# Investigating Dynamical Complexity and Fractal Characteristics of Bitcoin/US Dollar and Euro/US Dollar Exchange Rates around the COVID-19 Outbreak

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Asymmetric Multifractal Detrended Fluctuation Analysis (A-MF-DFA)

`(`${b}_{n,v}<0$

`),`the return time series have an upward (downward) trend within the $v$th segment.

`,`${M}^{+}+{M}^{-}=2{N}_{n}$ holds. Therefore, the $q$th order average fluctuation functions for the overall trend is written as:

`,`namely, the generalized Hurst exponent, is calculated by estimating the slope of the linear regression of $\mathrm{log}\left({F}_{q}\left(n\right)\right)$ versus $\mathrm{log}\left(n\right)$. The asymmetric generalized exponents ${h}^{+}\left(q\right)$ and ${h}^{-}\left(q\right)$ are calculated in a similar way from the relationship ${F}_{q}^{+}\left(n\right)\sim {n}^{{h}^{+}\left(q\right)}$ and ${F}_{q}^{-}\left(n\right)\sim {n}^{{h}^{-}\left(q\right)}.$ In this study, we consider $n$ ranging from $8$ to $N/4$ for the log-log linear regression to estimate the asymmetric generalized Hurst exponents.

#### 2.1.1. Asymmetric Multifractal Spectrum Parameters

`.`The multifractal spectrum is obtained by applying the first-order Legendre transform [39,46]:

`.`More specifically, the maximum of the multifractal spectrum $f\left(\alpha \right)$ is used to detect the correlation behavior in terms of persistence and anti-persistence. The spectrum ${\alpha}_{0}$ gives the maximum $f\left(\alpha \right)$, i.e., $f\left({\alpha}_{0}\right)=1$. At this spectrum, the measure provides information about the central tendency of the multifractal spectrum. If ${\alpha}_{0}<0.5$, then the correlations in the time series exhibit anti-persistent behavior (i.e., an increase is very likely to be followed by a decrease), if ${\alpha}_{0}>0.5$, then the correlations in the time series exhibit persistent behavior (i.e., an increase is very likely to be followed by an increase, and a decrease is very likely to be followed by a decrease), whereas if ${\alpha}_{0}=0.5$, then the time series displays characteristics of a standard non-correlated sequence [39,47,48]. By looking into the spectrum width, one can quantitatively detect the time series multifractality. Specifically, the width of the spectrum is estimated by the equation $\mathsf{\Delta}\alpha ={\alpha}_{max}-{\alpha}_{min}$, and it reflects the degree of multifractality of the time series. The larger values of $\mathsf{\Delta}\alpha $ are, the stronger the degree is and the more severe the fluctuations in the time series are. On the contrary, the smaller the values of $\mathsf{\Delta}\alpha $, the more the time series is close to a monofractal behavior, indicating less significant fluctuations in the time series. The spectrum width should be equal to zero for a completely monofractal time series [39,49,50]. The dominance of small or large fluctuations is also an interesting characteristic of time series. This information can be extracted from the skew asymmetry of the multifractal spectrum, which is defined by the equation [51] $A=\frac{L-R}{R+L}=\frac{-\Delta S}{W}$, where $R={\alpha}_{max}-{\alpha}_{0}$, $L={\alpha}_{0}-{\alpha}_{min}$, $\Delta S=R-L$, and $W=R+L=\mathsf{\Delta}\alpha ={\alpha}_{max}-{\alpha}_{min}$. If $A>0\left(LR\right),$ the spectrum is left-skewed, which means that the scaling behavior of large fluctuations dominates the multifractal behavior. On the contrary, if $A<0\left(LR\right),$ then the spectrum is right-skewed, where the scaling behavior of small fluctuations dominates. The case of $A=0$ indicates that the shape of multifractal spectra is symmetric [46,51].

#### 2.2. Fuzzy Entropy (FuzzyEn)

#### 2.3. Tsallis Entropy

`,`but may occur for ${S}_{{q}_{TS}}$, with a particular value of the index, ${q}_{TS}\ne 1$. Such systems are called non-extensive [56]. The cases ${q}_{TS}>1$ and ${q}_{TS}<1$ correspond to sub-additivity or super-additivity, respectively. As in the case of Rényi entropies, we may think of ${q}_{TS}$ as a bias parameter: ${q}_{TS}<1$ privileges rare events, while ${q}_{TS}>1$ highlights prominent events [58].

#### 2.4. Fisher Information Μeasure (FIM)

## 3. Data and Results

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**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 ${\alpha}_{0}$ 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 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 $\mathsf{\Delta}f\left(a\right)=f\left({\alpha}_{min}\right)-f\left({\alpha}_{max}\right)$. (

**e**,

**f**): Temporal evolution of the degree of truncation asymmetry, known as $\mathrm{C}-\mathrm{value}=\left|\mathsf{\Delta}f\left(a\right)\right|=\left|f\left({\alpha}_{min}\right)-f\left({\alpha}_{max}\right)\right|$. 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 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.

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**MDPI and ACS Style**

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

**AMA Style**

Zitis PI, Kakinaka S, Umeno K, Hanias MP, Stavrinides SG, Potirakis SM.
Investigating Dynamical Complexity and Fractal Characteristics of Bitcoin/US Dollar and Euro/US Dollar Exchange Rates around the COVID-19 Outbreak. *Entropy*. 2023; 25(2):214.
https://doi.org/10.3390/e25020214

**Chicago/Turabian Style**

Zitis, Pavlos I., Shinji Kakinaka, Ken Umeno, Michael P. Hanias, Stavros G. Stavrinides, and Stelios M. Potirakis.
2023. "Investigating Dynamical Complexity and Fractal Characteristics of Bitcoin/US Dollar and Euro/US Dollar Exchange Rates around the COVID-19 Outbreak" *Entropy* 25, no. 2: 214.
https://doi.org/10.3390/e25020214