**1. Introduction**

Financial markets are widely recognized as typical examples of complex dynamical systems [1]. Asset prices are created by a large number of nonlinear interactions between heterogeneous agents and complex events occurring in the external environment [2,3]. The properties observed in financial time series such as nonlinearity, long-range dependence [4,5], volatility clustering [6], fat tails [7,8], asymmetry [9], chaos [10,11], fractals and multifractals [12,13], and self-similarity [14] have attracted the interest of many scientists from different fields. In the last three decades, physicists have studied and developed models to understand the behaviors and interactions in financial systems, establishing an interdisciplinary research field known as Econophysics [15–17]. This term first appeared in the published article by Stanley et al. [18] when analyzing the Dow Jones index; they found

**Citation:** Zitis, P.I.; Kakinaka, S.; Umeno, K.; Hanias, M.P.; Stavrinides, S.G.; Potirakis, S.M. Investigating Dynamical Complexity and Fractal Characteristics of Bitcoin/US Dollar and Euro/US Dollar Exchange Rates around the COVID-19 Outbreak. *Entropy* **2023**, *25*, 214. https://doi.org/10.3390/ e25020214

Academic Editors: Stanisław Drozd˙ z,˙ Jarosław Kwapie ´n and Marcin W ˛atorek

Received: 8 December 2022 Revised: 15 January 2023 Accepted: 18 January 2023 Published: 22 January 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

that stock returns followed a power law distribution. Since then, significant progress has been made in the field of Econophysics [19].

The dynamics of financial markets are difficult to understand not only because of the complexity of their internal elements but also because of the many intractable external factors acting on them. A recent example of an external factor causing disruptions in global financial markets is the outbreak of the COVID-19 pandemic. At its roots, the COVID-19 crisis is not a financial or economic crisis, it is a health crisis. Nevertheless, through its effects on supply and demand conditions, it evolved rapidly to a large-scale financial and economic crisis. In March 2020, the US stock market hit the circuit breaker mechanism four times in a period of ten days. Since its inception in 1987, the breaker has only ever been triggered once, in 1997. At the same time as the US crash, stock markets in Asia and Europe plunged also. More specifically, Japan's stock market fell by more than 20%, while the UK's main index, FTSE, fell by about 10.87% on 12 March 2020. Additionally, during the pandemic period, most economies experienced exchange rate volatility and currency depreciation due to capital outflows and market sentiments. Typical examples are the Australian dollar hitting a 17-year low of AUD 0.59215 and the New Zealand dollar hitting an 11-year low of NZD 0.5850. Furthermore, the price of gold dropped about 3.53%. It is worth noting that although gold is considered a strong safe haven for most developed markets during financial crises, there are findings showing that during the pandemic it was a weak safe haven for investors in the stock market [20]. The impact of COVID-19 affected even the newer asset classes such as cryptocurrencies. The declines in value of the three leading cryptocurrencies (Bitcoin, Ethereum, and Litecoin) exceeded 50% during the pandemic period.

The exchange rate is crucial for maintaining an economy's external stability. As exchange rate directly associates with foreign debt, capital flows, trade balance, and export competitiveness, maintaining a stable exchange rate is one of the policymakers' major concerns. On the other hand, several researchers argue that specific characteristics of cryptocurrencies, including the independence from monetary policy and the non-correlation with traditional assets, increase their resilience during crisis periods such as the recent pandemic crisis [21–23]. However, there is also the opposing view which argues that monetary policy has a significant impact on the price of cryptocurrencies as well as that the cryptocurrencies do not have zero correlation with other asset classes. For example, Chaoqun Ma et al. [24] found a strong response of Bitcoin prices to unexpected monetary policy actions, while Khanh Quoc Nguyen [25] found that S&P 500 returns significantly affected Bitcoin returns during the pandemic period. Therefore, it is concluded that understanding the pandemic's impact and the possible changes it caused in the cryptocurrency and foreign exchange markets is crucial for both investors and risk managers as well as policymakers.

Particularly useful conclusions about the effects of COVID-19 on financial markets can be obtained by studying changes in the multifractality and complexity of financial time series during the period around the COVID-19 outbreak. In the field of Econophysics, extensive research has been conducted on these topics. For example, Mnif et al. [26] utilized the multifractal detrended fluctuation analysis (MF-DFA) approach to investigate the degree of cryptocurrency efficiency before and after the COVID-19 outbreak using a limited time period, until 19 May 2020. Their results indicated that the pandemic outbreak positively affected the efficiency of the five cryptocurrencies that they studied. Naeem et al. [9] examined the asymmetric efficiency of the cryptocurrencies Bitcoin, Ethereum, Litecoin, and Ripple, using 1-h data. In their analysis, the authors utilized the A-MF-DFA and their results showed that the price of cryptocurrencies exhibited significant asymmetric multifractality. Moreover, they found that uptrends showed stronger multifractality than downtrends. Additionally, applying the time-varying deficiency measure, they found that the pandemic outbreak had a negative impact on the efficiency of the cryptocurrencies that they analyzed. Kakinaka and Umeno [27], applying the A-MF-DFA, examined the asymmetric multifractality along with the market efficiency of two main

cryptocurrencies (Bitcoin, Ethereum) during the pandemic period, taking into consideration different investment horizons. Their empirical results showed that the outbreak of COVID-19 affected the efficiency of the two cryptocurrencies differently in the short- and long-term horizons. More specifically, after the outbreak of COVID-19, Bitcoin and Ethereum in the short term exhibited stronger multifractality, while in the long term exhibited weaker multifractality. In addition, they studied the asymmetric market patterns between small and large price fluctuations and between upward and downward trends. These results confirmed that the outbreak caused a significant change in the level of asymmetry in cryptocurrency markets. Aslam et al. [28] applied the MF-DFA to study the efficiency of foreign exchange markets during the initial period of the COVID-19 pandemic. In their analysis, they used high-frequency data of six major currencies, during the period from 1 October 2019 to 31 March 2020. Before calculating the MF-DFA, they examined the inner dynamics of multifractality through seasonal and trend decomposition using loess. Their results indicated that efficiency of foreign exchange markets during the COVID-19 outbreak declined. Mensi et al. [29] examined the effect caused by the COVID-19 crisis on the pricing efficiency and asymmetric multifractality of major asset classes (US Treasury bond, US dollar index, S&P500, Brent oil, Gold, and Bitcoin). In their article, they applied the permutation entropy on intraday data from 30 April 2019 to 13 May 2020. Their results indicated that after the outbreak of COVID-19, the efficiency of all asset classes that they studied was deteriorated, and in most cases this deterioration was significant. In addition, using the A-MF-DFA, they found evidence of asymmetric multifractality in all markets. Drozd˙ z et al. [ ˙ 30] studied the complexity of the cryptocurrency market in the period around the COVID-19 outbreak from three different perspectives. Their findings showed that throughout the time period analyzed, the returns of exchange rates were multifractal with intermittent signatures of bifractality that can be associated with the periods where the market was more volatile.

Lahmiri and Bekiros [31] investigated the time-varying characteristics of the informational efficiency in sixteen international stock markets and forty-five cryptocurrency markets before and during the pandemic period using the approximate entropy and Largest Lyapunov Exponent. Their results indicated that cryptocurrencies exhibited more irregularity and more instability during the pandemic period compared to international stock markets. Additionally, Lahmiri and Bekiros [32], applying Rényi entropy, analyzed the multiscale entropy function in the return time series of S&P500, Brent, WTI, Gas, Silver, Gold, Bitcoin, and VIX. Additionally, they analyzed the information sharing between these markets by estimating mutual information. Their results from Rényi entropy showed that for all market indices, disorder and randomness were more concentrated in less probable events. In addition, their results from the mutual information indicated that the information sharing network between markets has changed during the pandemic period. Wang J. and Wang X. [33] investigated the market efficiency of the S&P 500 Index, Gold, Bitcoin, and US Dollar Index during the extreme event of the COVID-19 pandemic using a multiscale entropy-based method. Their results indicated that, at all scales, the four markets' efficiency decreased abruptly and persistently during the period from February to March 2020. Market efficiency decreased the most in the S&P 500 Index and the least in the Bitcoin market. Additionally, their results showed that Bitcoin market efficiency was more resilient than the others during the extreme event. Fernandes et al. [34] investigated the informational efficiency and price disorder of five main cryptocurrencies (Ethereum, Bitcoin, Cardano, XRP, and BNB) before and during the pandemic period. In their article, the authors applied the permutation entropy and Fisher information measure to construct the Shannon–Fisher causality plane in order to map the cryptocurrencies and their respective locations in a two-dimensional plane. Their results indicated that all cryptocurrencies exhibited high but slightly varying informational efficiency during both periods. Additionally, their results showed that Cardano was the most efficient cryptocurrency. Kim and Lee [35] investigated the evolution of the complexity of the cryptocurrency market and analyzed the properties from the previous upward trend market in 2017 against the COVID-19 pandemic. In their article, the authors used three popular measures of complexity based on the nonlinear

analysis: sample entropy, approximate entropy, and Lempel–Ziv complexity. They studied the market complexity/unpredictability for forty-three cryptocurrency prices. They found that sample entropy, approximate entropy, and Lempel–Ziv complexity metrics of all markets could not generalize the COVID-19 effect of the complexity due to different patterns. Nevertheless, market unpredictability increased by the ongoing health crisis. Olbry´s and Majewska [36] applied sample entropy to evaluate changes in the regularity of returns of thirty-six U.S. and European stock market indices during periods of uncertainty. Specifically, the authors studied the period of the Global Financial Crisis as well as the period of the COVID-19 pandemic. Their results showed that entropy decreased during the periods of turbulence, indicating that the regularity and predictability of stock market indices returns increased during these periods. In the field of Econophysics, the study of the complexity and multifractality of financial time series during the pandemic is a challenging topic. However, to the best of our knowledge, until now there has not been an in-depth comparative analysis of the effects of the pandemic on the complexity and fractal characteristics of the returns of two completely different currencies, such as BTC/USD and EUR/USD, that play a key role in the modern financial system.

In this article, we present a study of 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 (i.e., 11 March 2020). We chose to analyze and compare the effects of the pandemic on the two most representative currencies from the cryptocurrency and forex markets, respectively. Although these two markets are completely different from each other, they play a significant role in the modern financial system. More specifically, we applied the A-MF-DFA to investigate the temporal evolution of the asymmetric multifractal spectrum parameters (*α*0, Δ*α*, *A*) before and after the outbreak of the pandemic. Although there are numerous studies that have followed a similar approach for the study of financial time series (e.g., [37–39]), as far as we are able to know, this is the first time that the temporal evolution of the specific parameters has been applied to BTC/USD and EUR/USD returns to study the period before and during COVID-19. At this point, it is important to mention that the analysis of the multifractal properties of financial time series has a wide contribution to the field of finance. For example, multifractality can be used to obtain better forecasts of tail risk as demonstrated by Batten et al. [40]. In addition, we examined the temporal evolution of four popular complexity measures. Although approximate and sample entropies are quite common for financial time series analysis [31,35,41], we chose to use Fuzzy entropy as it is considered as an upgraded alternative of approximate and sample entropy for evaluating the complexity, specifically for short time series contaminated by noise [42]. In combination, we chose to use the Shannon entropy as the standard information measure and Tsallis entropy as its non-extensive generalization, very closely related to multifractality. Additionally, we used another complexity measure, Fisher information. In financial data analysis, the application of Fisher information is very widespread for the construction of the Shannon–Fisher causality plane [34,43]. In the present article, we chose to investigate the temporal evolution of Fisher information as we believe that it can reveal useful elements for the evolution of the complexity of the dynamical system, providing a "mirror image" of the evolution of entropies, but also presenting the key difference of its so-called "locality" property (see Section 2.4). Our study attempts to provide a complete picture of the pandemic's impact in terms of the dynamical change of the complexity and the fractal characteristics of the two currencies. Additionally, our results provide useful conclusions about the behavior of two very different currencies during uncertainty periods. At the same time, interesting conclusions are drawn about the impact of *WHO* announcements and the reaction of investors to external events such as the pandemic. Our findings can help both investors and risk managers, as well as policymakers, to formulate a comprehensive response to the occurrence of such external events.
