*2.4. Volatility Clustering and Long Memory*

It takes some time for a market to completely absorb pieces of information that arrive there. This is a source of temporal market correlations that can be most easily observed in the price fluctuation amplitudes. Correlations are responsible for the phenomenon of volatility clustering, i.e., the existence of prolonged periods of fluctuations with elevated amplitude that are separated by quiet periods with more tamed fluctuations [75]. Volatility clustering is observed on all markets and can be quantified in terms of the autocorrelation function:

$$\mathbb{C}(\tau) = \langle r\_{\Delta t}(t)r\_{\Delta t}(t-\tau) \rangle\_{t,\prime} \tag{9}$$

where *τ* is the lag time. The autocorrelation functions calculated from the absolute logreturns for several individual cryptocurrencies and the average autocorrelation functions calculated for Groups I–III are presented in Figure 7 on a double-logarithmic scale. In each case, one can identify at least one range of lags over which *C*(*τ*) shows powerlaw decay. For BTC, ETH, and FUN, there is only one such range corresponding to 10 min ≤ *τ* ≤ 500 min with a relatively small upper end. The same refers to WAN but, in this case, the upper end exceeds *τ* ≈ 20,000 min (ca. two weeks). On the other hand, DOGE, PERL, and the average plots show two scaling regimes: the short-*τ* regime up to *τ* ≈ 500–1000 min (less than a day) and the long-*τ* regime for 1000 min < *τ* < 20,000 min. In each case, *C*(*τ*) falls to 0 around *τ* ≈ 100,000 min (more than 2 months). Compared to a more distant past, the scaling regions for BTC and ETH are shorter now (e.g., in the years 2016–2018, it reached *τ* = 1000 min [29]), which is consistent with the market time acceleration caused by an increased trading frequency. This overall picture for the cryptocurrency market does not depart much from the one observed in other financial markets. A power-law decaying autocorrelation function expressing the long memory of volatility is a common property that is a manifestation of the way that the market processes

information [27,76]. The time lag at which *C*(*τ*) reaches a statistically insignificant level is equal to the average length of a volatility cluster [76]. Due to the alternating character of market dynamics, where the high-volatility periods are interwoven with low-volatility periods, for larger time lags, the autocorrelation function becomes negative. Note that, due to the fact that volatility time series are unsigned, the long-range autocorrelations cannot be exploited for the related investment strategies.

**Figure 7.** Autocorrelation function *<sup>C</sup>*|*r*Δ*t*|(*τ*) of the absolute normalized log-returns <sup>|</sup>*r*Δ*t*(*t*)<sup>|</sup> (volatility) calculated for the selected individual cryptocurrencies—BTC, ETH, DOGE, FUN, PERL, and WAN as well as for the cryptocurrency Groups I-III characterized by specific range of the average intertransaction time: *δt* < 1*s* (Group I, dotted red), 1*s* ≤ *δt* < 2*s* (Group II, dotted blue), *δt* ≥ 2*s* (Group III, dotted green). *<sup>C</sup>*|*r*Δ*t*|(*τ*) has been averaged for each value of *<sup>τ</sup>* over all the cryptocurrencies belonging to a given group. Note the double-logarithmic scale.
