# Signatures of the Crypto-Currency Market Decoupling from the Forex
## 4 Multifractality in the Exchange Rates
The left-side of \(f(\alpha)\) is projected out by the positive \(q\)-values and as such reflects the organization of the large fluctuations, whereas the negative \(q\)-values determine the right-side of \(f(\alpha)\). Consequently, the left-sided asymmetry of \(f(\alpha)\) indicates a more pronounced multifractality at the level of large fluctuations, with the converse applying to a right-sided asymmetry. As Figure 6 reveals, the left-sided asymmetry emerges in the exchange rates involving BTC and ETH which, in turn, indicates that they develop a more pronounced multifractality at the level of large fluctuations and that their smaller fluctuations are noisier. The most asymmetric cases are the BTC/EUR (\(A_{\alpha}\approx 0.94\)) and the ETH/EUR (\(A_{\alpha}\approx 0.92\)), where, effectively, only the left wing of the width \(\Delta f(\alpha)\approx 0.35\) builds up. The BTC/USD (\(A_{\alpha}\approx 0.72\)) and BTC/USD  
Figure 5: Family of the \(q\)th-order fluctuation functions \(F_{q}(s)\) for different values of \(q\in[-4,4]\) in steps of 0.2 (the upper-most one represents \(q=4\)), calculated for the six exchange rates BTC/EUR, BTC/USD, ETH/EUR, ETH/USD, BTC/ETH and EUR/USD. Vertical dotted lines indicate the range of scales used in determining the generalized Hurst exponents \(h(q)\). Insets show the corresponding dependence of \(h(q)\) on \(q\). \((A_{\alpha}\approx 0.7)\) cases develop traces of a right wing in \(f(\alpha)\), but the overall width, \((\Delta f(\alpha)\approx 0.35)\), remains virtually unchanged. Notably, for BTC/ETH the right wing is even longer, however, the entire width extends to \(\Delta f(\alpha)\approx 0.42\). A commonly accepted and straightforward measure of correlations in time-series is provided by the Hurst exponent \(H\)[46], which, within the present formalism, corresponds to \(h(q=2)\). A finding of \(H>0.5\) indicates persistence, whereas \(H<0.5\) hallmarks anti-persistence. This parameter reveals the tendency to follow a trend and, as such, acquires particular relevance in the context of financial time series [6]. In order to verify the anticipated evolution of such characteristics over time for the dynamics of the cryptocurrency exchange rates, the Hurst exponent was estimated as \(H=h(q=2)\) following Equations (6) and (8) over 1-month (30 days) time windows, each comprising 259,200 data points (10 s returns). The results are visible in Figure 7. The Hurst exponent \(H(t)\) for the EUR/USD change rate was confirmed to dwell in the vicinity of 0.5, which hallmarks market maturity [47]. During the early trading period considered, i.e., between July 2016 and December 2017, the Hurst exponent \(H(t)\) for the exchange rates BTC/EUR, BTC/USD as well as for ETH/EUR and ETH/USD had a value significantly smaller than 0.5: this revealed the strong anti-persistence which is expected for such a high-risk, emerging market. After that, the same parameter gradually approached 0.5 from below, eventually arriving very close to this value by the end of the year 2017. This can be interpreted as a strong indication that the corresponding markets are approaching maturity [47], a fact that has recently been pointed out for BTC/USD rate [48]. At the same time, it is compelling to observe that the Hurst exponent for BTC/ETH exchange rate, while still markedly below 0.5 towards the end of the year 2017, started approaching this value later in the year 2018. Figure 6: (Color online) Singularity spectra \(f(\alpha)\) calculated for all six considered time-series of BTC/EUR, BTC/USD, ETH/EUR, ETH/USD, BTC/ETH and EUR/USD returns, setting \(q\in[-4,4]\). Insets show the \(q\)-dependence of the corresponding generalized Hurst exponents \(h(q)\).