*3.3. Currencies*

Unlike the stock indices, the return distributions for the exchange rates show the presence of increasingly thinner tails if the time scales increase, and hence, a faster convergence towards the normal distribution (Figure 9). If fitted by a power function, the differences between the individual exchange rates are smaller than those of the indices and generally exhibit an inverse cubic decay for smaller Δ*t*s: for Δ*t* = 1 s, they vary from *α* ≈ 3.0 (USD/JPY, GBP/USD) to *α* ≈ 3.4 (EUR/USD) and, for Δ*t* = 1 h, from *α* ≈ 4.2 (EUR/JPY) to *α* ≈ 5.5 (GBP/CHF) with the exception of GBP/JPY (*α* ≈ 2.8). The numbers are collected in Table 4. The scaling exponent *α* ≈ 3 was observed in many studies of the Forex data, including References [39,116,124,127,128]. If the stretched exponential function is used, the best-fitted parameter *β* reads for Δ*t* = 1 s from *β* = 0.37 (EUR/JPY) to *β* = 0.48 (GBP/JPY, USD/CHF) and, for Δ*t* = 1 h, from *β* = 0.49 (EUR/USD, USD/JPY) to *β* = 0.87 (GBP/USD). The mean values of the scaling exponent for the analyzed time scales are *α* ¯= 3.2 (1 s), *α*¯ = 3.1 (10 s), *α*¯ = 3.2 (1 min), *α*¯ = 3.7 (10 min), and *α*¯ = 5.0 (1 h). The inverse cubic scaling can therefore now be identified for scales shorter than 10 min. These scale are longer than those in the years 2004–2008 for the 1-min scale *α*¯ = 3.9 [124]. (It has to be noted, however, that those earlier results were obtained by fitting the *q*-Gaussian functions

instead of the power-law functions, which may make it difficult to compare the results properly even though the relation given by Equation (8) holds). We thus observe a slower convergence to the normal distribution now than before for Δ*t* = 1 h: *α*¯ = 5.9 (2004–2008) vs. *α* ¯ = 4.8 (2017–2020). However, if the obtained results are compared with those from the study [116] for the years 1987–1993, the acceleration becomes visible: *α*¯ = 3.9 (1987–1993) vs. *α* ¯ = 4.8 (2017–2020). In that case, the inverse cubic scaling was still visible for the 30-min scale, which is much longer than both in 2004–2008 and 2017–2020. This can be interpreted as the acceleration of the market time resulting in a faster convergence to the normal distribution in the 2000s, but later, this phenomenon of the effective time scale shortening disappeared.

**Figure 9.** Cumulative distribution functions of the major currency exchange rate returns: Swiss franc (CHF), euro (EUR), British pound (GBP), Japanese yen (JPY), and the U.S. dollar (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.


**Table 4.** Estimated tail exponent *α* and stretched exponent parameter *β* for the aggregated return distributions for the selected currency exchange rates and the cryptocurrency prices: BTC/USD and ETH/USD.
