*2.3. Price Impact*

At this point, it is worthwhile to consider a possible causal relation between the returns and the volume despite the fact that no clear relation can be seen between their cdfs. It revokes the empirically well-documented observation that volume can influence price changes (both on the level of the order book and the level of the aggregated transaction volume), which is known in the literature as the price impact [21,23,65–68]. In order to investigate this issue, for each cryptocurrency, two parallel time series corresponding to |*R*Δ*t*(*t*)| and *V*Δ*t*(*t*) were input into the *q*-dependent detrended cross-correlation coefficient *ρ<sup>q</sup>* measuring how correlated two detrended residual signals are across different scales [69]. The definition of the coefficient *ρq*, which allows one to quantify cross-correlations between two nonstationary signals, is based on the multifractal detrended cross-correlation analysis (MFCCA), whose algorithm can be sketched as follows [70].

In this particular case, there are two time series of length *T* and sampling intervals Δ*t*: |*R*Δ*t*(*ti*)| and *V*Δ*t*(*ti*) with *i* = 1, ... , *T*. One starts the procedure by dividing each time series into *Ms* = 2*T*/*s* non-overlapping segments of length *s* (called *scale*) going from both ends (· denotes the floor value). In each segment labelled by *ν*, both signals are integrated and polynomial trends *<sup>P</sup>*(*m*) ·,*s*,*<sup>ν</sup>* of degree *<sup>m</sup>* are removed:

$$\hat{R}\_{\Lambda t}(t\_j, s, \nu) = \sum\_{k=1}^{j} |R\_{\Lambda t}(t\_{s(\nu-1)+k})| - P\_{R, s, \nu}^{(m)}(t\_j), \tag{3}$$

$$\hat{V}\_{\Lambda t}(\mathbf{t}\_{\dot{j}}, \mathbf{s}, \nu) = \sum\_{k=1}^{\dot{j}} V\_{\Lambda t}(\mathbf{t}\_{\mathbf{s}(\nu-1)+k}) - P\_{V, \mathbf{s}, \nu}^{(m)}(\mathbf{t}\_{\dot{j}}), \tag{4}$$

where *j* = 1, . . . ,*s* and *ν* = 1, . . . , *Ms*. The detrended covariance is derived as

$$f\_{|\mathbb{R}|V}^{2}(s,\boldsymbol{\nu}) = \frac{1}{s} \sum\_{j=1}^{s} \left[\mathcal{R}\_{\Lambda t}(t\_{j},s,\boldsymbol{\nu}) - \langle \mathcal{R}\_{\Lambda t}(t\_{j},s,\boldsymbol{\nu})\rangle\_{j}\right] \left[\mathcal{V}\_{\Lambda t}(t\_{j},s,\boldsymbol{\nu}) - \langle \mathcal{V}\_{\Lambda t}(t\_{j},s,\boldsymbol{\nu})\rangle\_{j}\right],\tag{5}$$

where · *<sup>j</sup>* denotes the averaging over *j*. The detrended covariances calculated for all the segments *ν* are then used to determine the bivariate fluctuation function [70]:

$$F\_{\eta}^{|R|V}(s) = \left\{ \frac{1}{M\_{\mathfrak{s}}} \sum\_{\nu=1}^{M\_{\mathfrak{s}}} \text{sgn}[f\_{|R|V}^2(s,\nu)] |f\_{|R|V}^2(s,\nu)|^{\eta/2} \right\}^{1/q}. \tag{6}$$

Apart from the bivariate form given by the formula above, the univariate fluctuation functions *F*|*R*||*R*<sup>|</sup> *<sup>q</sup>* (*s*) and *FVV <sup>q</sup>* (*s*) can also be calculated but, in this case, the covariance functions become variances and do not need to be factorized into the sign and modulus parts as no negative value can occur.

The above elements of the formalism allow one to introduce the *q*-dependent detrended cross-correlation coefficient *ρq*(*s*) defined as [69]:

$$\rho\_q^{|R|V}(s) = \frac{F\_q^{|R|V}(s)}{\sqrt{F\_q^{|R||R|}(s) F\_q^{VV}(s)}}.\tag{7}$$

By manipulating the value of the parameter *q*, one can focus on the correlations between fluctuations in different size: the large fluctuations *q* > 2 or the small fluctuations *q* < 1. For *q* = 2, all the fluctuations in time series are considered with the same weights. For positive *q*, values of *ρ<sup>q</sup>* are restricted to the interval [−1, 1], with their interpretation being similar to the interpretation of the classic Pearson coefficient *C*: *ρ<sup>q</sup>* = 1 means a perfect correlation, *ρ<sup>q</sup>* = 0 means independence, and *ρ<sup>q</sup>* = −1 means a perfect anticorrelation. For negative *q*, the interpretation of the coefficient is more delicate and requires some experience [69]. Figure 3 presents the coefficient *ρq*(*s*) calculated in a broad range of scales *s* for the selected individual cryptocurrencies (BTC, ETH, DOGE, FUN, PERL, and WAN) and the average *ρq*(*s*) for Groups I-III. While different data sets are characterized by different strength of the detrended cross-correlations with Group I cross-correlated the strongest and Group 3 the weakest, there is an explicit division of scales into the short-scale range (*s* < 1000 min), where the correlations monotonously increase with increasing *s*, and the long-scale range (*s* > 1000 min), where one observes a kind of saturation-like behavior. In the latter, the correlations are characterized by 0.75 ≤ *ρq*(*s*) ≤ 0.95, which means that the cryptocurrency market does not differ from other financial markets and its volatility |*R*Δ*t*| and volume traded are strongly correlated. The two distinguished scale ranges are related to the information-processing speed of the market: it requires some amount of time for the investors to fully react to the incoming information and to build up the cross-correlations. One might view this result as a counterpart of the Epps effect for the detrended volatility–volume data [6,28,71–73]. The main difference between this market

and the regular financial markets is the relatively long cross-over scale (*s* ≈ 1000 min), which can be associated with its worse liquidity.

**Figure 3.** The *q*-dependent detrended cross-correlation coefficient *ρq*(*s*) of order *q* = 1 calculated for volatility |*R*Δ*t*(*t*)| and volume *V*Δ*t*(*t*) (with Δ*t* = 1 min) for the selected individual cryptocurrencies— BTC, ETH, DOGE, FUN, PERL, and WAN—where the cryptocurrency Groups I-III are characterized by a specific range of the average inter-transaction time: *δt* < 1*s* (Group I, dotted red), 1*s* ≤ *δt* < 2*s* (Group II, dotted blue), *δt* ≥ 2*s* (Group III, dotted green). The coefficient *ρq*(*s*) has been averaged over all the cryptocurrencies belonging to a given group.

The next question to be asked is if there exists any functional relationship between |*R*Δ*t*| and *V*Δ*t*. In order to address this question, *R*Δ*<sup>t</sup>* vs. *V*Δ*t* scatter plots for six selected cryptocurrencies were created; see Figure 4. In general, the cross-correlations identified by means of *ρq*(*s*) can also be confirmed visually on these plots: the larger the volume, the larger the volatility can be. However, no specific functional form of *R*Δ*t*(*V*Δ*t*) can be inferred from this picture. Therefore, it is instructive to change the presentation to the conditional probability plots of the form <sup>E</sup>[ *<sup>f</sup>*(|*r*Δ*t*|)|*v*Δ*t*], where the expectation value <sup>E</sup>[·] can be approximated by the mean · . From the perspective of a market with substantially limited liquidity, small price changes correspond to small transaction volumes and constitute market noise. Thus, one may expect that the most interesting relation between volatility and volume can be seen for large returns: |*r*Δ*t*(*t*)| 1.

**Figure 4.** Scatter plots of the returns *R*Δ*t*(*t*) and volume traded *V*Δ*t*(*t*) for a few selected cryptocurrencies (BTC, ETH, DOGE, FUN, PERL, and WAN). Each point corresponds to a specific 1 min long interval in the whole 3-year-long period of interest. The vertical dashed lines in each panel denote the 25th, 50th, and 75th quantile of the volume probability distribution function for a given cryptocurrency. Note the logarithmic horizontal axis and the varying axis ranges among the panels.

The values of the normalized volume traded *v*Δ*t*(*t*) were compartmentalized and, in each cell *vi*, a fixed fraction *p* 1 of the respective largest conditional volatility values was preserved for further analysis. A power-law function with the exponent *κ* is assumed to model a relation between the two quantities:

$$
\boldsymbol{v}\_{\Delta t} \sim |\boldsymbol{r}\_{\Delta t}|^\kappa , \quad |\boldsymbol{r}\_{\Delta t}| \sim \boldsymbol{v}\_{\Delta t}^\alpha . \tag{8}
$$

Figure <sup>5</sup> tests whether any of the relations of the form <sup>E</sup>[|*r*Δ*t*<sup>|</sup> *<sup>κ</sup>*|*v*Δ*t*] <sup>∼</sup> *<sup>v</sup>*Δ*<sup>t</sup>* hold for BTC if the following exponent values are selected: *κ* = 0.2, *κ* = 0.5, *κ* = 1, and *κ* = 2. The threshold value was chosen to be *p* = 0.1 because, for larger values, the relation becomes blurred and difficult to identify, whereas, for smaller values, too few data points can be considered, which amplifies the uncertainty. Looking at the graphs, one can reject the hypothesis that volatility and volume are related via *v*Δ*<sup>t</sup>* ∼ |*r*Δ*t*| <sup>2</sup> (i.e., *α* = 0.5) for all the sampling frequencies considered. In the case of the highest sampling frequency (Δ*t* = 1 min), the data are approximated the best for *κ* = 1 and, secondarily, for *κ* = 0.5 and *κ* = 0.2, over the

broad volume range 1 < *v*Δ*<sup>t</sup>* < 16. For Δ*t* ≥ 10 min, none of the values considered for *κ* work well, whereas, for Δ*t* = 5 min, two cases cannot be excluded: *κ* = 0.5 and *κ* = 0.2. This means that the likely functional form of the price impact cannot be inferred based on the available data. Figure 6 presents the analogous results for ETH. The square-root form of the price impact (corresponding to *κ* = 2) can also be rejected in this case. However, it cannot be decided which of the remaining models (*κ* ≤ 1) is the most likely.

**Figure 5.** Conditional expectation <sup>E</sup>[|*r*Δ*t*<sup>|</sup> *<sup>κ</sup>* <sup>|</sup>*v*Δ*t*] for BTC if only a *<sup>p</sup>*-fraction of the largest normalized returns *r*Δ*<sup>t</sup>* is preserved for each value of the normalized volume *v*Δ*t*. Each panel shows the results for a specific value of *κ* together with a corresponding fitted power-law model. Four cases of the sampling interval are presented: Δ*t* = 1 min, 5 min, 10 min, and 60 min. The error bars show the conditional standard deviation *σ*[|*r*Δ*t*| *<sup>κ</sup>* <sup>|</sup>*v*Δ*t*].

The fact that *κ* = 2 (*α* = 0.5) and likely *κ* ≤ 1 (*α* ≥ 1) for short sampling intervals is interesting because it makes the price impact function linear or superlinear (*α* ≥ 1): a result that differs from some earlier claims made for the regular financial markets, where the function was concave, at least for short and moderate sampling intervals [21,23]. There is also a discrepancy for the long sampling intervals because, in this case, the behavior reported for the regular markets was effectively linear, whereas here it remains undefined. It is noteworthy in this context that the superlinear (*α* > 1) price impact for large Δ*t* in Equation (8) could open the space for market manipulation [21], which, on the cryptocurrency trading platforms, can take the form of wash trading [18,74]. According to that, one can view the presented results as being in favor of the conclusion that full maturity is still ahead of the cryptocurrency market.

**Figure 6.** The same quantities as in Figure 5 for ETH.
