# Signatures of the Crypto-Currency Market Decoupling from the Forex
## 2 Data Specification and Properties  
The data set used in the present study consists of the exchange rates reflecting the actual trade involving the Bitcoin (BTC), Ethereum (ETH), Euro (EUR) and US dollar (USD). On this basis, the following six exchange rates are defined: BTC/USD, BTC/EUR, ETH/USD, ETH/EUR, BTC/ETH, together with the EUR/USD, herein taken as a standard benchmark. Since 1 July 2016, all cryptocurrency transactions are tracked at a frequency of \(\Delta t=10\) s, with the resulting time-series being recorded by Kraken [27], which is the world's largest Euro-to-Bitcoin exchange. Operating since September 2013 as one of the longest-trading, continuously-running Bitcoin exchanges, it has branches in Canada, the EU, and the US. Supported fiat pairs include the CAD, EUR, and USD. Notable supported cryptocurrencies encompass the BTC, ETH, LTC, BCH, XRP, XMR, DASH, XLM, DOGE, EOS, ICN, GNO, MLN, REP, USDT, ZEC, ADA and QTUM, allowing both fiat-to-crypto and crypto-to-crypto trades. In the present study, the four most liquid pairs, namely BTC/EUR, BTC/USD, ETH/EUR,ETH/USD, and the most liquid crypto-to-crypto pair, BTC/ETH, are considered. The Kraken API allowed seamless access to tick-by-tick data. The data were collected until 31 December 2018. The EUR/USD exchange rate is considered at the same \(\Delta t=10\) s frequency, and within the same period of time but, due to Forex market trading specifications, without weekends (the Forex does not operate between Friday 10 p.m. UTC and Sunday 10 p.m. UTC), as recorded by the Swiss forex bank Dukascopy [28]. Charts illustrating the time variation of these six exchange rates over the time-span under consideration are shown in Figure 1.  
In the corresponding logarithmic returns, one has \(r_{\Delta t}=\log(p(t+\Delta t))-\log(p(t))\), where \(\Delta t\) stands for the returns' time-lag gaps, and where and the time intervals during which some instruments were not traded have been removed (7 May 2017 22.30-23.45--DDoS attack on ETH/USD, 6.40 11 January 2018-14.30 13 January 2018 Kraken maintenance shutdown). Thus, the series of returns from Kraken comprise approximately \(N=7.6\) million observations for each of the considered time-series involving the BTC and ETH. For EUR/USD, there are about 5.6 million observations. Volatility clustering, a phenomenon reflecting the fact that large fluctuations tend to be followed by large fluctuations, of either sign, and small fluctuations tend to be followed by small fluctuations [29], is considered to be among the most characteristic financial stylized facts [5]. Such an effect is clearly seen in Figure 2, which shows the time-variation of the moduli of logarithmic returns for all six exchange rates under consideration. Remarkably, however, as demonstrated by the consecutive magnifications in the EUR/USD panel, the average time-span of the corresponding clusters in all the cases involving either the BTC or the ETH is about one order of magnitude longer than for the EUR/USD rate. In the former case, one can recognize about three high-low volatility cycles within the corresponding monthly insets of Figure 2, whereas in the latter case, five such cycles are clearly apparent within the one-week time-interval of the inset in the bottom panel. Figure 1: (Color online) Logarithm of the exchange rates BTC/EUR, BTC/USD, ETH/EUR, ETH/USD, BTC/ETH and EUR/USD over the period between 1 July 2016 and 31 December 2018. For improved visibility, the EUR/USD exchange rate was magnified by a factor of 100. A more formal quantification of the corresponding effects can be arrived at via the volatility (logarithmic return modulus) autocorrelation function \(C(\tau)=<|r(t)||r(t-\tau)|>_{t}\), as is shown in Figure 3 for all the six exchange rates under consideration.