Does Bitcoin exhibit the same asymmetric multifractal cross-correlations with crude oil, gold and DJIA as the Euro, Great British Pound and Yen?

Chaos, Solitons and Fractals 109 (2018) 195–205

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Does Bitcoin exhibit the same asymmetric multifractal cross-correlations with crude oil, gold and DJIA as the Euro, Great British Pound and Yen? Gabriel Gajardo a, Werner D. Kristjanpoller a,∗, Marcel Minutolo b a b

Departamento de Industrias, Universidad Tecnica Federico Santa Maria, Avenida España 1680, Valparaiso, Chile School of Business, Robert Morris University, 6001 Unviersity Boulevard, Moon Township, PA 15108, USA

a r t i c l e

i n f o

Article history: Received 6 October 2017 Revised 6 February 2018 Accepted 21 February 2018

Keywords: Multifractality Asymmetric cross-correlations Bitcoin Exchange rates Crude oil market Gold market

a b s t r a c t We applied MF-ADCCA to analyze the presence and asymmetry of the cross-correlations between the major currency rates and Bitcoin, and the Dow Jones Industrial Average (DJIA), gold price and the oil crude market. We find that multifractality exists in every cross-correlation studied, and there is asymmetry in the cross-correlation exponents under different trend of the WTI, Gold and DJIA. Bitcoin shows a greater multifractal spectra than the other currencies on its cross-correlation with the WTI, the Gold and the DJIA. Bitcoin shows a clearly different relationship with commodities and stock market indices which has to be taken into consideration when investing. This has to do with the years this currency has been traded, the characteristics of cryptocurrencies and its gradual adoption by financial organizations, governments and the general public.

1. Introduction Cryptocurrency is quickly becoming an important aspect of the global financial market. At the time of this writing, Cyrptocurrency Market Capitalization lists the total market capitalization of all cryptocurrency at approximately $144 billion dollars of which Bitcoin accounts for almost half of all the valuation. Bitcoin is currently trading at approximately $4,100 per coin and has a total capitalization of approximately $68 billion USD; a valuation which just five years ago would have been unthinkable. Major financial organizations are taking large positions in cryptocurrencies, retailers are taking coin for payment, and people are sending money abroad. In a 2016 piece, Harwick [1] finds that Bitcoin possesses some attributes that may make it a good complement to currencies of emerging markets. Some of the promise of cryptocurrency comes from the potential to reduce transaction costs, the security in the transaction, and potential reduction in exchange rate risk[2]. However, at present there has been little attention paid to how cryptocurrency behaves. Given the rising use of the cryptocurrencies, we propose to perform an analysis of the behavior of the value Bitcoin applying fractal theory and comparing it with the behavior of some ma-

∗

Corresponding author. E-mail address: [email protected] (W.D. Kristjanpoller).

https://doi.org/10.1016/j.chaos.2018.02.029 0960-0779/© 2018 Published by Elsevier Ltd.

© 2018 Published by Elsevier Ltd.

jor global currencies including the Euro, the Great Britain Pound, and the Japanese Yen. In particular, we propose to evaluate the behavior with respect to the presence and asymmetry of crosscorrelations between these currencies and three major financial assets: gold, crude oil and the DJIA index. The current study is motivated by the rise of the interest and potential of cryptocurrency in general and Bitcoin in particular. Analyzing the behavior of Bitcoin with respect to crude oil, gold and the DJIA can be contrasted if its behavior is similar to other currencies. If the results show that there is similarity with respect to other currencies then one may conclude that Bitcoin behaves like any other currency. However, if the results show differences then we might concluded that this an anomaly of the currency or that Bitcoin, and cryptocurrency by extension, does not behave as a currency and may be that its behavior is more similar to another financial asset. In fact, under the Internal Revenue Service (IRS) guidance, crytpocurrency is treated as property for U.S. Federal tax purposes (Notice 2014-21, 2014-16, IRBXXX). Even if the results are dissimilar one could conclude that we are facing a financial bubble or irrational exuberance. Recent movements in the exchange rate between Bitcoin and the U.S. Dollar highlight the importance of understanding the behavior of the asset. On December 17, 2017 the exchange rate between Bitcoin and the USD broke the $20,0 0 0 mark and in the same day closed at just above $19,0 0 0. In the month and a half since the all-time high, the rate has dropped back down to a low

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closing price of $7,100. The movement suggests that cryptocurrencies remain a speculators market and perhaps we have just witnessed the largest bubble in Bitcoin to date. Hence, the need to develop a better understanding of this instrument. Assuming that cryptocurrency in general, and Bitcoin in particular, one needs to apply a model that captures the potential nonlinearity of the instrument. Traditional techniques such as Ordinary Least Squares is limited in its ability to capture complex relationships between data. Cheung et al. [3] applied an augmented Dickey–Fuller test and found the presence of a number of shortlived bubbles and three large bubbles in their analysis of Bitcoin. Their approach was selected to test for explosive behavior in a given time series. Hence, a different approach that has the potential to capture the complexity and is more robust is sought. To this end, we propose to apply fractal analysis to the time series. One of the most widely used tests to determine the fractal dimension of a given time series is the Rescaled Range Analysis (R/S), introduced by Hurst [4,5]. One of the main benefits of the R/S analysis is that it is robust in that its behavior is related only by the long-term persistence dependence being able to detect nonperiodic cycles even if they have a length greater than the analyzed sample period. Additionally, R/S is able to detect long-term correlations in random processes. Mandelbrot and Wallis [6] using R/S found that many natural phenomena are not independent random processes; and, giving an interpretation to the exponent of Hurst H there is significant long-term correlation. One could use other approaches; however, there are limitations to others that fractal analysis overcomes. For instance, some limitations of traditional models, such as Fourier transform and spectral analysis, fail to qualify scaling behaviors. Peng et al. [7] developed the Fluctuation Analysis (FA) method and then the Detrended Fluctuation Analysis (DFA) method [8]. Because the mono-fractal scaling behavior cannot fully describe the uneven multi-fractal characteristics of the time series and signals, it is necessary to develop the Multi-Fractal Detrended Fluctuation Analysis (MF-DFA) [9,10]. The MF-DFA method has been applied in many fields [11–17]. Particularly in the field of finance MF-DFA has been applied to analyze stock market [18–28], exchange rates [29–39], interest rates [40], market efficiency [41–46], risk market valuation [47,48], financial crisis [49,50], investment strategies [51], gold market [52– 55], crude oil market [56–59], and the future market [60–62]. Since the behavior of financial assets may contain components of trends and asymmetry in the reaction to different impacts Alvarez-Ramirez et al. [63] introduced the asymmetric DFA (A-DFA) to analyze asymmetric correlations in the scaling behavior of the time series. Cao et al. [64] further extended the DFA (A-DFA) with the proposition of the asymmetric multi-fractal detrended fluctuation analysis (A-MFDFA). Further, Zhang et al. [65] introduced the asymmetric multi-fractal detrending moving average analysis (AMFDMA). Recently, Lee et al. [66] used A-MFDMA to analyze U.S. stock market indexes while Gajardo and Kristjanpoller [67] applied the cross-correlation version of the A-MFDFA, the A-MFDCCA, to study the Latin American stock markets and their relationship with the oil market. Analyzing the behavior of Bitcoin with respect to crude oil, gold and DJIA, one can be contrasted if its behavior to determine its similarity to the major currencies. If the results show that there is similarity with the other currencies a conclusion could be that Bitcoin behaves like any other currency; but, if the results show differences it can be concluded that this is an anomaly of a currency or that Bitcoin does not behave as a currency and may be that its behavior is more similar to another financial asset. Even if the results are dissimilar one might conclude that we are facing a financial bubble or irrational exuberance. This is the first study of Bitcoin’s multifractal properties. The comparison of the multifactral and asymmetric behavior of the currencies is carried out with

respect to three major financial assets, the price of gold, the price of crude oil and the DJIA stock index. The remainder of this paper is organized as follows. Section 2 describes the method used, Section 3 describes the data used in the analysis. Section 4 provides the results obtained. In the final section of the manuscript, we present our conclusions and recommendations. 2. Multifractal asymmetric detrended cross-Correlation analysis method Two time series xi and yi , i = 1, . . . , N, N is the length of the series. The following steps summarizes the approach [67]. First: Construct the profile

X (i ) =

i 

(xt − x¯ ), Y (i ) =

t=1

i 

(yt − y¯ ),

i = 1, . . . , N

(1)

t=1

Where x¯ and y¯ represent the average of the series in the whole period. Second: X(i) and Y(i) are separated into Ns ≡ [N/s] nonoverlapping windows of equal length s. Since the length of the series N is not necessarily a multiple of the time scale s, some part of the profile can remain at the end. In order to not discard this part, the same procedure is applied starting from the end of the series. This means that we obtain 2Ns segments. Third: The trends, Xv (i) and Yv (i) for each one of the 2Ns segments are estimated with a linear regression as: X v (i ) = aX v + bX v · i and Y v (i ) = aY v + bY v · i. This precedes the determination of the detrended covariance, calculated as follows

F ( v, s ) =

s 1 |X [(v − 1 )s + i] − X v (i )| · |Y [(v − 1 )s + i] − Y v (i )| s i=1

(2) for each segment v, v = 1, . . . , Ns and

F ( v, s ) =

s 1 |X [N − (v − Ns )s + i] − X v (i )| s i=1

· |Y [N − (v − Ns )s + i] − Y v (i )|

(3)

for each segment v, v = Ns + 1, . . . , 2Ns . Fourth: The qth order of the fluctuation function is obtained as follows for the different behavior of the trends in time series xt



Fq+

(s ) =

2N 1 s sign(bX v ) + 1 [F (v, s )]q/2 M+ 2

1/q

(4)

v=1

 Fq−

(s ) =

2N 1 s −[sign(bX v ) − 1] [F (v, s )]q/2 M− 2

1/q (5)

v=1

when q = 0, and



F0+

2N 1 s sign(bX v ) + 1 (s ) = exp [F (v, s )]q/2 2M + 2

1/q (6)

v=1



F0−

2N 1 s −[sign(bX v ) − 1] (s ) = exp [F (v, s )]q/2 2M − 2

1/q (7)

v=1

 Ns sign(bX v )+1  Ns −[sign(bX v )−1] for q = 0. M+ = 2v=1 and M− = 2v=1 are 2 2 the number of subtime series with positive and negative trends. We assume bX v = 0 for all v = 1, . . . , 2Ns , such that M+ + M− = 2Ns .

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The traditional MF-DCCA is implemented by computing the average fluctuation function for q = 0



Fq (s ) =

2N 1 s [F (v, s )]q/2 2Ns

1/q

(8)

v=1

and as follows when q = 0



2N 1 s Fq (s ) = exp ln[F (v, s )] 4Ns

 (9)

v=1

Fifth: The scaling behavior of the fluctuations is analyzed by observing the log-log plots of Fq (s) versus s for each value of q. In the case where the two series are long-range cross-correlated, Fq (s) will increase for large values of s, as a power law

Fq (s ) ∼ sHxy (q )

(10)

Fq+ (s ) ∼ sHxy (q )

(11)

Fq− (s ) ∼ sHxy (q )

(12)

+

Fig. 1. Cross-correlation exponent Hxy (q) vs. q for the currency series and WTI.

−

The scaling exponent Hxy (q) is known as the generalized crosscorrelation exponent, and describes the power-law relationship between two series. It is obtained by calculating the slope of the loglog plots of Fq (s) versus s through the method of Ordinary Least Squares (OLS). In the case of q = 2, the generalized cross-correlation exponent has similar properties and interpretation as the univariate Hurst exponent calculated by the DFA. If Hxy (2) > 0.5, the series are crosspersistent, so a positive (negative) change in one price is more statistically probable to be followed by a positive (negative) value of the other price. In the case where Hxy (2) < 0.5 the series are crossantipersistent, which means that a positive (negative) change in one price is more statistically probable to be followed by a negative (positive) change on the other price. For Hxy (2 ) = 0.5 only short-range cross-correlations (or no correlations at all) are present in the relationship between the series. To measure the asymmetric degree of the cross-correlations we can calculate, for every q, the following metric + − Hxy (q ) = Hxy (q ) − Hxy (q )

Fig. 2. Asymmetry degree Hxy (q) for the bivariate series when WTI has different trends.

(13)

The greater the absolute value, the greater the asymmetric behavior. If Hxy (q) is equal or close to zero, then the cross-correlations are symmetric for different trends of time series xt . If Hxy (q) is positive, it means that the cross-correlation exponent is higher when the time series xt has a positive trend than when it is negative. If it is negative, the cross-correlation exponent is lower when the time series xt has a positive trend than when it is negative. If the value of the generalized cross-correlation exponent Hxy (q) depends on the value of q, the cross-correlation between the two time series is multifractal. Just like in the MF-DCCA, for q > 0, + − Hxy (q), Hxy (q ) and Hxy (q ) describe the scaling behavior of large fluctuations, and for q < 0, describe the scaling behavior of small fluctuations. In this paper, we study the multifractal behavior under different trends of both the WTI and the stock market indices. 3. Data

Fig. 3. Multi-Fractal spectra v/s α .

For this study, we selected the daily closing price of the following currencies: the Euro (EUR); the Great Britain Pound (GBP); the Yen (YEN) and, Bitcoin (BTC); all prices expressed in American Dollar (USD). We analyzed the behavior of these currencies with respect to three financial assets: Gold (GOLD); Crude oil (WTI) and the Dow Jones Industrial Average (DJIA). The sample interval

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+ − Fig. 4. Plots of the Generalized Cross Correlation Exponent Hxy (q), Hxy (q ) and Hxy (q ) under different trends of the WTI.

Table 1 The descriptive statistics of return series. J-B represents the Jarque–Bera statistic, ADF denotes the Augmented Dickey– Fuller, Q(10) denotes the value of Ljung–Box–Pierce Q statistic with 10 lags and the ARCH(10) is the Engle’s ARCH test with 10 lags.

Mean (%) Std. Dev. (%) Median (%) Maximum (%) Minimum (%) Skewness Kurtosis Jarque–Bera ADF Q(10) ARCH(10) Observations ∗∗

Bitcoin

Dow Jones

Euro

GB Pound

Gold

WTI

YEN

0.444 6.195 0.308 48.478 −66.394 −1.042 25.023 30299.21∗ ∗ ∗ −40.228∗ ∗ ∗ 33.062∗ ∗ ∗ 218.978∗ ∗ ∗ 1486

0.045 0.812 0.040 4.153 −4.574 −0.264 6.102 613.15∗ ∗ ∗ −40.197∗ ∗ ∗ 19.288∗ ∗ 237.259∗ ∗ ∗ 1486

0.009 0.565 0.007 3.176 −3.025 0.054 5.482 382.28∗ ∗ ∗ −40.603∗ ∗ ∗ 15.479 66.884∗ ∗ ∗ 1486

−0.012 0.569 −0.006 2.347 −8.402 −2.306 35.762 67776.97∗ ∗ ∗ −38.534∗ ∗ ∗ 12.179 53.376∗ ∗ ∗ 1486

−0.023 1.088 −0.036 4.603 −9.821 −0.762 10.194 3347.89∗ ∗ ∗ −19.880∗ ∗ ∗ 3.222 25.400∗ ∗ ∗ 1486

−0.043 2.145 −0.011 11.621 −10.794 0.182 6.046 582.67∗ ∗ ∗ −17.256∗ ∗ ∗ 16.184∗ 231.564∗ ∗ ∗ 1486

0.024 0.621 0.010 3.464 −3.772 −0.170 6.965 980.53∗ ∗ ∗ 20.555∗ ∗ ∗ 8.684 38.364∗ ∗ ∗ 1486

Indicates rejection at the 5% significance level.

∗∗∗

Indicates rejection at the 1% significance level.

is from September 13th, 2005, to August 25th, 2017, which after cleansing left 1487 observations. To analyze the relationship between the currencies and the three financial assets, we calculate the daily return, rt , as the logarithmic difference between consecutive prices, rt = log(Pt ) − log(Pt−1 ). In Table 1, we present the descriptive statistics or the

daily returns for Bitcoin, DJIA, the Euro, GB Pound, Gold, WTI, and the Japanese Yen. One can see from the results that over the period under consideration, Bitcoin had a much higher average daily return than the other currencies and commodities. Additionally, during the period, while Bitcoin, the Euro, and the Yen had positive average daily returns, the Pound, Gold, and WTI had negative aver-

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199

Fig. 5. Log–log plots of F2 (n), F2+ (n ), F2− (n ) versus n when WTI has different trends.

Fig. 6. Cross-correlation exponent Hxy (q) vs. q for the currency series and Gold.

Fig. 7. Asymmetry degree Hxy (q) for the bivariate series when Gold has different trends.

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standard deviation of Bitcoin is 6.195, the next closest asset under consideration is WTI with a measure of 2.145 suggesting that Bitcoin is much more volatile than the other assets. Finally, looking at the range between the maximum daily return and the minimum daily return we note again that Bitcoin has a much greater range than the other assets further highlighting the overall volatility. 4. Empirical results In this section, we analyze the cross correlations for changes in the trends of the WTI, GOLD and DJIA for the four currencies studied. 4.1. Asymmetric behavior in the cross-correlations for different trends of crude oil prices

Fig. 8. Multi-Fractal spectra v/s α .

age daily returns. Additionally, we note that the standard deviation of Bitcoin is much higher than any other instrument. While the

In Fig. 1, the relationship between Hxy (q) and q is shown, with q varying from -10 to 10. It can be seen that Hxy (q) decreases as q is greater. Given that Hxy (q) decreases as q increase, we can state that Hxy(q) is not a constant, which indicates that multifractality exists in the cross-relations between all currencies and WTI and the scaling exponents. However,there is a difference in magnitude of the behavior of Bitcoin with respect to the WTI, since the Hxy (q) falls from 0.79 to 0.4 in the whole spectrum, which is much greater than the difference of the other currencies. We see in Fig. 1, that

+ − Fig. 9. Plots of the Generalized Cross Correlation Exponent Hxy (q), Hxy (q ) and Hxy (q ) under different trends of Gold.

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201

Fig. 10. Log–log plots of F2 (n), F2+ (n ), F2− (n ) versus n when Gold has different trends.

Fig. 11. Cross-correlation exponent Hxy (q) vs. q for the currency series and DJIA.

Fig. 12. Asymmetry degree Hxy (q) for the bivariate series when DJIA has different trends.

the rate of decreases is much sharper than the other assets and while Bitcoin starts much higher than the others, it falls to nearly

the same score and below the Euro. All the functions are monotonically decreasing.

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G. Gajardo et al. / Chaos, Solitons and Fractals 109 (2018) 195–205 Table 2 + − (2 ) and Hxy (2 ) for the bivariate series when the WTI has different Hxy (2), Hxy trends.

WTI-EUR WTI-GBP WTI-YEN WTI-BTC

Fig. 13. Multi-Fractal spectra v/s α .

In Fig. 2, we plot the asymmetry of Hxy(q) with respect to q for WTI. Of particular interest is that WTI has different trends with respect to each of the assets. As can be observed in the Fig. 3 the

Hxy (2)

+ (2 ) Hxy

− Hxy (2 )

Hxy (2)

0.514 0.514 0.514 0.561

0.448 0.508 0.498 0.572

0.602 0.529 0.546 0.551

−0.154 −0.021 −0.048 0.0206

behavior of Bitcoin with respect to the WTI is completely different than the other currencies. In particular it can be seen that it has a broader spectrum and shifted to the right, which implies that it has a deeper multifractal behavior than the other assets. In the analysis of Generalized Cross Correlation Exponent (Fig. 4), it can be seen that only the Bitcoin shows a slightly higher Hxy (q) when the WTI market is up. This can be confirmed by the asymmetry degree in Fig. 2, where Bitcoin shows the lower asymmetry degree for almost all fluctuations sizes. In the case of GBP a negative asymmetry is observed until q = 3, where its asymmetry is reversed. As can be observed in the log-log plots of F2 (n), F2+ (n ) and F2− (n ) versus n (Fig. 5), the monofractal case presents clear slope difference between F2+ (n ) and F2− (n ) for the EUR, which shows the presence of asymmetry in the cross-correlation with the WTI. Specifically the generalized cross-correlation exponent, Hxy (2), tends to be higher when the WTI is downwards. As can be

+ − Fig. 14. Plots of the Generalized Cross Correlation Exponent Hxy (q), Hxy (q ) and Hxy (q ) under different trends of DJIA.

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203

Fig. 15. Log–log plots of F2 (n), F2+ (n ), F2− (n ) versus n when DJIA has different trends.

seen in Table 2 in the case of Bitcoin both slopes are almost equal, ruling out the presence of asymmetry. The size of asymmetry is similar with the ones of GBP and YEN, but has opposite direction. 4.2. Asymmetric behavior in the cross-correlations for different trends of Gold price In Fig. 6, the relationship between Hxy (q) and q is shown, with the effect of Gold. Again, it can be seen that Hxy (q) decreases as q is greater for all the currencies, indicating that multifractality exists in the cross-relations between all currencies and Gold and all the functions are monotonically decreasing. But, as with the WTI, there is a difference in magnitude of the behavior of Bitcoin with respect to other currencies. Even in this case the differential is even greater, since it has the highest Hxy (q) for q = − 10 and has the lowest Hxy (q) for q = 10. As can be observed in Fig. 8, the behavior of Bitcoin with respect to Gold is different from the other currencies, as was observed with respect to the WTI. In particular it can be seen that it has a broader spectrum and shifted to the right, which implies that it has a deeper multifractal behavior. In this case the amplitude of the spectrum is greater than the case with respect to the WTI, showing greater multifractality. In the analysis of Generalized Cross Correlation Exponent (Fig. 9), it can be seen that the YEN shows a higher Hxy(q) when the Gold market is up for all fluctu-

ation sizes. Both the Bitcoin and the Euro have a similar behavior, showing higher values of Hxy (q) when the Gold market is down until q = 2 and q = 2, respectively, and then reversing its behavior. This behavior contrasts with the behavior of the GBP which for q < 2 shows similar behavior whether the Gold is up or down, but for q > 2 the Hxy (q) becomes higher under downward trends in Gold. In Fig. 7, it can be clearly observed that the asymmetry degree rises with fluctuation sizes, and that Bitcoin presents the most asymmetrical behavior. The log-log plots (Fig. 10), show a slope difference between F2+ (n ) and F2− (n ) indicating the presence of asymmetry in the cross-correlations for YEN, being higher for the case of upward trend of Gold. While for Bitcoin it happens that both tendencies are almost parallel, but the superior one is in the case of a downward market for Gold. On the other hand, in the case of the Euro and GBP the difference between its straight lines is almost nil, implying an almost absence of asymmetry with respect to the tendencies of the Gold. This can be confirmed by observing Table 3, where the asymmetry degree for Bitcoin is just 0.0 0 03. 4.3. Asymmetric behavior in the cross-correlations for different trends of Dow Jones Industrial Average index By analyzing the relationship between Hxy (q) and q (Fig. 11) incorporating the DJIA effect in Fig. 11, it can be seen that all func-

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G. Gajardo et al. / Chaos, Solitons and Fractals 109 (2018) 195–205 Table 3 + − (2 ) and Hxy (2 ) for the bivariate series when Hxy (2), Hxy the Gold has different trends.

Gold-EUR Gold-GBP Gold-YEN Gold-BTC

Hxy (2)

+ (2 ) Hxy

− Hxy (2 )

Hxy (2)

0.495 0.476 0.503 0.530

0.502 0.481 0.522 0.533

0.483 0.464 0.475 0.533

0.019 0.018 0.047 0.0 0 03

Table 4 + − (2 ) and Hxy (2 ) for the bivariate series when Hxy (2), Hxy DJIA has different trends.

DJIA-EUR DJIA-GBP DJIA-YEN DJIA-BTC

Hxy (2)

+ (2 ) Hxy

− Hxy (2 )

Hxy (2)

0.450 0.458 0.485 0.532

0.434 0.465 0.516 0.529

0.453 0.448 0.454 0.529

−0.019 0.016 0.062 0.0 0 0 02

tions are monotonically decreasing, with a similar behavior for the four currencies. It can also be noted in Fig. 13 that throughout the spectrum analyzed always the currency that has the greatest multifractality is Bitcoin. When analyzing the monofractal case of q = 2, the only coin that has an Hxy (2) greater than 0.5 is the Bitcoin, and it can be concluded that it has persistence in the monofractal context, whereas other currencies show anti-persistence. The behavior of Bitcoin with respect to the DJIA exhibits similar behavior to what was seen with respect to the WTI and the Gold. As can be seen in Fig. 13, it is the instrument that has the largest spectrum and its curve is shifted to the right. In this case, its spectrum is not as broad as with Gold. This evidence again confirms that Bitcoin has a greater multifractal behavior. In the analysis of Generalized Cross Correlation Exponent (Fig. 14), it can be seen that only Bitcoin shows a higher Hxy (q) when the DJIA market is down for every fluctuation size. EUR also shows a higher Hxy (q) for downward trends, but it is limited to large fluctuations and the difference is negligible. This small difference is also shown in GBP, but the cross-correlation exponent is always larger for upward trends of the DJIA. In Fig. 12, the differences in asymmetry are more clear. YEN shows a consistently higher cross-correlation exponent across fluctuations sizes. It even increases with q. On the other hand, other currencies experience a decrease in asymmetry, with the EUR turning its direction. Finally, the log-log plots of F2 (n), F2+ (n ) and F2− (n ) versus n (Fig. 15) with Table 4 show that there is a clear difference in the behavior of Bitcoin with respect to the DJIA than that of traditional currencies with respect to the DJIA. This difference consist in an nil asymmetry degree. EUR and GBP also show a small asymmetry degree, and YEN shows a clear asymmetry. 5. Conclusions The study presented consists on the first application of MFADCCA between Bitcoin and WTI, GOLD and DJIA. And the comparison in this behavior with other currencies. Our study resulted in the following findings: •

•

Multifractality exists in every cross-correlation studied, and there is asymmetry in the cross-correlation exponents under different trend of the WTI, Gold and DJIA. This asymmetry can appear across fluctuations sizes or only for small or large fluctuations. Bitcoin shows a greater multifractal spectra than the other currencies on its cross-correlation with the WTI. The crosscorrelation exponent shows a deviation from the asymmetric behavior of other currencies as the exponent is greater under upward trends of the WTI for all fluctuation sizes. However, the

•

•

•

•

•

•

•

•

•

•

asymmetry is much weaker and stable than the other currencies. The EUR shows the greatest asymmetry in the cross-correlation with the WTI, giving evidence of how this commodity affects that currency. It shows the same behavior as YEN, with a greater cross-correlation exponent for downward trends in the WTI, but for this currency the asymmetry is not as high. Bitcoin shows a greater multifractal spectra than the other currencies on its cross-correlation with Gold. All currencies show similar behavior under different trends of Gold market, they show a very small asymmetry degree under small fluctuations, but this degree increases with the size of the fluctuations. Bitcoin shows no asymmetry in the behavior of the crosscorrelation with this commodity for the monofractal case (q = 2), where the asymmetry changes direction. For small fluctuations the generalized exponent is greater under downward trends, but for large fluctuations is greater under upward trends, where it shows a high asymmetry degree. Cross-correlation with the DJIA is where Bitcoin shows more similar behavior in cross-correlation with the other currencies, which can be interpreted as Bitcoin being more affected by commodities than industry indices. The multifractal spectra is still the largest one, but the asymmetry, as with Gold, is practically zero. For currencies other than Bitcoin, asymmetries tends to appear large fluctuations of Gold, and small fluctuations of DJIA (except for YEN). But under different trends of WTI they show a more heterogeneous behavior. Bitcoin shows a clearly different relationship with commodities and stock market indices which has to be taken into consideration when investing. This has to do with the years this currency has been traded, the characteristics of cryptocurrencies and its gradual adoption by financial organizations, governments and the general public. This findings show the importance of the multifractal analysis in contrast with monofractal one. There are some important issues in asymmetry of the cross-correlation that are not detectable under a monofractal scheme. Our findings are supported by early work such as Baek and Elbeck [68] who found that the volatility of Bitcoin is driven largely by buyers and sellers, not the price of currencies themselves making the instrument highly speculative. Our work extended these findings to include the multifractal identification. The current work focuses solely on Bitcoin and may not be generalizable to all cryptocurrencies. In fact, further work should be conduct on the major currencies such as Ethereum, Ripple, and Litecoin to name a few. As acceptance and usage of cryptocurrency increases, how it performs and moves relative to other assets will likely change. Therefore, we suspect, the models by which we understand the asset are likely to change as well.

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