Power-Law Return-Volatility Cross Correlations of Bitcoin
PPower-Law Return-Volatility CrossCorrelations of BitcoinT. Takaishi
Hiroshima University of Economics, Hiroshima 731-0192 JAPAN
Abstract
This paper investigates the return-volatility asymmetry of Bitcoin. Wefind that the cross correlations between return and volatility (squaredreturn) are mostly insignificant on a daily level. In the high-frequencyregion, we find that a power-law appears in negative cross correlationbetween returns and future volatilities, which suggests that the cross cor-relation is long ranged. We also calculate a cross correlation betweenreturns and the power of absolute returns, and we find that the strengthof the cross correlations depends on the value of the power. a r X i v : . [ q -f i n . S T ] F e b Introduction
It has long been known that return and volatility are negatively correlated,and early studies[1, 2] attempt to explain the return-volatility asymmetry as aleverage effect: a drop in the value of a stock increases finance leverage or debt-to-equity ratio, which makes the stock riskier and increases the volatility. Theother promising explanation for the return-volatility asymmetry is the volatilityfeedback effect discussed in [3, 4]: if volatility is priced, an anticipated increasein volatility raises the required return, leading to an immediate stock pricedecline. Although the two effects suggest the same negative correlations, thecausality is different[5].Comparing the two effects empirically, Baekaert et al.[5] and Wu[6] arguethat the dominant determinant is the volatility feedback effect. However, thestudies using GARCH-type models[7, 8, 9] suggest that volatility increases moreafter negative returns than positive ones, which favors the leverage effect.To discuss the full temporal structure of return-volatility asymmetry, usingsquared returns as a proxy of volatility, Bouchaud et al.[10] calculate the return-volatility correlation function and find that returns and future volatilities arenegatively correlated. On the other hand, reverse correlations, i.e., correlationsbetween future returns and volatilities are found to be negligible. The resultsare fitted to an exponential function, and it is concluded that the correlationsare short ranged. In addition, the decay times are estimated to be about 10(50) days for stock indices (individual stocks).While for most developed markets, negative correlations between returnsand future volatilities are found, an interesting phenomenon is observed in Chi-nese markets. Qiu et al.[11] calculate the return-volatility correlation functionfor equities in the Chinese market and find that returns and future volatili-ties are ”positively” correlated, which is called the anti-leverage effect. Furtherstudies[12, 13] also support the anti-leverage effect in the Chinese market.In this study, we focus on the return-volatility asymmetry of the Bitcoin mar-ket. Since the first proposal of cryptocurrency in 2008[14], the Bitcoin system,based on a peer-to-peer network and blockchain technology, developed quickly,and Bitcoin has become widely recognized as a payment medium. In recentyears, a large body of literature has investigated various aspects of Bitcoin, e.g.,hedging capabilities[15], inefficiency[16, 17, 18, 19], multifractality[20], extremeprice fluctuations[21], liquidity and efficiency[22, 23], transaction activity[24],complexity synchronization[25], long memory[26], and so forth.Although the return-volatility asymmetry of Bitcoin has been investigatedusing various models, such as asymmetric GARCH-type and stochastic volatil-ity, it seems that a consistent picture of the return-volatility asymmetry ofBitcoin has not yet been obtained. For instance, while Bouoiyour et al.[27] ob-serve a volatility asymmetry that reacts to negative news rather than positive,Katsiampa[28] and Baur et al.[29] find an inverted volatility asymmetry that re- The correlations are fitted with an exponential function of α exp( − t/τ ), and the decaytime is defined by τ . | r | d , is dependent on the value of power d , and, typ-ically, the maximum autocorrelations are obtained at d ≈ d ≈ . We use Bitcoin tick data (in dollars) traded on Bitstamp from January 10, 2015to January 23, 2019 and downloaded from Bitcoincharts . Let p t i ; t i = i ∆ t ; i =1 , , ..., N be the time series of Bitcoin prices with sampling period ∆ t . Wedefine the return, R i , by the logarithmic price difference, namely, R i +1 = log p t i +1 − log p t i . (1)In this study, we consider high-frequency returns with ∆ t = 2 min, and wealso consider daily returns. We further calculate the normalized returns by r i = ( R i − ¯ R ) /σ R , where ¯ R and σ R are the average and standard deviation of R i , respectively. We calculate the cross correlation, CC d ( j ), between returnsand the d -th power of absolute returns at lag j as CC d ( j ) = E [( r t − µ r )( | r t + j | d − µ | r | d )] σ r σ | r | d , (2)where µ r and µ | r | d are the averages of r i and | r i | d , and σ r and σ | r | d are thestandard deviations of r i and | r i | d , respectively. E [ O j ] in Eq.(2) stands forthe average over N − j values of O j . For d = 2, Eq.(2) reduces to the usualdefinition of the return-volatility correlation that uses squared returns as a proxyof volatility[12, 13] except the normalization. http://api.bitcoincharts.com/v1/csv/
40 -20 0 20 40lag j-0.2-0.100.10.2 d=2.0
Figure 1: Cross correlation CC d ( j ) for daily returns as a function of lag j at d = 2 .
0. Error bars of data points represent one sigma errors calculated by thejackknife method.We calculate CC d ( j ) for d = 0 . . j s, CC d ( j )at d = 2 evaluates the relationships between returns and future volatilities. Thereverse correlations, i.e., relationships between future returns and volatilities,are obtained for negative j s. First, in Figure 1, we show the cross correlation, CC d ( j ), of the daily returns for d = 2 .
0. The cross correlations are mostly consistent with zero for both positiveand negative lags, j , except for j = 0 and 1, at which negative correlations areobserved. For other d s, similar results are obtained. Thus, at the daily level,the cross correlations are mostly insignificant, except for contemporaneous andsmall, positive lags.Next, in Figure 2, we show the cross correlation, CC d ( j ), calculated with2-min, high-frequency returns for d = 2 .
0. For positive j s, we find negativecross correlations lasting from small to large lags, which is consistent with theresults observed for developed markets[10, 38]. For negative j s, we observe pos-itive, but smaller, cross correlations at several small lags. For larger (negative)lags, the cross correlations are consistent with zero. For the contemporaneouscorrelations, i.e., j = 0, we observe negative cross correlations.To examine the scaling properties of the cross correlations at positive lags ,we plot negative values of the results, i.e., − CC d ( j ) in Figure 3 in log-log scale.We fit the cross correlations with the power law function of κj − γ and theexponential function of α exp( − j/τ ) in a range of j = [1 , κ , γ , α , Since the cross correlations at negative lags quickly become consistent with zero at verysmall lags, we only consider those at positive lags.
40 -20 0 20 40lag j -0.1-0.0500.05 d=2.0
Figure 2: Cross correlation CC d ( j ) for high-frequency returns as a function oflag j at d = 2 .
0. Error bars of data points represent one sigma errors calculatedby the jackknife method.Table 1: Results of fitting to a quadratic function, γ ( d ) = αd + βd + ρ . Thevalues in parentheses represent the asymptotic standard errors of the fittingparameters. α β ρ Bitcoin 0.0184(13) 0.0470(35) 0.5630(18)and τ are fitting parameters. The fitting results of the power law (exponential)function are depicted by the red (green) curve in Figure 3. We find that thecross correlations are better described by the power law function than by theexponential function. In particular, we recognize that the exponential functiondoes not adequately describe the data points of cross correlations at small lags.This finding is different from the results of previous studies that observe expo-nential behavior in the cross correlation[10, 11, 13]. The exponential behaviorin the cross correlation indicates that the cross correlation quickly disappearsas the lag increases, i.e., the correlation is short ranged. On the other hand,the power law behavior that we observe indicates that the cross correlationdecreases slowly with the lag, i.e., the correlation is long ranged.In Figure 4, we plot the results of γ as a function of d and find that γ increaseswith d . We fit the results to a quadratic function, γ ( d ) = αd + βd + ρ , where α, β ,and ρ are fitting parameters; the fitting results are listed in Table 1. From thefitting results, we recognize that for d →
0, the power γ seems to approach thevalue around 0.56. To investigate the strength of the cross correlations, we plot κ as a function of d in Figure 5. More precisely, κ represents the strength of the More precisely, the power law exponent, γ , should be γ < γ <
1. As seen inFigure 4, the condition of γ < d <
10 100lag j0.00010.0010.010.1 - CC d ( j ) power law fitexponential fit (a) d=1 - CC d ( j ) power law fitexponential fit (b) d=2 Figure 3: Cross correlation CC d ( j ) at d = 1 and 2 in log-log scale. (a) d=1 and(b) d=2. Error bars of data points represent one sigma errors calculated by thejackknife method. For better visibility, noisy data that are consistent with zerowithin 1.5 sigma errors are omitted from the pictures. The red (green) solidcurve represents the power law (exponential) fitting to the data. The reducedchi square: (a) 0.796 (power law) and 1.09 (exponential), (b) 1.13 (power law)and 1.33 (exponential). 5 d γ quadratic fitting Figure 4: Fitting results of γ as a function of d , where γ is a parameter of thepower law function of κj − γ . Error bars of the results represent the asymptoticstandard errors of the parameter. The solid curve represents the quadraticfitting to γ .cross correlations at lag j = 1. We find that κ is a convex function and that themaximum strength is obtained around d ≈ .
4. Thus, the correlation CC d (1)at d ≈ . d = 2. At the daily level, cross correlations are mostly insignificant for Bitcoin. By ex-amining high-frequency Bitcoin returns, we find that returns and future volatil-ities are negatively correlated and the cross correlations between returns andfuture volatilities show power law behavior. We calculate cross correlationsbetween returns and the d -th power of absolute returns and find that the maxi-mum cross correlation is obtained at d ≈ .
4. Thus, we were able to obtain clearevidence on the cross correlation by choosing other values of d , rather than thetraditional value of d = 2.Our findings on cross correlations suggest that, in modeling asset time series,we should more seriously consider models that produce power law behavior inthe cross correlations.For example, Ref.[40] proposes a fractional random walk model combinedwith a simple auto-regressive conditional heteroskedastic model, denoted as FR-WARCH, and finds that the FRWARCH model exhibits a power law in the crosscorrelations.There exist universal properties, such as volatility clustering and no auto-correlations in returns, that appear across various assets. These properties arecalled the stylized facts (e.g., [41]). The existence of stylized facts suggests that6 d κ Figure 5: Fitting results of κ as a function of d , where κ is a parameter of thepower law function of κj − γ . κ corresponds to the strength of a cross correlationat lag j = 1. The error bars of the results represent the asymptotic standarderrors of the parameter.the price formation is governed by certain common dynamics. If Bitcoin has adifferent property in the cross correlation from other assets, there could exit adifferent type of dynamics in Bitcoin. To come to a definite conclusion aboutwhether the power law behavior only appears in Bitcoin, it would be desirableto examine other assets in detail. Acknowledgement
Numerical calculations for this work were carried out at the Yukawa InstituteComputer Facility and at the facilities of the Institute of Statistical Mathemat-ics. This work was supported by JSPS KAKENHI (Grant Number JP18K01556)and the ISM Cooperative Research Program (2018-ISMCRP-0006).
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