**3. Data and Results**

The cryptocurrency market is a relatively new and emerging market, meaning that the trading mechanism is unique and makes it very different from traditional markets. More than 21, 800 different cryptocurrencies are currently traded around the world with an estimated total market capitalization of over USD 843 billion (see, e.g., https://coinmarketcap. com/ (accessed on 7 December 2022)). On the other hand, the foreign exchange market is the largest financial market worldwide, with transactions amounting to trillions of US dollars daily [70]. In this article, we focused on the analysis of the two most representative currencies of these two markets, i.e., the BTC/USD and EUR/USD. Our analyses were applied to the daily logarithmic returns (*rt* = ln *pt* − ln *pt*−1, where *pt* denotes the price at time *t*) of the BTC/USD and EUR/USD during the period from 1 May 2019 to 20 January 2021. In an announcement by the *WHO* on 11 March 2020, the outbreak of COVID-19 was declared a global pandemic. Therefore, we considered the period from 1 May 2019 to 11 March 2020 as the pre-announcement period, and the period from 12 March 2020 to 20 January 2021 as the post-announcement period. All financial time series were taken from Yahoo Finance (http://finance.yahoo.com/ (accessed on 7 December 2022)).

In our study, we investigated the temporal evolution of complexity and fractal characteristics by using overlapping sliding windows (with window length equal to 512 samples and slide step equal to 1 sample). First, we investigated the temporal evolution of the multifractal spectrum parameters (*α*0, Δ*α*, *A*) before and after the outbreak of the pandemic. Then, for the same time period, we extended our analysis by examining the temporal evolution of Fuzzy entropy, Tsallis entropy, Shannon entropy, and Fisher information.

At this point, we should mention that for the calculation of Tsallis entropy we have chosen to use the value *qTS* = 1.8 for the non-extensive parameter, *qTS*. On one hand, for financial time series *qTS* has been found to take values *qTS* ∼ 1.6 − 1.8 [3], which has been discussed within the framework of the similarities in scaling properties and universality related to observables of extreme events from different disciplines (e.g., financial crisis, earthquake, epileptic seizure, magnetic storm, solar flare) [3,55,60,61]. On the other hand, from the time series analysis point of view, the selection of the *qTS* value for the calculation of the temporal variation of Tsallis entropy practically only affects the "separation" between the lower and higher complexity parts of the analyzed time series (e.g., min to max entropy values ratio, in direct analogy to the signal to noise ratio), while for the herein analyzed time series it was found that any *qTS* value in the range ∼ 1.6 − 1.8 leads to approximately the same results.

Figure 1c,d, depict the temporal evolution of *α*<sup>0</sup> values under different market trends of the BTC/USD and EUR/USD returns, respectively. By analyzing the overall trend of the BTC/USD returns, it is observed that the values of the *α*<sup>0</sup> fluctuate around 0.6 (Figure 1c). These results indicate that the returns time series is characterized by long-range correlations, both before and after the onset of COVID-19. By analyzing the downtrend markets of the BTC/USD returns, it is observed that the values of the *α*<sup>0</sup> fluctuate over 0.6 both before and after the outbreak of the pandemic, indicating persistent behavior. In the uptrend markets of the BTC/USD returns, the values of *α*<sup>0</sup> fluctuate between 0.5 and 0.65 for almost throughout the analysis period. An exception is a short period of time after the *WHO* announcement, where *α*<sup>0</sup> values fell below 0.5.

**Figure 1.** Comparative asymmetric multifractal analysis of BTC/USD (left panels) and EUR/USD (right panels) under different market trends. (**a**,**b**): Exchange rates and Returns. (**c**,**d**): Temporal evolution of *α*<sup>0</sup> parameter. (**e**,**f**): Temporal evolution of width of the multifractal spectrum. (**g**,**h**): Temporal evolution of the asymmetry parameter *A* values. The red vertical dash line corresponds to the date of the *WHO* announcement in which COVID-19 was declared a global pandemic (i.e., 11 March 2020). The period from 1 May 2019 to 11 March 2020 corresponds to the pre-announcement period, while the period from 12 March 2020 to 20 January 2021 corresponds to the post-announcement period.

Figure 1d depicts the temporal evolution of the *α*<sup>0</sup> values of the EUR/USD returns under different market trends. In this case, we observe that for the overall trend the *α*<sup>0</sup> values fluctuate around 0.4 during the whole pre-announcement period, while after the announcement they present a progressive increase approaching very close to *α*<sup>0</sup> = 0.5 at the end of the considered analysis period. This suggests that the time series exhibit a "different" power-law correlation, such that large and small time series are more likely to alternate (anti-persistent behavior). It is worth mentioning that the downtrend *α*<sup>0</sup> values remain at the anti-persistent side except for the very last part of the analyzed period, while the uptrend *α*<sup>0</sup> values, although < 0.5 for the whole pre-announcement period, present an alternating behavior after the *WHO* announcement, taking values *α*<sup>0</sup> > 0.5 for two two-month-long periods.

Figure 1e,f illustrate the width of the multifractal spectrum under different market trends of the BTC/USD and EUR/USD returns, respectively. As already mentioned in Section 2.1.1, the width of the multifractal spectrum Δ*α* is a measure of the degree of multifractality. If a time series presents a smaller width of the multifractal spectrum, this indicates that the time series has lower heterogeneity, i.e., lower fluctuations and lower market risk [58]. The results show that throughout the period analyzed, the width of the multifractal spectrum receives higher values for the BTC/USD returns compared to the EUR/USD returns. Therefore, it can be concluded that EUR/USD is relatively more stable than BTC/USD. In addition, we observe that after the outbreak of the pandemic, the width of the multifractal spectrum increased for both BTC/USD returns and the EUR/USD returns for the overall trend. This suggests that after the outbreak of the pandemic, both currencies reacted similarly in terms of multifractality when observed from an overall trend point of view. The degree of multifractality increased, and, therefore, the fluctuations became more intense and the market risk increased. However, in terms of asymmetric multifractality, this is not always the case. When focused on downside markets of BTC/USD, the degree of multifractality decreased after the outbreak. More interestingly, downtrend multifractality was higher than the uptrend multifractality during the period before COVID-19, but during the pandemic the uptrend multifractality became higher. These findings reveal that the incremental multifractality in BTC/USD is due to intense fluctuations and higher heterogeneity during price increases, but not during price declines. In EUR/USD, it appears that the downside markets play a more important role in increasing multifractality. Nevertheless, both market trends may have had some impact in the post-announcement period increase in multifractality. It is important to note that the increase in multifractality in BTC/USD returns during COVID-19 is consistent with the existing literature as other studies have reached the same conclusion (e.g., [26,27]).

Figure 1g,h depict the temporal evolution of the asymmetry parameter *A* values under different market trends of the BTC/USD and EUR/USD returns, respectively. In the time period before the onset of the pandemic, the asymmetry parameter *A* of BTC/USD returns appears to have been consistently below 0, indicating relative dominance of the small fluctuations. Immediately after the date of the *WHO* announcement, there was a sharp change in the values of *A* in all market trends. Specifically, for both overall trend and uptrend markets, the values of *A* of the BTC/USD returns remain above 0 for the entire period after the outbreak of the pandemic. This sharp change shows a transition from a period where small fluctuations predominate (before the pandemic) to a period where large fluctuations predominate (during the pandemic). In the downtrend markets, the values of parameter *A* are almost at 0 for the entire period after the outbreak of the pandemic, indicating that the spectrum became practically symmetrical (Figure 1g). On the other hand, the values of the asymmetry parameter *A* of the EUR/USD returns for the uptrend markets are almost equal to 0 for the whole period before the outbreak of the pandemic. This fact indicates that the spectrum is practically symmetrical. On the contrary, analyzing the overall and downward trends of the market, we observe that the values of the asymmetry parameter *A* are below 0, almost for the entire period before the outbreak of the pandemic. Therefore, it is concluded that in the overall and downward trends of

the markets, they are dominated by the small fluctuations in EUR/USD returns before the outbreak of the pandemic. Immediately after the announcement date, the values of the asymmetry parameter *A* of the EUR/USD returns exceeded 0 in all market trends. This result shows that EUR/USD returns after the outbreak of the pandemic are dominated by large fluctuations (Figure 1h).

Figure 2c,d indicate that the effect of the announcement was, for all cases (for both BTC/USD and EUR/USD returns and for all three considered market trends), a sharp change towards right-truncation, which means that after the *WHO* announcement the multifractal structure in the time series became more insensitive to the local fluctuations with small magnitudes. On the other hand, Figure 2e,f show that the behavior of BTC/USD and EUR/USD returns was different concerning the degree of truncation asymmetry, indicated by the so-called C − value. Specifically, EUR/USD returns present C − values quite close to 0 before the *WHO* announcement, which means that the underlying system then presented the lowest possible undulation or instability. After the *WHO* announcement, the picture changed and for all market trends an increase in the undulation or instability of the underlying system is observed. In contrast, BTC/USD returns present a general trend (although with notable fluctuations for the overall and uptrend markets) towards a decrease in the C − values after the *WHO* announcement, which means that a trend for the decrease in the undulation or instability of the underlying system is observed. It is noted that the downtrend market after the *WHO* announcement presents C − values closer to 0, as compared with the uptrend and overall markets, indicating lower undulation or instability.

**Figure 2.** Comparative asymmetric multifractal analysis of BTC/USD (left panels) and EUR/USD (right panels) under different market trends. (**a**,**b**): Exchange rates and Returns. (**c**,**d**): Temporal evolution of truncation Δ*f*(*a*) = *f*(*αmin*) − *f*(*αmax*). (**e**,**f**): Temporal evolution of the degree of truncation asymmetry, known as C − value = |Δ*f*(*a*)| = | *f*(*αmin*) − *f*(*αmax*)|. The red vertical dash line corresponds to the date of the *WHO* announcement in which COVID-19 was declared a global pandemic (i.e., 11 March 2020). The period from 1 May 2019 to 11 March 2020 corresponds to the pre-announcement period, while the period from 12 March 2020 to 20 January 2021 corresponds to the post-announcement period.

Moreover, we analyzed the temporal evolution of some complexity measures. Figure 3c,d illustrate the temporal evolution of the Fuzzy entropy of the BTC/USD and EUR/USD returns, respectively. As it has already been mentioned in Section 2.2., smaller values of Fuzzy entropy indicate a greater chance that a set of data will be followed by similar data (regularity). Conversely, larger values of Fuzzy entropy point to a lower chance of similar data being repeated (irregularity). As we observe in Figure 3c,d, the values of Fuzzy entropy dropped sharply in both BTC/USD and EUR/USD returns immediately after the *WHO* announcement. This fact indicates that in the pre-announcement period, both BTC/USD and EUR/USD returns were characterized by a higher degree of disorder and randomness, i.e., by higher complexity. In contrast, in the period during the pandemic, the values of Fuzzy entropy decreased, suggesting that the returns were characterized by a higher degree of order and lower complexity. Therefore, it is concluded that the pandemic led investors to behave in an "organized" (similar) way that thereby reduced the complexity of the two markets.

**Figure 3.** Comparative analysis of BTC/USD (left panels) and EUR/USD (right panels). (**a**,**b**): Exchange rates and Returns. (**c**,**d**): Temporal evolution of Fuzzy entropy. (**e**,**f**): Temporal evolution of Tsallis entropy. (**g**,**h**): Temporal evolution of Shannon entropy. (**i**,**j**): Temporal evolution of Fisher information. The red vertical dash line corresponds to the date of the *WHO* announcement in which COVID-19 was declared a global pandemic (i.e., 11 March 2020). The period from 1 May 2019 to 11 March 2020 corresponds to the pre-announcement period, while the period from 12 March 2020 to 20 January 2021 corresponds to the post-announcement period.

Corresponding results are obtained by also studying two quite popular complexity measures, i.e., the Shannon entropy (Figure 3g,h) and Tsallis entropy (Figure 3e,f). More specifically, the time variations of the Shannon entropy as well as the Tsallis entropy (for a given *qTS*) quantify the dynamical changes of the information content and the complexity of the system. Smaller values characterize time series with lower complexity and randomness, as well as higher information content and order. Conversely, larger values characterize time series with higher complexity, disorder and randomness, as well as lower information content. As we observe in Figure 3e–h, during COVID-19, the values of Tsallis and Shannon entropies were reduced in both BTC/USD and EUR/USD returns, indicating that the complexity of the two markets was reduced and the information content was increased. It is important to note that all the entropy measures we applied quickly adapted to market conditions, showing a sharp decrease immediately after the *WHO* announcement, with Shannon entropy being the exception in the case of BTC/USD. Additionally, it is of particular interest that the entropy values remained at low levels throughout the pandemic period we analyzed, showing that the effects of the pandemic were not short-term. Additionally, concerning the study of Lahmiri and Bekiros [32], although not the main finding of their analyses, it is nevertheless important to note that their results showed a decrease in Rényi entropy (and consequently a decrease in randomness) for the BTC/USD market during the pandemic compared to before.

In addition, we applied one more complexity measure, the Fisher information. Fisher information is a useful method to study non-stationary and complex time series. Fisher information is used as a measure of the degree of order of a system, behaving inversely to entropy, i.e., when the order increases, the entropy decreases, while the Fisher information increases. Moreover, unlike entropy, it is sensitive to changes in the shape of the probability distribution corresponding to the measured variable. Figure 3i,j illustrate the temporal evolution of the Fisher information of the BTC/USD and EUR/USD returns, respectively. We observe that immediately after the *WHO* announcement, the values of Fisher information increased in both BTC/USD and EUR/USD returns, indicating an increase in the order of the two markets.

At this point, it has to be mentioned that the observed decrease in randomness after the *WHO* announcement, indicated by all the applied complexity measures, is fully compatible with the corresponding increase of multifractality. Specifically, the more random a time series is, the more unifractal its scaling is, which means that a more multifractal time series can be considered as being farther away from "randomness" [71].

From the interpretation of our results in financial terms, useful conclusions are revealed. More specifically, in analyzing the values of *α*<sup>0</sup> for overall trend, as we have already mentioned, we observe that the BTC/USD returns show persistent behavior, while the EUR/USD returns exhibit anti-persistent behavior almost throughout the time period we studied them (Figure 1c,d). A persistent or anti-persistent market return series is characterized by a long memory effect. Therefore, what happens today, theoretically, will impact the future in a nonlinear fashion. For example, if a persistent market return change has been up (down) in the last period, then the changes will continue to be positive (negative) in the next period. On the other hand, anti-persistent markets are "mean-reverting." If the market return was up (down) in the previous period, it is more likely to be down (up) in the next period [72]. The long-memory characteristic in asset return is a fascinating topic for investors, risk managers, and scholars since appropriate return modeling is crucial for asset allocation and risk control. For example, existence of long memory in asset returns indicates that historical returns changes could be predictors of future returns changes [73]. Then, analyzing the Δ*α* and *A* parameters, we observe that in the post-announcement period, mainly in the case of the EUR/USD, the degree of multifractal returns increased, and, therefore, fluctuations became more intense and market risk increased (Figure 1e,f). At the same time, we observe that in the post-announcement period, returns were dominated by large fluctuations (Figure 1g,h). Therefore, it is concluded that in the post-announcement period, EUR/USD returns experienced intense and large fluctuations. Similar behavior is observed for the overall trend and uptrend markets of the BTC/USD returns. Regarding the downtrend markets of the BTC/USD returns, it appears that during the pandemic period there were less intense fluctuations compared to the pre-pandemic period without small or large fluctuations in returns dominating. The analysis of the truncation asymmetry degree (Figure 2e,f), moreover, revealed that the *WHO* announcement had different impacts on BTC/USD and EUR/USD returns concerning the undulation or instability of the underlying system. For EUR/USD returns, the post-announcement period was characterized by an increase in the undulation or instability of the underlying system, whereas for BTC/USD

returns, the opposite behavior was generally observed. Analyzing the complexity measures (Fuzzy entropy, Tsallis entropy, Shannon entropy, and Fisher information) (Figure 3c–j), we observe a sharp decline in complexity (i.e., increase in the order and information content, and decrease in randomness) in the returns of both BTC/USD and EUR/USD in the post-announcement period. This fact, in financial terms, suggests that the pandemic led investors to behave in an "organized" (similar) way that thereby reduced the complexity of the two markets. In other words, after the outbreak of the pandemic, it seems that investors behaved like a herd. Therefore, it is concluded that although the fluctuations were larger and more intense after the outbreak of the pandemic, this was not carried out in a random way as investors seem to have behaved in an "organized" way; however, this behavior for BTC/USD returns was generally followed by a decrease in undulation or instability of the underlying system, while the opposite happened for EUR/USD returns.

Additionally, it is worth noting that the majority of the measures that we studied showed a strong change for both BTC/USD and EUR/USD returns immediately after the *WHO* announcement (11 March 2020), in which COVID-19 was mentioned for the first time as a pandemic. This fact indicates that the behavior of the system changed immediately after the *WHO s* announcement, although the discussions about COVID-19 being a public health emergency of international concern had begun weeks before. Therefore, although many researchers accept the date of 2 January 2020 as the beginning of the COVID-19 pandemic crisis (e.g., [74–76]), we consider the most suitable start date of the pandemic to be 11 March 2020.
