Exploring asymmetric multifractal cross-correlations of price-volatility and asymmetric volatility dynamics in cryptocurrency markets
EExploring asymmetric multifractal cross-correlations of price-volatility andasymmetric volatility dynamics in cryptocurrency markets
Shinji Kakinaka ∗ and Ken Umeno (Dated: February 5, 2021)We explore the scaling properties of price-volatility nexus in cryptocurrency markets and ad-dress the issue of the asymmetric volatility effect within a framework of fractal analysis. TheMF-ADCCA method is applied to examine nonlinear interactions and asymmetric multifractalityof cross-correlations between price and volatility for Bitcoin, Ethereum, Ripple, and Litecoin. Fur-thermore, asymmetric reactions in the volatility process to price fluctuations between uptrend (bull)and downtrend (bear) regimes are discussed from the viewpoint of cross-correlations quantified bythe asymmetric DCCA coefficient, which is a different approach from the conventional GARCH-class models. We find that cross-correlations are stronger in downtrend markets than in uptrendmarkets for maturing Bitcoin and Ethereum. In contrast, for Ripple and Litecoin, inverted reac-tions are present where cross-correlations are stronger in uptrend markets. Our empirical findingsuncover the dynamics of asymmetric volatility structure and provide new guidance in investigatingdynamical relationships between price and volatility for cryptocurrencies. I. INTRODUCTION
Since Bitcoin was first released in 2009 by a pseudony-mous publisher Satoshi Nakamoto [1], it has been a prin-cipal subject of discussion within the global financialmarket. Differently from other financial assets, Bitcoinis supported by a decentralized financial system wherecentral banks and central management systems are notin control. Anonymity and well-founded security are ex-pected to be assured, and the unique system of the block-chain technology has attracted intensive attention fromfinancial practitioners and researchers. The market cap-italization of Bitcoin has grown explosively over a stag-gering 58 billion dollars at the end of 2020 along withthe other competing cryptocurrencies such as Ethereum,Ripple, Litecoin, and other minor coins. They have alsoemerged and grown rapidly in recent years dominatingmore than half of the cryptocurrency market capital-ization, so analyzing coins other than Bitcoin has alsobecome important. Researchers have discussed whethercryptocurrencies should be classified as currencies, fi-nancial assets, an expedient medium of exchange, ora technological-based product [2–6], but they have notcome to a complete conclusion. Moreover, Bitcoin is lesscorrelated with conventional assets, commodities, andthe U.S. dollar, making it useful as a diversified invest-ment for hedging purposes [7]. Given these idiosyncraticcharacteristics, modeling the price and volatility dynam-ics of Bitcoin, along with other cryptocurrencies, playsan important role in various types of financial analyses.Numerous studies have shown that financial assets areincredibly complex and have nonlinear dynamic systemswith scaling properties, power-law distributions with fat-tails, long-range dependencies, and volatility clustering. ∗ Corresponding author, Department of Applied Mathematicsand Physics, Graduate School of Informatics, Kyoto University,Japan.
Such stylized facts are seen in cryptocurrency marketsas well [8–11]. Bitcoin returns do not follow a ran-dom walk behavior and the market is significantly inef-ficient, but the efficiency becomes higher in recent pe-riods [12] and the market heads to maturity [9]. Al-though some studies suggest that market efficiency holdsfor certain periods, the returns do not generally satisfythe efficient market hypothesis (EMH) [13–15]. Jiang etal. (2018) [16] investigate how the Hurst exponent variesthrough a rolling window approach and conclude thatthe Bitcoin market has a high degree of inefficiency overtime. Bariviera (2017) [13] uses a dynamical approach ofdetrended fluctuation analysis (DFA) proposed by Penget al. (1994) [17], which can be applied to nonstationarydata and provides more reliable estimates of the Hurst ex-ponent compared to the traditional rescaled range analy-sis [18]. He finds long-memory and captures the underly-ing dynamic process generating the prices and volatilityin the Bitcoin market.Multifractal detrended fluctuation analysis (MFDFA)is a development of the DFA algorithm, overcoming thelimitations of DFA by quantitatively describing multi-fractal structures in terms of the generalized scalingexponents [19]. The analysis of financial series usingthe MFDFA has lead to a breakthrough in the fieldof econophysics as an effective approach to detect in-efficiency, multifractality, and long-memory in a non-linear way. Many studies have applied the MFDFAand found evidence that cryptocurrency markets havestrong multifractality originating from correlations andfat-tails [20–24]. The generalization of the DFA tocross-correlations between bivariate series is known asthe detrended cross-correlation analysis (DCCA) intro-duced by Podobnik and Stanley (2008) [25]. Its ex-tension to multifractality, the multifractal detrendedcross-correlation analysis (MFDCCA, MFDXA, MF-X-DFA) [26], is developed and implemented for the empir-ical studies of cryptocurrencies, stock prices, and crudeoil markets [15, 27–29]. This method is a combinationof the MFDFA and the DCCA, thus describing multi- a r X i v : . [ q -f i n . S T ] F e b fractal characteristics of cross-correlated nonstationaryseries. A different generalization of the DFA, the asym-metric DFA (A-DFA), was proposed by Alvarez-Ramirezet al. (2009) [30] since financial assets may show dif-ferent behavior in reaction to trends. The multifrac-tal version was later proposed, namely, the asymmetricMFDFA (A-MFDFA) [31, 32], and the further extensionto cross-correlations is known as the multifractal asym-metric DCCA (MF-ADCCA) [33]. The method of MF-ADCCA is a versatile tool that takes into account boththe asymmetric structure and the multifractal scalingproperties between the two series. An empirical studyby Garjardo et al. (2018) [34] applies the MF-ADCCAto price behaviors of Bitcoin and leading conventionalcurrencies, suggesting the presence and asymmetry ofcross-correlations between them. Using the same ap-proach, Kristjanpoller and Bouri (2019) [35] find evi-dence of asymmetric multifractality between the maincryptocurrencies and the world currencies. These studiesshow that the MF-ADCCA approach is powerful for un-covering complex systems in cryptocurrency markets [36].One of the fundamental issues in financial markets isthe behavior of the relationship between price and volatil-ity. It is known that the conditional variance of equityreturns are more affected by negative news compared topositive news [37]. In this sense, negative returns in-crease the volatility by more than positive returns dueto the leverage effect, also known as the asymmetricvolatility effect [38]. This is related to the trading of in-formed and uninformed investors posing different impactson the return process. Specifically, uninformed traderslead to higher serial correlations in returns that makethe volatility increase, whereas informed traders suggestno autocorrelation [39]. Many studies have traditionallyapplied the asymmetric Generalized Autoregressive Con-ditional Heteroscedasticity (GARCH) models to analyzethe asymmetric reactions of volatility to returns. Baurand Dimpfl (2018) [38] use the TGRACH model and findan intriguing aspect of cryptocurrency price behavior dif-ferently from other traditional assets— the presence ofinverse-asymmetric volatility effect. In other words, pos-itive shocks increase the volatility by more than nega-tive shocks, where speculative investments made by unin-formed noise traders are dominant after positive shocks.Cheikh et al. (2020) [40] employ a flexible model of ST-GARCH and also report similar results of a positivereturn-volatility relationship.Although the anomalies in volatility dynamics justifythe application of GARCH class models [41], they focusmore on the linear correlations of returns and volatil-ity fluctuations without the scaling properties. Applyingthe DFA based analysis is advantageous to accomplisha more essential understanding of the dynamics of priceand volatility because it accounts for the widely acknowl-edged nonlinearity and scaling properties. In addition,the method does not require the specification of a statis-tical model, so it is easy to implement and can directlycalculate the scaling exponents. In this paper, we firstly employ the MF-ADCCA ap-proach to investigate the asymmetric multifractal prop-erties of cross-correlations between price fluctuations andrealized volatility fluctuations in cryptocurrency mar-kets. Our empirical findings reveal traces of asymmet-ric multifractality and its dependence on the directionsof price movements. Secondly, we address the finan-cial issue of asymmetric volatility effect, crucial for fi-nancial investors and regulators. We use a frameworkof fractal analysis, which differs from the conventionalGARCH-class models where how one-time price shockinfluences volatility is analyzed. We show that the asym-metric volatility effect can be examined by the assess-ment of asymmetric cross-correlations between bull andbear markets over various time scales from short to longtime periods. The levels of cross-correlations are quanti-tatively investigated for each time scale relying on theasymmetric DCCA coefficient [42] based on the ideaof the DCCA coefficient [43, 44]. We find the pres-ence of different reactions of volatility to price in majorcryptocurrencies, where price-volatility is more stronglycross-correlated under negative market trends comparedto positive market trends. However, for the relativelyminor ones, cross-correlations are stronger under posi-tive market trends, implying that the inverse-asymmetricvolatility effect is present. Our findings not only providenew insights into the nature of the price-volatility dy-namics, but also contribute to other relevant issues suchas volatility spillovers, speculative trading, and the ma-turity of cryptocurrency markets.The rest of this paper is organized as follows. Sec-tion 2 describes the data used in the analysis. Section3 explains the two nonlinear dynamical methods used inthe analysis, the MF-ADCCA approach and the asym-metric DCCA coefficient, to further investigate volatilitydynamics. Section 4 present the results and discussionsof the empirical analysis. Section 5 concludes. II. DATA
In this study, we use cryptocurrency price data tradedon https://poloniex.com/ for five major coins of Bit-coin (BTC), Ethereum (ETH), Ripple (XRP), and Lite-coin (LTC) all against Tether (USDT), which is a cryp-tocurrency designed to maintain the same value as theUS dollar. We analyze the period starting from June 1st,2016, and ending on December 28th, 2020. This periodincludes the cryptocurrency boom at the end of 2018 andthe crash at the beginning of 2019. Recently the pricesof many cryptocurrencies are increasing, for instance, theBitcoin price marked a record-breaker of over $ r t Date v t (a) BTC r t Date v t (b) ETH r t Date v t (c) XRP r t Date v t (d) LTC FIG. 1. The series of daily returns r t and the series of volatility changes v t calculated from 5 minute intervals for (a) Bitcoin,(b) Ethereum, (c) Ripple, and (d) Litecoin.TABLE I. Descriptive statistics for cryptocurrency markettime series (returns)BTC ETH XRP LTCMean(%) 0.233 0.234 0.233 0.198Median(%) 0.243 0.042 -0.152 -0.067Std. Dev.(%) 4.174 5.701 7.397 5.934Max.(%) 23.814 25.274 104.605 60.051Min.(%) -50.435 -58.697 -68.039 -47.796Skewness -1.107 -0.704 2.182 0.877Kurtosis 15.557 9.800 37.335 12.740Jarque-Bera a ∗∗∗ . ∗∗∗ ∗∗∗ ∗∗∗ a *** denotes statistical significance at 1% level. price data as follows:RV t = (cid:88) j r t,t j , (1)where r t,t j is the return or the logarithm price differ- TABLE II. Descriptive statistics for cryptocurrency markettime series (volatility change)BTC ETH XRP LTCMean(%) 0.032 0.008 0.022 0.063Median(%) -2.800 -3.578 -3.871 -2.377Std. Dev.(%) 37.617 35.970 40.764 35.212Max.(%) 201.193 144.555 281.472 205.722Min.(%) -152.324 -137.893 -158.280 -127.422Skewness 0.719 0.640 0.546 0.629Kurtosis 2.476 1.458 2.389 2.126Jarque-Bera a . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ a *** denotes statistical significance at 1% level. ence calculated from δt -minute sampling intervals, and t j = jδt denotes the time on day t . Since it is well-knownthat RV t is expected to converge to the integrated volatil-ity σ t in the limit δt →
0, we can consider the realizedvolatility as an alternative volatility measure. In thisstudy, we use the 5-minute sampling interval because itavoids strong bias derived from extremely high frequen-cies and maintains an accurate measure of σ t [45, 46].The log-price increments (returns) and log-volatility in-crements (volatility changes) are respectively calculatedas r t = ln p t − ln p t − , (2) v t = ln ˆ σ t − ln ˆ σ t − , (3)where ˆ σ t = √ RV t , and p t denotes daily closing price.The data length of r t and v t equally consists of N = 1669for all the series we use in the analysis. The series areshown in Fig. 1.In Table I, we report the descriptive statistics for thereturns of the investigated cryptocurrencies. As we cansee from the results of the Jarque-Bera test, all of themare remarkably far from the Gaussian distribution withhigh values of kurtosis and a certain degree of skewness.Similarly, descriptive statistics for the volatility changesin Table II tell us that the series also have non-gaussianbehavior, but tend to have higher variance and lower kur-tosis in comparison with the return series. III. METHODOLOGYA. Multifractal asymmetric detrendedcross-correlation analysis
This subsection presents a slightly modified version ofthe MF-ADCCA method of Cao et al. (2014) [33], wherethe asymmetric proxy is not directly the return series butthe price index, similar to the index-based MFDFA pro-posed by Lee et al. (2017) [32]. This method describesthe asymmetric cross-correlations between two time se-ries { x t : t = 1 , . . . , N } and { y t : t = 1 , . . . , N } in termsof whether the aggregated index shows a positive incre-ment or a negative increment.First, we start by constructing the profiles from theseries X ( k ) = k (cid:88) t =1 ( x t − ¯ x ) , t = 1 , . . . , N,Y ( k ) = k (cid:88) t =1 ( y t − ¯ y ) , t = 1 , . . . , N where ¯ x and ¯ y is the average over the entire returnseries respectively. We also calculate the index proxyseries I ( k ) = I ( k −
1) exp( x k ) for k = 1 , . . . , N with I (0) = 1, used for judging the positive and negative di-rections of the index series afterwards. Next, the profiles X ( k ), Y ( k ), and the index proxy I ( k ) are divided into N s = (cid:98) N/s (cid:99) non-overlapping segments of length s . Thedivision is repeated starting from the other end of theseries to consider the entire profile, since N is unlikely to be a multiple of s and there may be remains in theprofile. Thus, we have 2 N s segments in total for eachseries.We next move on to the procedure of detrending theseries. For each segment v = 1 , . . . , N s of length s ,the local trend of the profiles are calculated by fittinga least-square degree-2 polynomial ˜ X v and ˜ Y v , which isused to detrend X ( k ) and Y ( k ), respectively. At thesame time we determine the local asymmetric directionof the index series by estimating the least-square linear fit˜ I v ( i ) = a I v + b I v i ( i = 1 , . . . , s ) for each segment. Positive(upward) or negative (downward) trends depend on thesign of the slope b I v .Then the detrended covariance for each of the 2 N s seg-ments is calculated as: f ( s, v ) = 1 s s (cid:88) i =1 (cid:12)(cid:12)(cid:12) X (( v − s + i ) − ˜ X v ( i ) (cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12) Y (( v − s + i ) − ˜ Y v ( i ) (cid:12)(cid:12)(cid:12) for v = 1 , . . . , N s and f ( s, v ) = 1 s s (cid:88) i =1 (cid:12)(cid:12)(cid:12) X ( N − ( v − N s ) s + i ) − ˜ X v ( i ) (cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12) Y ( N − ( v − N s ) s + i ) − ˜ Y v ( i ) (cid:12)(cid:12)(cid:12) for v = N s + 1 , . . . , N s .The upward and downward q -th order fluctuation func-tions are calculated by taking the average over all seg-ments as: F + q ( s ) = (cid:40) M + 2 N s (cid:88) v =1 b I v )2 (cid:2) f ( s, v ) (cid:3) q/ (cid:41) /q ,F − q ( s ) = (cid:40) M − N s (cid:88) v =1 − sgn( b I v )2 (cid:2) f ( s, v ) (cid:3) q/ (cid:41) /q , (4)for any real value q (cid:54) = 0, and F +0 ( s ) = exp (cid:40) M + 2 N s (cid:88) v =1 b I v )2 ln (cid:2) f ( s, v ) (cid:3)(cid:41) ,F − ( s ) = exp (cid:40) M − N s (cid:88) v =1 − sgn( b I v )2 ln (cid:2) f ( s, v ) (cid:3)(cid:41) , (5)for q = 0. M + = (cid:80) N s v =1 1+sgn( b Iv )2 and M − = (cid:80) N s v =1 1 − sgn( b Iv )2 respectively represent the numbers ofsegments with positive and negative trends under the as-sumption of b I v (cid:54) = 0 for all v = 1 , . . . , N s , such that M + + M − = 2 N s . The q -th order fluctuation func-tions for the overall trend corresponds to the MF-DCCAmethod shown as: F q ( s ) = (cid:40) N s N s (cid:88) v =1 (cid:2) f ( s, v ) (cid:3) q/ (cid:41) /q , (6)for q (cid:54) = 0 and when q = 0, F ( s ) = exp (cid:40) N s N s (cid:88) v =1 ln (cid:2) f ( s, v ) (cid:3)(cid:41) . (7)If the series x k and y k are long-range power-law cross-correlated, then the q -th order fluctuation functions fol-low a power-law of the forms F + q ( s ) ∼ s h + xy ( q ) , F − q ( s ) ∼ s h − xy ( q ) , and F q ( s ) ∼ s h xy ( q ) . The long-range power-lawcorrelation properties are represented in terms of the scal-ing exponent also known as the generalized Hurst expo-nent.The scaling exponent can easily be calculated by per-forming a log-log linear regression. However, the per-formance of the regression more or less depends on thechoice of which range of scales to be implemented. Asrecommended in Thompson and Wilson (2016) [47], weemploy the scale ranging from s min = max(20 , N/ s max = min(20 s min , N/
10) and using 100 points in theregression, in order to avoid biases and maintain the va-lidity of the estimation.In cases of no cross-correlations, h xy ( q ) = 0 . h xy ( q ) > .
5, the cross-correlations between theseries are persistent with long-memory. On the contrary, h xy ( q ) < . h + xy ( q ) and h − xy ( q ), but the scaling exponents of cross-correlationsare individually measured for positive and negative in-crements.The order q implies to what degree the various magni-tudes of fluctuations are to be evaluated. Scaling expo-nents for q >
0, where the fluctuation function F q ( s ) isdominated by large fluctuations, reflect the behavior oflarger fluctuations. Scaling exponents for q < h xy ( q ) is in-dependent of q , then the cross-correlation of the series ismonofractal since the scaling behavior of the detrendedcovariance F ( s, v ) is identical for all segments. On theother hand, if the value differs depending on q , small andlarge behaviors have different scaling properties and thecross-correlations of the series are multifractal. It shouldbe mentioned that when q = 2, h xy ( q ) corresponds to theHurst exponent.The features of the multifractality can be further ex-plored by the R´enyi’s exponent given as τ xy ( q ) = qh xy ( q ) − . (8)If τ xy ( q ) is a linear function of q , the cross-correlation ofthe series is monofractal but otherwise, it is multifractal.From the Legendre transform, the singularity spectra isobtained as follows: α = h xy ( q ) + qh (cid:48) xy ( q ) f xy ( α ) = q ( α − h xy ( q )) + 1 , (9) where α is the singularity of the bivariate series. Thesingularity spectrum width ∆ α = α max − α min representsthe degree of multifractality of the bivariate series, where α max and α min are respectively the α values at the max-imum and minimum of f xy ( α ) support. In the case ofmonofractality, ∆ α heads to zero, and thus the singular-ity spectra is theoretically just a point. The discussionsabove can also be expanded to examine the multifractalproperties for asymmetric cases of generalized Hurst ex-ponents h + xy ( q ) and h − xy ( q ). Thus the asymmetric casesof the R´enyi’s exponent, τ + xy ( q ) and τ − xy ( q ), and the sin-gularity spectra, f + xy ( α ) and f − xy ( α ), can be calculated aswell.It must be noticed that when the series x j and y j areidentical, the method aims to study the properties of au-tocorrelations and the method is consistent with the A-MFDFA. B. Asymmetric DCCA coefficient
The cross-correlation between two series with a highdegree of non-stationarity and self-similarity can bequantified via the DCCA coefficient, which utilizes theanalysis based on the DCCA. An asymmetric extensionof the cross-correlation coefficient is proposed by Cao etal. (2018) [42], where the coefficient of the bivariate seriesfor the cases of when prices increase and decrease can beexamined separately.The cross-correlation coefficient of Zebende (2011) [43]and Podobonik et al. (2011) [44] between the series { x t : t = 1 , . . . , N } and { x t : t = 1 , . . . , N } is derived relyingon the fluctuation functions calculated from overlapping N − s segments of length s + 1 as F ( s ) = 1 N − s N − s (cid:88) i =1 f ( s, i ) , (10)where f ( s, i ) is the detrended covariance of theresiduals for each segment defined as f ( s, i ) = 1 s + 1 s (cid:88) k = i ( R x ( k ) − ˜ R x ( k ))( R y ( k ) − ˜ R y ( k )) , (11)where R x ( k ) = (cid:80) kt =1 x t , R y ( k ) = (cid:80) kt =1 y t , and thedegree-2 polynomial fits ˜ R x ( k ) and ˜ R y ( k ) are used todetrend R x ( k ) and R y ( k ). When we are considering thecorrelations between two identical series, the fluctuationfunction F ( s ) is reduced to F ( s ) defined in termsof the DFA method.Based on the fluctuation functions above, the cross-correlation coefficient is defined as follows: ρ DCCA ( s ) = F ( s ) F DFA x ( s ) F DFA y ( s ) , (12)where F x ( s ) and F y ( s ) is the DFA fluctuationfunction for each of the series x and y , respectively. Notethat with a given scale of s , F ( s ) represents cross-correlation features at that scale, and F ( s ) repre-sents autocorrelation features of each series at scale s .This coefficient, therefore, reveals the degree of cross-correlations for various scales.In the asymmetric DCCA coefficient, the cross-correlation coefficients for the upward and downwardtrends are taken in consideration as follows: ρ DCCA+ ( s ) = F ( s ) F DFA x + ( s ) F DFA y + ( s ) ,ρ DCCA − ( s ) = F − ( s ) F DFA x − ( s ) F DFA y − ( s ) , (13)where F ( s ) and F − ( s ) are obtained from thecalculation process in line with equations (4) and (5) us-ing the detrended covariance of the residuals for each seg-ment in equation (11). The coefficients ρ DCCA , ρ DCCA+ ,and ρ DCCA − range from -1 to 1, and the value equal to 1indicates the existence of a perfect cross-correlation, and-1 indicates perfect anti-cross-correlation. IV. RESULTS AND DISCUSSIONS
This section explores asymmetric multifractal featuresof price-volatility cross-correlations and quantifies theircoupling levels to clarify the presence of asymmetricvolatility effects in cryptocurrency markets.
A. Multifractal and asymmetric properties ofcross-correlations
Before we conduct the MF-ADCCA method, we firsttest the presence of cross-correlations between pricechanges and volatility changes to confirm that applica-tion of DFA-based methods is appropriate for the analy-ses. We apply the statistic test proposed by Podobnik etal. (2009) [48] to check the presence of cross-correlationsbetween the bivariate series. The cross-correlation statis-tic for the series { x i } and { y i } of equal length N is definedas: Q cc ( m ) = N m (cid:88) i =1 X i N − i , (14)where X i is the cross-correlation function defined as: X i = (cid:80) Nk = i +1 x k y k − i (cid:113)(cid:80) Nk =1 x k (cid:80) Nk =1 y k . (15)Since the statistic Q cc ( m ) is approximately χ ( m ) dis-tributed with m degrees of freedom, it can be used to testthe null hypothesis that the first m cross-correlation coef-ficients are nonzero. If the value of Q cc ( m ) is larger than the critical value of χ ( m ), the null hypothesis is rejectedand thus the series have a significant cross-correlation. m c r o ss - c o rr e l a t i on s t a t i s t i cs Q cc ( m ) BTCETHXRPLTCcritical values
FIG. 2. Cross-correlation statistics between the return seriesand volatility change series of the four major cryptocurrencies.The black line represents the critical values at the 5% level ofsignificance.
The cross-correlation test statistics in eqs.(14) and (15)for price changes and volatility changes of the four cryp-tocurrencies are calculated with various degrees of free-dom m , ranging from 1 to 500. The results are shown inFig. 2 together with the critical values of the χ ( m ) dis-tribution at the 5% level of significance. We find that forall the investigated cryptocurrencies, the statistic Q cc ( m )deviates from the corresponding critical value, indicatingthat there are nonlinear cross-correlations between pricechanges and volatility changes. For XRP and LTC, thetest statistic deviates from the critical value more thanthose of BTC and ETH. This implies the presence ofstronger nonlinear cross-correlations in the minor cryp-tocurrencies compared to the major ones.Now that we have verified the existence of nonlinearcross-correlations in the bivariate series, we next an-alyze the multifractal properties of asymmetric cross-correlations for each cryptocurrency via the MF-ADCCAmethod. Figure 3 shows the q -th order fluctuation func-tions calculated from the returns and volatility changeswith various q ranging from -10 to 10. The fluctuationfunctions under different situations of bull and bear mar-kets (uptrend and downtrend) are also depicted with theoverall trend. For all cases, we observe that the fluctu-ation functions generally follow a power-law against thescale, which means that the nonlinear cross-correlationsbetween the bivariate series have a long-range power-law property. Therefore the MF-ADCCA is expected tobe an effective method for analyzing cross-correlations,along with the asymmetry between uptrend and down-trend cross-correlations.Fig. 3 shows that the behavior of power-law cross-correlations varies among market situations of differenttrends. To measure the degree of asymmetry of the cross- ln( s ) l n ( F q ( s )) price-volatility (overall) 3.0 3.5 4.0 4.5 5.0 ln( s ) l n ( F + q ( s )) price-volatility (uptrend) 3.0 3.5 4.0 4.5 5.0 ln( s ) l n ( F q ( s )) volatility-volatility (downtrend) (a) BTC ln( s ) l n ( F q ( s )) price-volatility (overall) 3.0 3.5 4.0 4.5 5.0 ln( s ) l n ( F + q ( s )) price-volatility (uptrend) 3.0 3.5 4.0 4.5 5.0 ln( s ) l n ( F q ( s )) volatility-volatility (downtrend) (b) ETH ln( s ) l n ( F q ( s )) price-volatility (overall) 3.0 3.5 4.0 4.5 5.0 ln( s ) l n ( F + q ( s )) price-volatility (uptrend) 3.0 3.5 4.0 4.5 5.0 ln( s ) l n ( F q ( s )) volatility-volatility (downtrend) (c) XRP ln( s ) l n ( F q ( s )) price-volatility (overall) 3.0 3.5 4.0 4.5 5.0 ln( s ) l n ( F + q ( s )) price-volatility (uptrend) 3.0 3.5 4.0 4.5 5.0 ln( s ) l n ( F q ( s )) volatility-volatility (downtrend) (d) LTC FIG. 3. Log-log plots of F q ( s ) , F + q ( s ) and F − q ( s ) versus time scale s for the four cryptocurrency series of (a) BTC, (b) ETH, (c)XRP, and (d) LTC. We show the cases of q = − , − , − , . . . ,
10. The power-law relations indicate that the bivariate serieshave long-range power-law cross-correlations. correlations, we calculate the metric defined as∆ h xy ( q ) = h + xy ( q ) − h − xy ( q ) , (16)for given q . The greater the value, the greater theasymmetric behavior in terms of different trends. If ∆ h xy ( q ) > h xy ( q ) < h xy ( q ) is theoretically zero and the two q gene r a li z ed H u r s t e x ponen t s overalluptrenddowntrend (a) BTC q gene r a li z ed H u r s t e x ponen t s overalluptrenddowntrend (b) ETH q gene r a li z ed H u r s t e x ponen t s overalluptrenddowntrend (c) XRP q gene r a li z ed H u r s t e x ponen t s overalluptrenddowntrend (d) LTC FIG. 4. Relationship between the generalized Hurst exponents and the order q for the cases of (a) BTC, (b) ETH, (c) XRP,and (d) LTC.TABLE III. Asymmetric degree of cross-correlations in termsof uptrend and downtrend price movements shown togetherwith the multifractal degree for each of the four cryptocur-rencies. ∆ h xy ( −
10) ∆ h xy (2) ∆ h xy (10) D xy BTC 0.065 0.082 0.132 0.231ETH 0.027 0.131 0.158 0.119XRP 0.043 0.050 0.008 0.169LTC 0.040 0.055 0.058 0.214 series have symmetric cross-correlations. We calculatethe cases of q = −
10 (small fluctuations), q = 2 (cor-responding to the Hurst exponent), and q = 10 (largefluctuations). We clearly find in Table III that regard-less of small and large fluctuations, ∆ h xy ( q ) is positivefor all the investigated cryptocurrencies. Fig. 4 also sup-ports these findings where h + xy ( q ) is always larger than h − xy ( q ). For all four cryptocurrencies, cross-correlationsof price-volatility in the uptrend markets have slightlyhigher persistency at all levels of fluctuations comparedto those in the downtrend markets.The presence of multifractality can be examined by looking into whether or not the generalized Hurst expo-nents are dependent on its order q . Given that h xy ( q )decreases as q increases in Fig. 4, h xy ( q ) is not con-stant for q and hence multifractal behaviors exsist in thecross-correlations between the bivariate series. To nu-merically explain the deviation from monofractality, weuse the market efficiency measure (MDM) of Wang etal. (2009) [49] defined as D xy = 12 ( | h xy ( − − . | + | h xy (10) − . | ) . (17)If D xy is zero or close to zero, then the relationship be-tween the series are efficient. Larger values of D xy indi-cate higher inefficiency, and smaller values indicate lowerinefficiency. This metric is useful to determine the rank-ing of the (in)efficiency degree [50]. From the resultsshown in Table III, we suggest that BTC is the most in-efficient with D xy = 0 . D xy = 0 . f xy ( α ), f + xy ( α ), and f − xy ( α )in Fig. 5 where the asymmetric market trends are consid-ered. The spectra are quite broad as expected, and thewidth differs with respect to cryptocurrencies and market s i ngu l a r i t y s pe c t r a overalluptrenddowntrend (a) BTC s i ngu l a r i t y s pe c t r a overalluptrenddowntrend (b) ETH s i ngu l a r i t y s pe c t r a overalluptrenddowntrend (c) XRP s i ngu l a r i t y s pe c t r a overalluptrenddowntrend (d) LTC FIG. 5. Singularity spectra f xy ( α ), f + xy ( α ), and f − xy ( α ) for the cases of (a) BTC, (b) ETH, (c) XRP, and (d) LTC.TABLE IV. The asymmetric degree of singularity spectra A α and α , α max , and α min with respect to price-volatility coupling.We show also the asymmetric spectrum parameter and the values α for asymmetric market trends (uptrend and downtrend).Overall Uptrend Downtrend A α α α max α min A + α α +0 α +max α +min A − α α − α − max α − min BTC -0.151 0.387 0.699 0.157 -0.285 0.415 0.694 0.259 -0.105 0.338 0.623 0.108ETH 0.081 0.386 0.491 0.263 0.150 0.439 0.513 0.338 -0.100 0.320 0.498 0.173XRP 0.246 0.400 0.544 0.163 0.332 0.430 0.571 0.149 0.121 0.362 0.533 0.145LTC -0.418 0.362 0.742 0.206 -0.325 0.384 0.740 0.203 -0.339 0.329 0.692 0.150 trends. In particular, among the investigated markets,spectrum width in BTC for the overall trends present thelargest degree of multifractality of ∆ α = 0 . α = 0 . A α = ∆ α L − ∆ α R ∆ α L +∆ α R ,where ∆ α L = α − α min , ∆ α R = α min − α , and α is the value of α at the maximum of the singular spec-trum [51]. Note that the asymmetry here is not in termsof different market trends but the distortion of the sin-gularity spectrum f xy ( α ). The metric A α provides infor-mation to identifying the compositions of the bivariateseries. If A α > A α < q > q < A α = 0, the left and right sides ofspectrum width are equivalent, and thus the small andlarge fluctuations operate equally on multifractality.We report in Table IV the asymmetric spectrum pa-rameter A α and values of α for the three different markettrends: overall, uptrend, and downtrend. In real-worldfinance data, left-sided spectrum is more common andreasonable to have peculiar features in the larger fluc-tuations and a noise-like behavior in the smaller fluc-tuations [51]. However, we find uncommon features in0some cases. The asymmetric spectrum parameters A α , A + α , and A − α take negative values in BTC and LTC mar-kets, and the right-skewed spectra explain that smallerevents play a more important role in the underlying mul-tifractality. In contrast, larger events contribute moreto the multifractal behavior in the XRP market becausethe asymmetric spectrum parameters take positive val-ues. It is interesting to mention that regardless of themarket trends, multifractality of the price-volatility cou-pling for the above three markets exhibit the same behav-ior of asymmetry f ( α ) either with the right- or left-sidedspectrum, i.e., the same distortion. Among the inves-tigated markets, only the ETH market shows differentmultifractal properties depending on the market trends,where large fluctuations are dominant in bull markets butsmall fluctuations are dominant in bear markets. We findthat the LTC shows the strongest right-skewed f ( α ) withthe smallest values of A α for all market trends, whereasthe XRP shows the strongest left-skewed f ( α ) with thelargest values of A α for all market trends. In addition tothe right-skewed property of BTC and LTC, they tendto have wider spectrum width ∆ α , demonstrating thatthese markets exhibit a highly complex price-volatilitybehavior. B. Assessment of asymmetric volatility
We next apply the DCCA coefficient analysis to quan-tify the asymmetric cross-correlations between price andvolatility and to examine the asymmetry volatility ef-fects in cryptocurrency markets. Fig. 6 depicts the coef-ficients for the overall trend, ρ DCCA ( s ), and the upward(bull) and downward (bear) market trends, ρ DCCA+ ( s )and ρ DCCA − ( s ), for various time scales s ranging from10 to 334 days. For the entire period (overall trend), thecoefficients are not so far from zero at all time scales. An-alyzing the case of the overall trend appears to suggestthat price and volatility are less interrelated. However,once we consider uptrend and downtrend, we can real-ize different pictures of the markets. The asymmetricDCCA coefficient approach enables us to separately fig-ure out the interrelationship between price and volatilityunder bull and bear regimes.As shown in Fig. 6, the coefficients are positive foruptrend markets but negative for downtrend markets.This reconfirms that both positive and negative pricechanges have a certain degree of the impact on volatil-ity. When | ρ DCCA − ( s ) | > | ρ DCCA+ ( s ) | is satisfied, i.e.,price and volatility are more strongly cross-correlated indowntrend markets than in uptrend markets, we can saythat asymmetric volatility is present at a certain timescale. On the contrary, when | ρ DCCA − ( s ) | < | ρ DCCA+ ( s ) | holds, we can say that inverse-asymmetric volatilityis present where uptrend markets have stronger price-volatility cross-correlations.The results provide clear evidence that for BTCand ETH, price and volatility are more strongly cross- correlated in downtrend markets than in uptrend mar-kets. In particular, we find ρ DCCA − ( s ) ≈ − . ρ DCCA+ ( s ) ≈ ρ DCCA − ( s ) ≈ − . ρ DCCA+ ( s ) ≈ .
15 for ETH under approximately one-month time scales ( s ≈ s ≈ ρ DCCA − ( s ) ≈ − .
25 and ρ DCCA+ ( s ) ≈ . | ρ DCCA+ ( s ) | is always larger than | ρ DCCA − ( s ) | for all time scales.Uninformed noise traders are more dominant when theprice increases than when the price decreases. More in-terestingly, LTC exhibits different volatility-asymmetrydepending on which time scale the cross-correlation anal-ysis was implemented at. At relatively longer time scales( s > | ρ DCCA+ ( s ) | is larger than | ρ DCCA − ( s ) | , but atrelatively shorter time scales, | ρ DCCA − ( s ) | surpasses thevalue of | ρ DCCA+ ( s ) | . In other words, inverse-asymmetricvolatility is present at scales of s < s = 150. Such explanations cannot be con-firmed by the conventional GARCH-class models, whichgenerally focus on the effects of one-time price shock onvolatility. The approach in this study challenges to a dy-namical effects of price on volatility while accounting forthe directions of price trends with various time scales. Fo-cusing on the time scales, we find that cross-correlationsfor uptrend markets are generally constant; however, theloss of cross-correlations is observed for downtrend mar-kets at longer time scales.To make sure that our empirical findings and discus-sions toward the asymmetric behavior of price-volatilityare reasonable, we implement two types of GARCH mod-els that explain the asymmetric effects on volatility. Thefirst model is the exponential GARCH (EGARCH) writ-ten as follows [52]:ln σ t = ω + α | r t − | σ t − + α r t − σ t − + β ln σ t − , (18)with r t = ε t σ t , where σ t is the conditional variance attime t , and ε t denotes an error term with i.i.d. stan-dard Gaussian noise N (0 , σ t = (cid:26) ω + α r t − + βσ t − , r t − ≥ ω + ( α + α ) r t − + βσ t − , r t − < α r t − is operated only when the marketgoes downwards. For convenience, both models are withone lag of the innovation ( p = 1) and one lag of volatility( q = 1), since the selection of optimized lags often pro-duces similar estimation results. The GARCH modelsabove are represented by the parameters ω , α , α , and1
50 100 150 200 250 300 lags (scales)
DCC A c r o ss - c o rr e l a t i on c oe ff i c i en t DCCA : overall +DCCA : uptrend
DCCA : downtrend (a) BTC
50 100 150 200 250 300 lags (scales)
DCC A c r o ss - c o rr e l a t i on c oe ff i c i en t DCCA : overall +DCCA : uptrend
DCCA : downtrend (b) ETH
50 100 150 200 250 300 lags (scales)
DCC A c r o ss - c o rr e l a t i on c oe ff i c i en t DCCA : overall +DCCA : uptrend
DCCA : downtrend (c) XRP
50 100 150 200 250 300 lags (scales)
DCC A c r o ss - c o rr e l a t i on c oe ff i c i en t DCCA : overall +DCCA : uptrend
DCCA : downtrend (d) LTC
FIG. 6. DCCA cross-correlation coefficients ρ DCCA ( s ), ρ DCCA+ ( s ), and ρ DCCA − ( s ) between the price-volatility relationshipsunder various scales s for the cases of (a) BTC, (b) ETH, (c) XRP, and (d) LTC. The scales represent the lag of days.TABLE V. Estimate results of conditional variance for cryptocurrencies. The EGARCH model and GJR-GARCH model areestimated over the period from June 3, 2016 to December 28, 2020. Standard errors of estimates are reported in parentheses.Note that ***, **, and * denote 1%, 5%, and 10% significance levels, respectively. Q (10) is the square Q-statistic and thep-values are presented in brackets.EGARCH GJR-GARCHBTC ETH XRP LTC BTC ETH XRP LTC ω − . ∗∗∗ − . ∗∗∗ − . ∗∗∗ − . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ (0.0661) (0.0712) (0.0422) (0.0303) (0.0000) (0.0000) (0.0000) (0.0000) α . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ (0.0208) (0.0185) (0.0182) (0.0129) (0.0160) (0.0172) (0.0269) (0.0101) α -0.0586 ∗∗∗ -0.0071 0.0828 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ -0.0256 ∗∗∗ (0.0080) (0.0097) (0.0097) (0.0060) (0.0165) (0.0171) (0.0257) (0.0082) β . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ (0.0083) (0.0106) (0.0060) (0.0040) (0.0176) (0.0195) (0.0141) (0.0096)Log likelihood 3079.33 2553.73 2412.35 2419.25 3083.58 2556.66 2428.99 2466.02AIC -3.6818 -3.0524 -2.8831 -2.9775 -3.6869 -3.0559 -2.9030 -2.9473 Q (10) 2.4476 8.4555 4.7584 1.7640 4.2086 10.143 3.2243 1.2243[0.992] [0.584] [0.907] [0.998] [0.937] [0.428] [0.976] [1.000] β . Among these parameters, α is the responsible onethat determines the asymmetric responses of volatilityto market shocks. Significantly positive values of α forthe EGARCH model imply that positive shocks increasevolatility more than negative shocks. Note that the pos- itive/negative direction is reversed for the GJR-GARCHmodel, where negative shocks increase the volatility morewhen α is positive. As shown in Table V, the values of α are negative in EGARCH and positive in GJR-GARCHfor BTC and ETH, and hence the volatility is increased2more by negative shocks. We find opposite results for theXRP and ETH because α are positive in EGARCH andnegative in GJR-GARCH. In such cases, volatility is in-creased more by positive shocks. Note that for the ETH,the asymmetric parameter is insignificant and thus theasymmetric effect cannot be statistically confirmed bythe implemented GARCH models.The results of GARCH models appear to highlight thatour empirical findings of the DCCA coefficient analysis onasymmetric cross-correlations are relevant to the asym-metric volatility dynamics. Both asymmetric-GARCHmodels and detrended cross-correlation analysis revealconsistent results of the underlying asymmetric/inverse-asymmetric volatility in cryptocurrency markets. Paststudies have shown inverse-asymmetric volatility dynam-ics of cryptocurrency markets [38, 40]. Our findings con-clude that unlike in the earlier periods, the volatility forthe two major coins of BTC and ETH, are no longerinverse-asymmetric. One possible reason may be the lesspresence of uninformed noise traders and increasing par-ticipants of informed traders in the market, and the ma-jor coins head to maturity in recent years. Minor coinsare immature with inverse-asymmetric volatility wherespeculated noise traders still dominate the market andplay a significant role in raising the volatility especiallyduring the uptrend periods. The appearance of strongernegative cross-correlation of price-volatility at shorterscales for LTC could be a key clue to explain the pro-cess that minor coins are expected to head for maturemarkets. V. CONCLUSION
This paper examined the nexus between daily priceand volatility in addition to the impact of price fluctu-ations on volatility in cryptocurrency markets. The ap-proach of MF-ADDCA revealed that price-volatility ex- hibit power-law cross-correlations as well as multifractalproperties. The asymmetric cross-correlations were stud-ied together with the overall market trend, and we foundthat the bivariate series have slightly different proper-ties between positive and negative market trends. Themultifractal features of those were investigated in detailby using the generalized Hurst exponents and the singu-lar spectrum. We pointed out that when prices goes up,directions of price-volatility series are more likely to befollowed by the current direction of either series, becauseuptrend markets are always more persistent compared todowntrend markets. Distinctive features of how smalland large fluctuations operate on multifractality werealso discovered and reported by investigating the spec-trum distortion among the cryptocurrencies and markettrends.More importantly, the level of the asymmetric cross-correlations for each cryptocurrency was quantitativelyevaluated by employing the asymmetric DCCA coeffi-cient. Our empirical findings showed that dependingon market directions, the level of cross-correlations dif-fers. We found that cross-correlations are stronger inbear markets than in bull markets for the maturing ma-jor coins (BTC and ETH), whereas the opposite resultswere observed for the still-developing minor coins (XRPand LTC). As long as price-volatility is our subject, weprovided evidence that such an approach enables us todiscuss whether asymmetry/inverse-asymmetry volatilitydynamics are present with various time scales, which isan intriguing financial phenomenon crucial for investors,financial investors and regulators. The detection of asym-metric volatility works well since the results were in linewith the conventional asymmetric GARCH-class models.Taking the advantage of getting to understand the multi-fractal features, power-law cross-correlations, and scalingbehaviors of price-volatility within various time scales,our approach can be an alternative to the financial ap-proach for discussing dynamical volatility behaviors incryptocurrency markets. [1] S. Nakamoto,
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