**4. Conclusions**

The detection of dynamical complexity in time series originated from various complex systems, including the disciplines of physics, finance, and medicine, and is one of the foremost problems in science. The measurement of complexity includes nonlinear statistics methods to extract hidden patterns as well as exploring multifractality, randomness, and information flows. Hence, complexity provides important information regarding the order or disorder states of a system under scrutiny. In the field of finance, the detection of the dissimilarity of complexity between order and disorder states (e.g., before and after the occurrence of extreme events) could shed light on the mechanisms associated with investor reaction to these events.

In this article, we studied the temporal evolution of the multifractality and complexity of BTC/USD and EUR/USD returns for the period before and after the *WHO* announcement that declared COVID-19 a global pandemic. In our study, we first examined the asymmetric multifractality through the analysis of the multifractal spectrum parameters as obtained by the A-MF-DFA method. Then, we extended our analysis by applying Fuzzy, Tsallis, and Shannon entropies as well as the Fisher information measure. Our results can be summarized as follows: (i) For the entire time period that we studied (i.e., before and during the pandemic), the behavior of BTC/USD returns was persistent in all trends of the market. On one hand, in the period before the outbreak of the pandemic, the behavior of EUR/USD returns was anti-persistent in all trends of the market. On the other hand, in almost the entire period after the outbreak of the pandemic, the returns of the EUR/USD exhibited anti-persistent behavior in the overall trend and downtrend markets, while the uptrend market presented an alternating behavior, including short periods of persistent dynamics. (ii) Throughout the period analyzed, the width of the multifractal spectrum received higher values for the BTC/USD returns compared to the EUR/USD returns. This implies that the multifractality of the BTC/USD returns was higher than the multifractality of the EUR/USD returns. In addition, after the outbreak of the pandemic, in the overall trend and uptrend markets, the width of the multifractal spectrum increased for both BTC/USD returns and EUR/USD returns. In the case of BTC/USD, the downtrend multifractality was higher in the pre-announcement period. In EUR/USD, it appears that the downtrend markets played an important role in increasing multifractality. Nevertheless, both market trends may have had some impact on the post-announcement period increase

in multifractality. (iii) In the pre-announcement period, small fluctuations in BTC/USD returns for all market trends dominated. In contrast, in the post-announcement period, large fluctuations in BTC/USD returns for overall trend and uptrend markets dominated, while in downtrend markets the spectrum became practically symmetrical. On the other hand, although in the uptrend markets the spectrum of EUR/USD returns was almost symmetrical, the returns in the overall trend and downtrend markets were dominated by small fluctuations for almost the entire pre-announcement period. During the pandemic period, the returns of the EUR/USD were dominated by large fluctuations in all market trends. (iv) For both BTC/USD and EUR/USD returns and all market trends, a sharp change towards becoming more insensitive to the local fluctuations with small magnitudes was observed after the *WHO* announcement. Nevertheless, 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. (v) Fuzzy entropy, non-extended Tsallis entropy, Shannon entropy, and Fisher information showed a sharp decrease in the degree of complexity immediately after the *WHO* announcement for both BTC/USD and EUR/USD. This fact shows that in the post-announcement period, the order and the information content of the systems increased, i.e., the randomness and complexity in the returns of the two currencies decreased. Therefore, in financial terms, we conclude that investors seem to have behaved in an "organized" way, as a herd. In addition, our analyses show that the date of the *WHO* announcement (11 March 2020) could be considered as the most appropriate date for the start of the pandemic. This element could be useful in future research studies.

The main finding that is revealed from our study is that immediately after the *WHO* announcement, the returns of both BTC/USD and EUR/USD presented a decrease in complexity and corresponding increase in multifractality, both indicating that they became less random compared to the pre-announcement period. Hence, it seems that although they are two such different currencies, which both play a key role in the modern financial system, they reacted in a similar way in response to the pandemic.

**Author Contributions:** Conceptualization, P.I.Z. and S.M.P.; methodology, P.I.Z., S.K., K.U., M.P.H., S.G.S. and S.M.P.; software, P.I.Z., S.K. and S.M.P.; validation, P.I.Z., M.P.H., S.G.S. and S.K.; formal analysis, P.I.Z. and S.K.; investigation, P.I.Z., S.K. and S.M.P.; data curation, P.I.Z.; writing—original draft preparation, P.I.Z., S.K. and S.M.P.; writing—review and editing, K.U., M.P.H., S.G.S. and S.M.P.; visualization, P.I.Z. and S.K.; supervision, S.M.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Data Availability Statement:** All financial time series used in this article are publicly available from Yahoo Finance (http://finance.yahoo.com/, accessed on 7 December 2022).

**Conflicts of Interest:** The authors declare no conflict of interest.
