
32] stopped or was only a temporary effect. Such effects were already reported before for Asian markets [
35,
69] as well as in this work regarding the stocks, so they may be a source of the observed behavior. On the other hand, the results for the individual stock groups show that the market time acceleration can still be ongoing, but it is masked at the level of indices owing to the cross-correlations among the stocks that are now stronger and developing faster than even during the years 2004–2006 [
32]. That particular time interval (2004–2006) was characterized by a volatility much smaller than in recent years, which witnessed large market events such as the flash crash on 5 February 2018, the coronavirus-related unsteadiness in early 2020 and the subsequent rally ending with new record highs of S&P500 in August, the oil price drop in April 2020, etc. Large events, especially large falls, elevate the market correlation level, which can influence the statistical properties of data, including the distribution of returns. The auto- and cross-correlations are involved in an interesting interplay between two opposite-acting factors. The first factor is the market time flow speed, which works for market efficiency by shortening the period when the market autocorrelations are admissible. This factor shifts gradually the low-
behavior and the central limit theorem’s realm to ever shorter time scales. The second factor is the asset cross-correlation strength, which causes thickening of the tails and decreases in
and
. It also violates the assumption of random variable independence and prevents the CLT from affecting the aggregated returns. This interplay and its consequences are interesting enough to be worthy of some more attention in future analyses. In particular, they can be responsible for the reported behavior of the return distributions in different time periods and suppressing the effects of the market time acceleration.
Currency exchange rates also no longer feel the market time acceleration such as that during 2004–2006 [
32], but now, not only is there no further time scale shortening but also a moderate step backwards is observed: the inverse cubic scaling is seen at longer time scales than in 2004–2006 but is still significantly shorter than that during the years 1987–1993 [
116]. The cryptocurrencies (BTC and ETH) show the same crossover scale as before—equal to 1 h [
131]. Since this market is relatively young, it underwent a phase of strong market time acceleration after 2013, and now, it seems to be stabilized. It is still the market that shows the most exemplary inverse cubic scaling behavior across different scales out of all the markets analyzed in this work. Gold price CFDs show a clear difference between the present results and the distribution tails over the years 1969–1999 [
37] and 2012–2018 [
36] with increased tail slope during the recent years. In contrast, there is no clear change in the tail slope regarding silver, high-grade copper, and crude oil.
It should be noted, however, that the CFD contract price quotations analyzed here are not precisely the same as the related asset spot price quotations, which the authors of other works dealt with. This difference may partially account for the difference in the outcomes. Finally, the COVID-19 pandemic outburst that took place in March–April 2020 in the U.S. constituted a strong perturbation to all the markets, caused large-amplitude price fluctuations, and led to a strong increase in the cross-correlations among many assets. For example, it resulted in decreasing distribution tail slopes for the CFD returns for crude oil and gold. Even more significant were the bitcoin fluctuations, which become Lévy stable for the pandemic-outburst period.
In general, our results indicate that the monotonous shift in the time scales at which different types of dynamics can be observed in the financial data as well as the related continuously accelerating market time from past to present are oversimplified. In fact, there can be an underlying long-term trend of this type, but it is “decorated” with short-term phases of abrupt acceleration and, then, deceleration and stagnation. Our results indicate that the real market dynamics consists of continuous alternation of different regimes with different statistical properties that can form the overall impression of the market evolution direction. Together with the aforementioned problem of how the asset cross-correlations and the shortening autocorrelations compete against each other in shaping the statistical properties of data, it opens an intriguing direction for future work.