*3.4. Cryptocurrencies*

The cryptocurrency market is strongly related to Forex and its significance has risen steadily since its beginning [38,39]. The most important assets traded on this market in terms of their capitalization and volume are bitcoin (BTC) and ethereum (ETH). Their return distributions are shown in Figure 10. For Δ*t* up to 10 min, the power-law function approximates the data well, with the tail exponent displaying the same inverse cubic scaling for BTC and ETH: *α* ≈ 2.8 (1 s), *α* ≈ 3.1 (10 s), *α* ≈ 3.2 (1 min), *α* ≈ 3.3 (10 min). In contrast, for Δ*t* = 1 h, the crossover is observed and the tail exponent rises to *α* ≈ 3.7 for BTC and to 4.2 for ETH. (The stretched exponential function does not fit the data in the tail region, except on the 1 h scale for ETH) A good agreemen<sup>t</sup> between the empirical distributions of the cryptocurrency price returns expressed in major regular currencies and the inverse cubic scaling paradigm has already been reported [38–40,128], and it was interpreted as a sign of maturation (it used to be even more heavy-tailed with *α* ≈ 2.2 before 2014 [38,73]). A crossover for the same scale Δ*t* = 1 h has also been reported [128].

A more time-resolution-oriented analysis [129] showed that the BTC dynamics can actually be compounded with alternating phases of fluctuations with different statistical properties. For example, during the COVID-19 outburst in March 2020, the BTC returns were characterized by *α* ≈ 1.8, which corresponds to the Lévy stability, but typically, the scaling index resided between 2.0 and 3.5 during the years 2019–2020 [129]. In contrast, the Lévy-stable distribution of daily returns with *α* ≈ 1.3 was reported to fit the BTC empirical data (the years 2011–2017) in [130], but there, the model was fitted to the whole distributions, not only the tails, and the analyzed period also covered the early years of the cryptocurrency market when it was immature. These may be a source of the discrepancy mentioned above.

**Figure 10.** Cumulative distribution functions of the bitcoin-dollar exchange rate returns (BTC/USD) and the ethereumdollar exchange rate returns (ETH/USD). 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.

### *3.5. Commodities*

Figure 11 shows the return distributions for the commodity CFDs (see also Table 5). Out of the commodities considered in our study, gold (XAU) has the strongest decay with increasing Δ*t*: *α* ≈ 2.6 (1 s), *α* ≈ 3.1 (10 s), *α* ≈ 3.6 (1 min), *α* ≈ 3.6 (10 min), and *α* ≈ 4.2 (1 h). This has to be compared to *α* ≈ 2.5 for daily returns covering the years 1969-1999 [37] and 5-min returns covering the years 2012–2018 [36]. It is instructive to address why the tail for Δ*t* = 1 s is so thick that *α* falls significantly below 3. We therefore identified a period of the largest gold price fluctuations, which occurred at the COVID-19 outburst in the U.S. during March and April, and removed it from the time series. The resulting distributions are shown in Figure 12 (left panel) for Δ*t* = 1 s, 1 min, and 1 h (dashed lines) together with the complete ones (solid lines). It is evident that, for Δ*t* = 1 s, the thickest part of the tail becomes thinner *α* ≈ 3.2, while no significant alternation is observed for Δ*t* = 1 min (*α* ≈ 3.6) and 1 h (*α* ≈ 4.2). The distribution tails roughly agree in this case with the inverse cubic scaling on time scales that become increasingly short with time.

Compared to gold, silver (XAG) shows stronger invariance under the time scale change: *α* increases from 2.8 (1 s) to 3.0 (1 h), while it was 2.5 (positive tail) and 2.8 (negative tail) for the daily returns over the years 1969–1999 [37]. High-grade copper return distributions (HG) reveal the most wandering behavior: its scaling exponent goes from *α* ≈ 3.7 (1 s) through *α* ≈ 3.6 (10 s), *α* ≈ 3.1 (1 min), and *α* ≈ 3.9 (10 min) to *α* ≈ 2.7 (1 h) compared to *α* ≈ 2.6 (negative tail; daily data) and *α* ≈ 2.8 (positive tail) for the years 1971–1999 [37].


**Table 5.** Estimated tail exponent *α* and stretched exponent parameter *β* for the aggregated distributions of the CFDs returns for the selected commodities: high-grade copper (HG), crude oil (CL), silver (XAG), and gold (XAU).

**Figure 11.** Cumulative distribution functions of the CFD returns for commodities: high-grade copper (HG), the U.S. crude oil (CL), silver (XAG), and gold (XAU). 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.

Crude oil return distributions (CL) have the thickest tails with the scaling exponent 2.0 ≤ *α* ≤ 2.6 and with no signature of the CLT convergence. In parallel to what was observed for gold, we removed the COVID-19 outburst period from the time series and calculated the distributions again—see Figure 12 (right panel). Such incomplete signals are characterized by *α* = 3.0 for Δ*t* = 1 s and *α* ≈ 2.7 for Δ*t* = 1 h. These numbers have to be compared with *α* ≈ 2.9 (negative tail) and *α* ≈ 3.1 (positive tail) reported for the WTI oil daily returns covering the years 1988–1998, with *α* ≈ 2.0 (negative tail) and *α* ≈ 2.8 (positive tail) reported for the crude oil daily returns covering the years 1983–1999 [37], and with *α* ≈ 3.0 (negative tail) and *α* ≈ 3.1 (positive tail) for the WTI oil 5-min returns covering the years 2012–2018 [36]. As those values do not differ much from each other, there is no evidence that the crude oil returns change their global dynamics over time. However, during the market turbulence that happened during the COVID-19 pandemic, the dynamics did change considerably, which was manifested by thickening the

distribution tail for all considered time scales. It is noteworthy that the crude oil was the asset that was affected the strongest by the pandemic: in April 2020, the price of the May series of WTI oil futures even dropped below 0.

**Figure 12.** Cumulative distribution functions of the CFD returns for gold (XAU) and the U.S. crude oil (CL) after removing the COVID-19 outburst period (March–April 2020). 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.
