Time-varying properties of asymmetric volatility and multifractality in Bitcoin
TTime-varying properties of asymmetric volatility andmultifractality in Bitcoin
Tetsuya Takaishi , Hiroshima University of Economics, Hiroshima, Japan* [email protected]
Abstract
This study investigates the volatility of daily Bitcoin returns and multifractal propertiesof the Bitcoin market by employing the rolling window method and examinesrelationships between the volatility asymmetry and market efficiency. Whilst we find aninverted asymmetry in the volatility of Bitcoin, its magnitude changes over time, andrecently, it has become small. This asymmetric pattern of volatility also exists in higherfrequency returns. Other measurements, such as kurtosis, skewness, average, serialcorrelation, and multifractal degree, also change over time. Thus, we argue thatproperties of the Bitcoin market are mostly time dependent. We examineefficiency-related measures: the Hurst exponent, multifractal degree, and kurtosis. Wefind that when these measures represent that the market is more efficient, the volatilityasymmetry weakens. For the recent Bitcoin market, both efficiency-related measuresand the volatility asymmetry prove that the market becomes more efficient.
Bitcoin, advocated by Satoshi Nakamoto [1], was launched in 2009 as the firstdecentralized cryptocurrency. Its system is based on a peer-to-peer network. Whilstmany other cryptocurrencies have been created since its launch, and the cryptocurrencymarket has grown rapidly, Bitcoin remains the dominant cryptocurrency in terms ofFebruary 16, 2021 1/27 a r X i v : . [ q -f i n . S T ] F e b arket capitalization. Fig 1 represents the market capitalizations of the largest 10cryptocurrencies. Bitcoin dominates about 70% of the total capitalization of 10cryptocurrencies. Fig 1.
Market capitalization of the largest 10 cryptocurrencies (as of 31 August). Thesolid line represents a Pareto chart. The data are taken from”https://coinmarketcap.com/.”In recent years, Bitcoin has attracted interest of many researchers. Various aspectsof Bitcoin, including hedging capabilities [2], bubbles [3], liquidity and efficiency [4],Taylor effect [5], structural breaks [6], transaction activity [7], complexitysynchronization [8], long memory effects [9], price clustering [10], rough volatility [11]power-law cross-correlation [12], market structure [13] have been investigated.Similar to other assets, stylized facts [14, 15], such as volatility clustering, fat-tailedreturn distribution, and long memory in absolute returns, are observed in Bitcoin(e.g., [16, 17]). An aggregational Gaussianity that the fat-tailed return distributionschange to the Gausian distribution on large time scales is another stylized fact observedin various assets [14, 15]. This aggregational Gaussianity is also observed in Bitcoin anda minimum time scale required to recover the Gaussianity is estimated to be twoweeks [17]. The skewness is negative on a short time scale, but it moves to zero on alarger time scale [17].Despite the similar stylized facts in Bitcoin, researchers report a distinct property:inverted volatility asymmetry. By using generalized autoregressive conditionalFebruary 16, 2021 2/27eteroscedasticity (GARCH) models [18–21], several studies report an ”invertedasymmetry” in which volatility reacts more to positive returns than negativeones [22–25]. This contrasts sharply with the observation that the volatility of stocksreacts to negative returns more than positive ones [26–28].However, some studies report that there is no significant asymmetry involatility [17, 29]. An opposite result to the inverted asymmetry, that is, the samevolatility reaction as stocks, is documented [30]. Moreover, whilst inverted asymmetry isalso observed in other cryptocurrencies [31, 32], the asymmetry in Bitcoin isinsignificant [31]. Therefore, a consistent picture of volatility asymmetry in Bitcoin hasnot been obtained. We infer that the discrepancy observed in the volatility asymmetryis caused, in part, by the time-varying property of volatility asymmetry. We attributethe differing conclusions to the various data periods used in earlier studies. Thepossibility of time-varying asymmetry has been already noted [22], and it has beenreported that whilst inverted asymmetry is observed before 2014, no significantasymmetry is observed after 2014.This study aims to investigate time-varying properties of the Bitcoin market,especially volatility asymmetry. In emprical finance, one of popular models to analyzevolatility is the GARCH model [18, 19] which can successfully capture some stylizedfacts such as volatility clustring and fat-tailed distribution. There exist many variants ofthe GARCH model designed to capture more properties of financial time series [33]. Toassess the volatility asymmetry, we use the threshold GARCH (TGARCH) model [21],which is widely accepted in empirical finance and has been used in the previous studieson the Bitcoin volatility asymmetry [17, 22, 25, 30–32, 34]. We estimate model parametersby employing the rolling window method, which enables us to see time variation. Therolling window method is commonly used in econometrics or empirical finance fortime-series analysis in a limited amount of financial data. We also investigate the Hurstexponent and multifractality of the Bitcoin market by the multifractal detrendedfluctuation analysis (MF-DFA) method [35], which is a powerful method to studymultifractal properties and has been applied for various assets in econophysics; see, forexample, [36]. The Hurst exponent and multifractality of the Bitcoin time series areexamined intensively in connection with the market efficiency (e.g., [17, 37–45]). Herenote that according to the efficient market hypothesis [46], there are three types ofFebruary 16, 2021 3/27arket efficiencies: weak, semi-strong, and strong forms. Since we use the time-seriesdata only, the market efficiency in this study means the weak-form market efficiency.A remarkable feature that the Bitcoin time series exhibits is presence ofanti-persistency, that is, the Hurst exponent less than 1/2 [37]. The anti-persistencymeans that the time series reverses its moving direction more often than a random timeseries. On the other hand, the time series with the Hurst exponent greater than 1/2persists the same moving direction more than a random time series. Thisanti-persistency behavior, however, turns out to be temporary. It is observed that theHurst exponent and the multifractality degree vary over time, and the anti-persistencyappears repeatedly [40, 44]. Then, it seems that the Hurst exponent approaches thevalue of 0.5, which might be an indication toward a maturity market [37, 47].Since the efficient market should be free from any type of volatility asymmetry thatresults in predicting a certain market property to help gaining profits, the volatilityasymmetry is expected to induce some inefficiency and to relate with efficiency-relatedmeasures such as multifractality. Thus, we also examine a possible relationship betweenvolatility asymmetry and efficiency-related measures. We find that efficiency-relatedmeasures are related to the volatility asymmetry. When the efficiency-related measuresindicate that the market is more efficient, the volatility asymmetry weakens.To fully understand the dynamics of Bitcoin time series, we need to investigatevarious aspects of Bitcoin. This study investigates not only the volatility asymmetrybut also the multifractality, and combines them to advance the understanding ofproperties of the Bitcoin market. Our results reveal that the Bitcoin market efficiencyhas improved in recent years. It has been claimed that the market efficiency is relatedwith market size and economic development [48]. In accordance with this, our resultssuggest that the Bitcoin market is more mature than ever.The rest of this paper is organized as follows. Section 2 describes the methodology.In Section 3, we describe the data, and in Section 4, we present the empirical resultsand discuss the results. Finally, we conclude our study in Section 5.February 16, 2021 4/27
Methodology
Let p t i ; t i = i ∆ t ; i = 0 , , , ..., N be the Bitcoin price time series with sampling period∆ t . We define the return r t i +1 by the logarithmic price difference: r t i +1 = 100 × (log p t i +1 − log p t i ) . (1)Since in empirical finance the volatility analysis using the GARCH-type models mainlyfocuses on daily volatility, here we also focus on daily returns, i.e. ∆ t = 1440 − min .The TGARCH model [21] is used to investigate volatility asymmetry in the Bitcoinmarket; the return r t i and the volatility σ t i at t i are modelled as follows: r t i = µ + c r t i − + (cid:15) t i , (2) σ t i = ω + α(cid:15) t i − + βσ t i − + γ(cid:15) t i − I ( (cid:15) t i − ) , (3)where (cid:15) t i is defined by (cid:15) t i = σ t i η t i , and I ( (cid:15) t i − ) is an indicator function, implying thatit is 1 if (cid:15) t i − < η t i is an unobservable random variable from anindependent and identically distributed (IID) process. Here, we use the Student tdistribution as an IID process. To check the robustness on choice of distributions, weuse the normal distribution and the generalized error distribution [20].It is empirically well-known that stock return volatility increases after negativereturns more than positive returns [26, 27]. This volatility asymmetry is called ”theleverage effect” and causes a negative correlation between stock returns and volatility.To capture the leverage effect, various GARCH-type models with the volatilityasymmetry are introduced, e.g. [20, 21, 49–51]. For the TGARCH model, the volatilityasymmetry is measured by the γ parameter in Eq (3), and when the leverage effectexists, the γ parameter takes a positive value. On the other hand, for the invertedvolatility asymmetry observed in the Bitcoin market the γ parameter takes a negativevalue and volatility reacts more to positive returns than negative ones, leading to theinverted volatility asymmetry. c is the coefficient of an autoregressive model of order 1(AR(1)) that captures the serial correlation.To investigate the time-varying properties of volatility asymmetry, we use the rollingwindow method to estimate the model parameters. For the parameter estimation, we useFebruary 16, 2021 5/27he ”urgarch” package of R. First, we set a window size of 548 days ( ≈ one and a halfyears). We choose this window size because the smaller window size such as one yearleads to more noisy results in estimating model parameters, and perform a parameterestimation for the window containing the first 548 data samples for the time series.Next, we shift the window 30 days ( ≈ one month) and perform a parameter estimationfor the data in the next window. We repeat this process until the end the time series.Multifractal analysis is a useful method to quantify properties of complex system,and it has been applied in many different fields, e.g. [52–56]. Multifractal analysis is alsopopular in studies of financial markets and multifractal properties have been intensivelystudied [56]. To investigate the multifractal properties of the Bitcoin market, we applythe MF-DFA method [35]. The MF-DFA method is described by the following steps.(i) Determine the profile Y ( i ). Y ( i ) = i (cid:88) j =1 ( r t j − (cid:104) r (cid:105) ) , (4)where (cid:104) r (cid:105) stands for the average of returns.(ii) Divide the profile Y ( i ) into N s non-overlapping segments of equal length s ,where N s ≡ int ( N/s ). Since the length of the time series is not always a multiple of s , ashort time period at the end of the profile may remain. For this part, the sameprocedure is repeated starting from the end of the profile. Therefore, total 2 N s segments are obtained.(iii) Calculate the variance. F ( ν, s ) = 1 s s (cid:88) i =1 ( Y [( ν − s + i ] − P ν ( i )) , (5)for each segment ν, ν = 1 , ..., N s and F ( ν, s ) = 1 s s (cid:88) i =1 ( Y [ N − ( ν − N s ) s + i ] − P ν ( i )) , (6)for each segment ν, ν = N s + 1 , ..., N s . Here, P ν ( i ) is the fitting polynomial to removethe local trend in segment ν ; we use a cubic order polynomial.February 16, 2021 6/27iv) Average over all segments and obtain the q th order fluctuation function. F q ( s ) = (cid:40) N s N s (cid:88) ν =1 ( F ( ν, s )) q/ (cid:41) /q . (7)For q = 0, the averaging procedure in Eq(7) cannot be directly applied. Instead, weemploy the following logarithmic averaging procedure. F ( s ) = exp (cid:34) N s N s (cid:88) ν =1 ln( F ( ν, s )) (cid:35) . (8)(v) Determine the scaling behavior of the fluctuation function. If the time series r t i are long-range power-law correlated, F q ( s ) is expected to be the following functionalform for large s : F q ( s ) ∼ s h ( q ) . (9)In calculating the fluctuation function F q ( s ), we take q varying between -25 and 25,with a step of 0.2. The scaling exponent h ( q ) is called the generalized Hurst exponent,and the usual Hurst exponent is given by h (2). When h ( q ) is constant, the time series iscalled ”monofractal.” For example, the random Gaussian time series is monofractal [35].On the other hand, when h ( q ) varies depending on q , the time series is called”multifractal.”Following [57], we define the multifractality degree ∆ h ( q ) by∆ h ( q ) = h ( q min ) − h ( q max ) , (10)where q min = − q and q max = q .Since for the random Gaussian time series, ∆ h ( q ) takes zero, the magnitude of ∆ h ( q )is expected to relate with the strength of the market inefficiency.We also calculate the singularity spectrum f ( α ), which is another way tocharacterize a multifractal time series. It is defined by α ( q ) = h ( q ) + qh (cid:48) ( q ) , (11) f ( α ) = q [ α − h ( q )] + 1 , (12)February 16, 2021 7/27here α ( q ) is the H¨older exponent or singularity strength [35]. The multifractalitydegree through f ( α ) is defined by∆ α ( q ) = α ( q min ) − α ( q max ) , (13)which takes zero for monofractal. In this study we take q = [ − , | q | is large, the moments may diverse and the estimated results areunreliable [36]. Thus, to avoid such a situation, following [47], for the calculations ofthe multifractal degree ∆ h ( q ) and ∆ α ( q ), we take q = 4. The strength of ∆ α ( q ) is alsorelated with the market inefficiency, and we call ∆ h ( q ) and ∆ α ( q ) including kurtosis”efficiency-related measures.” We use the same rolling window method to investigatetime-varying properties of multifractality and take the same window size (548 days)with the volatility analysis by the TGARCH model to compare the results. We use Bitcoin tick data (in dollars) traded on Bitstamp from September 10, 2011, toJune 06, 2020, downloaded from Bitcoincharts (http://api.bitcoincharts.com/v1/csv/).Due to a hacking incident, no data are available from January 4, 2015 to January 9,2015. For these missing data, we treat them as the price is unchanged.
Fig 2 illustrates daily (∆ t = 1440-min) price and returns r t j constructed from theBitcoin tick data. We eliminate the data that are larger than 40, i.e., r t i >
40, asoutliers. This manipulation keeps the results almost unchanged except kurtosis. In ourdata set, we find four outliers. Fig 2(d) shows the volatility series s t defined by s t = s t − + | r t | − ¯ r , where r t and ¯ r stand for the return at t and the average of r t ,respectively. The volatility series introduced in [58, 59] can be utilized to identify thevolatility clustering. Namely, the increasing (decreasing) trend of the volatility seriesimplies the existence of the high (low) volatility clustering. Such trends indicating highand low volatility clusterings are seen in Fig 2(d).February 16, 2021 8/27
012 2014 2016 2018 2020 p r i ce p r i ce -60-40-2002040 r e t u r n -202468 vo l a tilit y s e r i e s (a)(b)(c)(d) Fig 2. (a): Daily price, (b): daily price (semi-log plot), (c): returns and (d) volatilityseries.Table 1 provides descriptive statistics for the whole sample of returns, and we find apositive average, high kurtosis, and negative skewness. We also explore the timevariation in these using the rolling window method.
Table 1.
Descriptive statistics for the whole sample of daily returns.SD stands for ”standard deviation.” The values in parentheses indicate one sigma errorsestimated by the Jackknife method.Mean SD Kurtosis Skewness Nobs0.27(11) 4.63(41) 11.9(22) -0.267(99) 3188
Fig 3 illustrates the average, standard deviation (SD), kurtosis, and skewness calculatedwith a 548-day rolling window. Interestingly, they vary considerably over time. Whilstthe kurtosis before 2017 is very high (i.e., more than 10), it decreases after 2017.Recently, it has taken a value of around 6, which is still higher than the GaussianFebruary 16, 2021 9/27urtosis. The origin of high kurtosis could be a fat-tailed return distribution that meanshigher price variations are observed more often. At the early stage of Bitcoin market,the tail index µ of the cumulative return distribution is found to be µ ≈
2, which isreferred to as the inverse square law [60]. The similar tail indces are have also beenreported in [61] This is sharply contrast to the well-known inverse cubic law for otherassets [62–65], in which the tail index µ is µ ≈
3. The inverse square law observed in theBitcoin market, however, is not permanent. The recent Bitcoin data up to 2017 showthat the tail index comes close to 3, which suggests that the Bitcoin market is becomingmore mature [47]. Further studies [66, 67] also indicate the change of the tail index to 3.It is also worth noting that the recent COVID-19 pandemic considerably affects thecryptocurrency market and as a result the market experiences a volatile period in whichthe tail index varies [13].These observations imply that the tail index µ of the cumulative return distributionin the Bitcoin market varies over time and moves from µ ≈ µ ≈ Fig 4 illustrates the aggregational Gaussianity of returns, namely the kurtosis of returnssampled at ∆ t as a function of ∆ t . The figure is plotted in log-log scale, and we findthat the kurtosis decreases according to a power-law up to ∆ t ≈ ≈ twoweeks, with an exponent ∼ − .
62, and the kurtosis of returns at ∆ t longer than twoFebruary 16, 2021 10/27
013 2014 2015 2016 2017 2018 2019 2020 -0.500.511.52 a v e r a g e S D ku r t o s i s s k e w n e ss Fig 3. Fig 3. Average, SD, kurtosis, and skewness as a function of time.SD stands for standard deviation. These are calculated using the rollingwindow method with a window size of 548 days. Bars in the data pointsrepresent one sigma error, estimated by the Jackknife method. weeks is consistent with the Gaussian kurtosis, in agreement with the previousresult [17]. This finding suggests that the time series of returns at ∆ t longer than twoweeks most likely becomes the random Gaussian time series. Table 2 presents the TGARCH parameters estimated for the whole dataset. The γ parameter is negative, which indicates an inverted asymmetry in the volatility. Itsmagnitude, however, is not large, which is consistent with the result of [31]. This isprobably because the strength of time-varying asymmetry of γ weakens in theparameter estimation for the whole dataset which may include both periods of positiveFebruary 16, 2021 11/27
10 100 1000 10000 1e+05 ∆ t (min) ku r t o s i s ~ ∆ t -0.62 Fig 4.
Kurtosis at various sampling period ∆ t as a function of ∆ t . Bars in the datapoints represent one sigma error, estimated by the Jackknife method. The red linedisplays the Gaussian kurtosis (= 3).and negative asymmetries. In the following, we find that the γ parameter varies overtime considerably. Table 2.
TGARCH parameter estimates for the whole dataset. ν is the shapeparameter of the Student t distribution.The values in parentheses indicate standard errors. α β ω γ c µ ν γ ismostly negative, we find some exceptions. For instance, parameter γ takes positive orzero values around 2015; and after 2019, its magnitude becomes small or consistent withzero. We also observe a strong inverted asymmetry between 2016 and 2018. For therobustness check on the IID distribution in Eq (2), we perform the parameterestimation with the normal and generalized error distributions, and find that the similarasymmetric volatility patterns to that from the Student t distribution are obtained.Therefore, the choice of distributions in the IID process is irrelevant.February 16, 2021 12/27
013 2014 2015 2016 2017 2018 2019 202000.10.20.30.40.5 α β ω Fig 5.
Estimation results of α, β , and ω . The error bars show the standard errors.The AR(1) parameter c , which captures serial correlation, also varies considerably.It is argued that non-zero serial correlation implies that uninformed investors dominatein trading and that price changes due to uninformed investors will increase volatilitymore than price changes caused by informed investors [68]. In line with [68], it isclaimed that non-zero AR(1) coefficients are found for cryptocurrencies; and theinverted asymmetry due to uniformed investors is consistent with phenomena such asfear of missing out, pump and dump schemes, and the disposition effect [31]. Ourresults of c indicate that there are periods in which c is consistent with zero, whichsuggests that the Bitcoin market is not always dominated by uninformed investors. Wefind both strong inverted asymmetry and non-zero c from 2016 to 2018. Thus, theperiod from 2016 to 2018 is considered to be dominated by uninformed investors,thereby affecting the volatility asymmetry. As seen in Fig 2, in this period, the Bitcoinprice increases considerably and recorded the highest value on December 2017.Consequently, Bitcoin price movement in this period is associated with the invertedasymmetry induced dominantly by uninformed investors.We also estimate the model parameters for high-frequency returns (6h and 12h). Fig7 represents the results of γ together with those from the daily returns. We find thathigh-frequency returns exhibit similar variation with daily returns except that the 6hreturns for which no significant inverted asymmetry is seen before 2015. These resultsFebruary 16, 2021 13/27
013 2014 2015 2016 2017 2018 2019 2020-0.2-0.100.10.2 γ c -0.200.20.40.6 µ Fig 6.
Estimation results of γ, c and µ . The error bars show the standard errors.imply that whilst the asymmetric volatility pattern remains for higher frequencyreturns, the detail of the asymmetry pattern depends on the frequency of returns. -0.200.20.4 γ Daily12h6h
Fig 7.
Asymmetry parameter γ for 6h, 12h, and daily returns. The error bars show thestandard errors.To calculate the multifractal degree ∆ h ( q ) and ∆ α ( q ), we first determine h ( q ) byfitting the fluctuation function F q ( s ) to Eq(9) in a range of s = [20 , h ( q ) and ∆ α ( q ) by Eq(10) and (13), respectively. As representatives, Fig 8(a)-(b) shows fluctuation functions F q ( s ) calculated using the first window data and h ( q ), respectively.February 16, 2021 14/27 s F q ( s ) -20 -10 0 10 20 q h ( q ) (a)(b) Fig 8. (a) Fluctuation functions F q ( s ) (b) The generalized Hurst exponent h ( q ). In Fig8 (a), the results are plotted from q = −
25 (bottom) to q = 25 with a step of 1.0.Fig 9 (a) represents the Hurst exponent h (2) as a function of time, showing someanti-persistent periods ( h (2) < / h (2) around 2020 is consistent with orslightly above 1/2, which suggests that the recent Bitcoin market is becoming moreefficient. This finding is consistent with the results of kurtosis and skewness in Fig 3that shows low kurtosis and insignificant skewness for the recent Bitcoin market,meaning that the return distribution becomes more Gaussian shaped than before. For∆ h ( q ) and ∆ α ( q ), we show the results of q = 4 in Fig 9 (b) as a function of time. Twomeasures of the multifractal degree calculated here show the same time-varying pattern.Broadly speaking, the multifractal degree decreases with time, which suggests that theBitcoin market gradually approaches the efficient market. Using the Amihud illiquiditymeasure [69], it is argued that the market inefficiency in the cryptocurrency market iscaused by illiquidity and the illiquidity is related with anti-persistency [4]. For theBitcoin market, liquidity in terms of the Amihud illiquidity measure turns out to beimproving [44], which agrees with the improved efficiency of the Bitcoin market inFebruary 16, 2021 15/27ecent years. h ( ) m u lti fr ac t a l d e g r ee ∆ h(4) ∆α(4) (a)(b) Fig 9. (a) the Hurst exponent h (2), (b) multifractal degrees ∆ h (4) and ∆ α (4). Theseresults are obtained by the rolling window method with a 548-day window and a step of1 day.In Fig 10 (a)-(b), we examine the relationship between kurtosis and theefficiency-related measures (a) h (2) and (b) ∆ α (4). Fig 10 (a) shows that the results of h (2) near 1/2 takes kurtosis smaller values, for example, less than 7. However, mostlarger kurtoses more likely correspond to h (2) far from 1/2. In Fig 10 (b), we find thatthe smaller kurtoses opt to take small ∆ α (4). Therefore, the results of Fig 10 (a)-(b)imply that in the more efficient market, the kurtosis takes smaller values close to theGaussian one. Here, note that since returns at short scale are usually fat-taileddistributions, the kurtosis of daily returns may not reach the Gaussian one, and itrather reaches a certain minimum value ( >
3) at the efficiency-improved market.Fig 11 (a)-(c) illustrates relationships between the asymmetric parameter γ and threeFebruary 16, 2021 16/27 .2 0.3 0.4 0.5 0.6 0.7 h(2) ku r t o s i s ∆α(4) ku r t o s i s (a)(b) Fig 10. (a) Kurtosis versus h (2) and (b) Kurtosis versus ∆ α (4).measurements, (a) h (2), (b) ∆ α (4), and (c) kurtosis, and some remarks are in order. InFig 11 (a), it is observed that in an anti-persistent domain ( h (2) < / γ takes negative value more than positive ones, and the parameter γ nearzero tends to cluster in a region that h (2) takes values near 1/2 or slightly above 1/2.In Fig 11 (b), we find that in a region of small ∆ α (4), for example, ∆ α (4) < .
5, theparameter γ mostly takes values near zero. However, for larger ∆ α (4), the parameter γ tends to take negative values. Fig 11 (c) indicates that for a region near the Gaussiankurtosis (e.g., kurtosis < γ comes to values near zero. Moreover,mostly the strong negative γ comes to a region with higher kurtoses. The overall resultsfrom Fig 10 (a)-(c) indicate that for the market being more efficient, that is, for h (2)near 1/2, small ∆ α (4), and kurtosis closer to the Gaussian one, the volatilityFebruary 16, 2021 17/27symmetry likely disappears. This is consistent with the consequence of the efficientmarket that any predictable patterns such as the asymmetric volatility should not exist. Correct understanding of the market state is of great importance for investors whochange trading strategy depending on the state of the market. This study couldcontribute to offer such information. For example, according to the efficient markethypothesis [46], the technical analysis is not supported on the efficient market. Ourresults imply that the recent Bitcoin market is being efficient and it might be difficultto make high profits by the technical analysis. By monitoring the market efficiency ofBitcoin, if the Bitcoin market becomes inefficient substantially again, one could use thetechnical analysis to gain profits. Another suggestion from the efficient markethypothesis is that on the efficient market, the most efficient portfolio is a marketportfolio consisting of every asset weighted in proportion to its market capitalization.Of course, it is difficult to form a completely diversified portfolio in practice [70].However, it may be possible to make an approximate diversified portfolio including theefficient Bitcoin. Furthermore, although not all cryptocurrencies are efficient, when thecryptocurrency markets become more mature and more efficient, an index or portfolioconsisting of cryptocurrencies could be a proxy of a fully diversified portfolio on thecryptocurrency markets.
We use the rolling window method to investigate time-varying properties of Bitcoin. Wefind that various measurements, such as volatility asymmetry, kurtosis, skewness, serialcorrelation, and multifractality, are time varying. Thus, the Bitcoin market may haveinherently variable properties. Although the inverted asymmetry is observed in Bitcoinand the strong inverted asymmetry is found around 2016-2018, the recent volatilityasymmetry is weak. The magnitude of the volatility asymmetry may relate with themarket state, especially the market efficiency.To investigate a relationship between the volatility asymmetry and the marketefficiency, we examine efficiency-related measures: the Hurst exponent, multifractalFebruary 16, 2021 18/27egree, and kurtosis. We find that when these efficiency-related measures indicate thatthe market is more efficient, the volatility asymmetry is more likely to weaken. Theefficiency-related measures indicate that the recent Bitcoin market has become moreefficient.Whilst we use one of the GARCH-type models, that is, TGARCH model, which iscommonly used in volatility analysis of Bitcoin, other types of volatility models such asstochastic volatility model [71–73] exist. It might be interesting to investigate whetherthese models lead to the similar results on the volatility asymmetry.In addition to the inverted asymmetry, other remarkable properties are observed inthe Bitcoin market. For example, whilst it is claimed that for stock markets, time seriesof the log-volatility increments shows ”monofractal” anti-persistence behavior [74–76],for the Bitcoin market ”multifractal” anti-persistence behavior is reported [11]. Thisdifference is important to construct a correct volatility model with monofractal ormultifractal behavior. Another interesting property is observed in return-volatilitycross-correlation. For stock markets, it is found that return-volatility cross-correlationfunction exhibits an exponential decay that indicates that the return-volatilitycross-correlation is short-ranged [77–79]. However, for the Bitcoin market, thereturn-volatility cross-correlation function shows a power law meaning that thecross-correlation is long-ranged [12]. The mechanism that originates these remarkableproperties including the inverted asymmetry are not established yet. Toward thecomplete understanding of the Bitcoin dynamics, we should further study theseproperties in more detail in future direction.
Acknowledgment
Numerical calculations for this work were carried out at the Yukawa Institute ComputerFacility and at the facilities of the Institute of Statistical Mathematics.
References
1. Nakamoto S. Bitcoin: A Peer-to-Peer Electronic Cash System; 2008.February 16, 2021 19/27. Dyhrberg AH. Hedging capabilities of Bitcoin. Is it the virtual gold? FinanceResearch Letters. 2016;16:139–144.3. Cheah ET, Fry J. Speculative bubbles in Bitcoin markets? An empiricalinvestigation into the fundamental value of Bitcoin. Economics Letters.2015;130:32–36.4. Wei WC. Liquidity and market efficiency in cryptocurrencies. Economics Letters.2018;168:21–24.5. Takaishi T, Adachi T. Taylor effect in Bitcoin time series. Economics Letters.2018;172:5–7.6. Thies S, Moln´ar P. Bayesian change point analysis of Bitcoin returns. FinanceResearch Letters. 2018;27:223–227.7. Koutmos D. Bitcoin returns and transaction activity. Economics Letters.2018;167:81–85.8. Fang W, Tian S, Wang J. Multiscale fluctuations and complexity synchronizationof Bitcoin in China and US markets. Physica A. 2018;512:109–120.9. Phillip A, Chan J, Peiris S. On long memory effects in the volatility measure ofcryptocurrencies. Finance Research Letters. 2019;28:95–100.10. Urquhart A. Price clustering in Bitcoin. Economics Letters. 2017;159:145–148.11. Takaishi T. Rough volatility of Bitcoin. Finance Research Letters.2020;32:101379.12. Takaishi T. Power-law return-volatility cross-correlations of Bitcoin. EPL(Europhysics Letters). 2020;129(2):28001.13. Dro˙zd˙z S, Kwapie´n J, O´swi¸ecimka P, Stanisz T, W¸atorek M. Complexity ineconomic and social systems: cryptocurrency market at around COVID-19.Entropy. 2020;22(9):1043.14. Cont R. Empirical Properties of Asset Returns: Stylized Facts and StatisticalIssues. Quantitative Finance. 2001;1:223–236.February 16, 2021 20/275. Kwapie´n J, Dro˙zd˙z S. Physical approach to complex systems. Physics Reports.2012;515(3-4):115–226.16. Chu J, Nadarajah S, Chan S. Statistical analysis of the exchange rate of Bitcoin.PloS one. 2015;10(7):e0133678.17. Takaishi T. Statistical properties and multifractality of Bitcoin. Physica A.2018;506:507–519.18. Engle RF. Autoregressive conditional heteroscedasticity with estimates of thevariance of United Kingdom inflation. Econometrica: Journal of the EconometricSociety. 1982; p. 987–1007.19. Bollerslev T. Generalized Autoregressive Conditional Heteroskedasticity. Journalof Econometrics. 1986;31:307–327.20. Nelson DB. Conditional Heteroskedasticity in Asset Returns: A New Approach.Econometrica. 1991;59:347–370.21. Glosten LR, Jaganathan R, Runkle D. On the Relation Between the ExpectedValue and the Volatility of the Nominal Excess on Stocks. Journal of Finance.1993;48:1779–1801.22. Bouri E, Azzi G, Dyhrberg AH. On the return-volatility relationship in theBitcoin market around the price crash of 2013. Economics: The Open-Access,Open-Assessment E-Journal. 2017;11:1–16.23. Katsiampa P. Volatility estimation for Bitcoin: A comparison of GARCH models.Economics Letters. 2017;158:3–6.24. Stavroyiannis S, Babalos V. Dynamic properties of the Bitcoin and the USmarket. Available at SSRN 2966998. 2017;.25. Ardia D, Bluteau K, R¨uede M. Regime changes in Bitcoin GARCH volatilitydynamics. Finance Research Letters. 2019;29:266–271.26. Black F. Studies of Stock Market Volatility Changes. 1976 Proceedings of theAmerican Statisticsl Association, Business and Economic Statistics Section. 1976;p. 177–181.February 16, 2021 21/277. Christie AA. The stochastic behavior of common stock variances: Value, leverageand interest rate effects. Journal of Financial Economics. 1982;10:407–432.28. Wu G. The determinants of asymmetric volatility. The Review of FinancialStudies. 2001;14(3):837–859.29. Dyhrberg AH. Bitcoin, gold and the dollar–A GARCH volatility analysis.Finance Research Letters. 2016;16:85–92.30. Bouoiyour J, Selmi R. Bitcoin: A beginning of a new phase. Economics Bulletin.2016;36(3):1430–1440.31. Baur DG, Dimpfl T. Asymmetric volatility in cryptocurrencies. EconomicsLetters. 2018;173:148–151.32. Fakhfekh M, Jeribi A. Volatility dynamics of crypto-currencies’ returns: Evidencefrom asymmetric and long memory GARCH models. Research in InternationalBusiness and Finance. 2020;51:101075.33. Bollerslev T, Chou RY, Kroner KF. ARCH modeling in finance: A review of thetheory and empirical evidence. Journal of econometrics. 1992;52(1-2):5–59.34. Kyriazis NA, Daskalou K, Arampatzis M, Prassa P, Papaioannou E. Estimatingthe volatility of cryptocurrencies during bearish markets by employing GARCHmodels. Heliyon. 2019;5(8):e02239.35. Kantelhardt JW, Zschiegner SA, Koscielny-Bunde E, Havlin S, Bunde A, StanleyHE. Multifractal detrended fluctuation analysis of nonstationary time series.Physica A. 2002;316(1):87–114.36. Jiang ZQ, Xie WJ, Zhou WX, Sornette D. Multifractal analysis of financialmarkets. Rep Prog Phys. 2019;82(12):125901.37. Urquhart A. The inefficiency of Bitcoin. Economics Letters. 2016;148:80–82.38. Bariviera AF, Basgall MJ, Hasperu´e W, Naiouf M. Some stylized facts of theBitcoin market. Physica A. 2017;484:82–90.39. Bariviera AF. The inefficiency of Bitcoin revisited: A dynamic approach.Economics Letters. 2017;161:1–4.February 16, 2021 22/270. Alvarez-Ramirez J, Rodriguez E, Ibarra-Valdez C. Long-range correlations andasymmetry in the Bitcoin market. Physica A. 2018;492:948–955.41. Kristoufek L. On Bitcoin markets (in) efficiency and its evolution. Physica A.2018;503:257–262.42. da Silva Filho AC, Maganini ND, de Almeida EF. Multifractal analysis of Bitcoinmarket. Physica A. 2018;512:954–967.43. Zhang W, Wang P, Li X, Shen D. The inefficiency of cryptocurrency and itscross-correlation with Dow Jones Industrial Average. Physica A.2018;510:658–670.44. Takaishi T, Adachi T. Market efficiency, liquidity, and multifractality of Bitcoin:A dynamic study. Asia-Pacific Financial Markets. 2020;27:145–154.45. Dimitrova V, Fern´andez-Mart´ınez M, S´anchez-Granero MA, Trinidad Segovia JE.Some comments on Bitcoin market (in) efficiency. PloS one. 2019;14(7):e0219243.46. Fama EF. Efficient capital markets: A review of theory and empirical work. Thejournal of Finance. 1970;25(2):383–417.47. Dro˙zd˙z S, G¸ebarowski R, Minati L, O´swi¸ecimka P, W¸atorek M. Bitcoin marketroute to maturity? Evidence from return fluctuations, temporal correlations andmultiscaling effects. Chaos: An Interdisciplinary Journal of Nonlinear Science.2018;28(7):071101.48. Zunino L, Bariviera AF, Guercio MB, Martinez LB, Rosso OA. On the efficiencyof sovereign bond markets. Physica A. 2012;391(18):4342–4349.49. Sentana E. Quadratic ARCH Models. Review of Economic Studies.1995;62:639–661.50. Heston SL, Nandi S. A closed-form GARCH option valuation model. The reviewof financial studies. 2000;13(3):585–625.51. Takaishi T. Rational GARCH model: An empirical test for stock returns.Physica A. 2017;473:451–460.February 16, 2021 23/272. Ivanov PC, Amaral LAN, Goldberger AL, Havlin S, Rosenblum MG, Struzik ZR,et al. Multifractality in human heartbeat dynamics. Nature.1999;399(6735):461–465.53. Stanley HE, Meakin P. Multifractal phenomena in physics and chemistry. Nature.1988;335(6189):405–409.54. Lennartz S, Livina V, Bunde A, Havlin S. Long-term memory in earthquakes andthe distribution of interoccurrence times. EPL (Europhysics Letters).2008;81(6):69001.55. Ihlen EA, Vereijken B. Multifractal formalisms of human behavior. Humanmovement science. 2013;32(4):633–651.56. Jafari G, Pedram P, Hedayatifar L. Long-range correlation and multifractality inBach’s inventions pitches. Journal of Statistical Mechanics: Theory andExperiment. 2007;2007(04):P04012.57. Zunino L, Tabak BM, Figliola A, P´erez D, Garavaglia M, Rosso O. A multifractalapproach for stock market inefficiency. Physica A. 2008;387(26):6558–6566.58. Trinidad Segovia JE, Fern´andez-Mart´ınez M, S´anchez-Granero MA. A novelapproach to detect volatility clusters in financial time series. Physica A.2019;535:122452.59. Nikolova V, Trinidad Segovia JE, Fern´andez-Mart´ınez M, S´anchez-Granero MA.A Novel Methodology to Calculate the Probability of Volatility Clusters inFinancial Series: An Application to Cryptocurrency Markets. Mathematics.2020;8(8):1216.60. Easwaran S, Dixit M, Sinha S. Bitcoin dynamics: the inverse square law of pricefluctuations and other stylized facts. In: Econophysics and data driven modellingof market dynamics. Springer; 2015. p. 121–128.61. Beguˇsi´c S, Kostanjˇcar Z, Stanley HE, Podobnik B. Scaling properties of extremeprice fluctuations in Bitcoin markets. Physica A. 2018;510:400–406.February 16, 2021 24/272. Gopikrishnan P, Meyer M, Amaral LN, Stanley HE. Inverse cubic law for thedistribution of stock price variations. The European Physical JournalB-Condensed Matter and Complex Systems. 1998;3(2):139–140.63. Gopikrishnan P, Plerou V, Amaral LAN, Meyer M, Stanley HE. Scaling of thedistribution of fluctuations of financial market indices. Physical Review E.1999;60(5):5305.64. Plerou V, Gopikrishnan P, Amaral LAN, Meyer M, Stanley HE. Scaling of thedistribution of price fluctuations of individual companies. Physical review E.1999;60(6):6519.65. Pan RK, Sinha S. Self-organization of price fluctuation distribution in evolvingmarkets. EPL (Europhysics Letters). 2007;77(5):58004.66. Dro˙zd˙z S, Minati L, O´swi¸ecimka P, Stanuszek M, W¸atorek M. Signatures of thecrypto-currency market decoupling from the Forex. Future Internet.2019;11(7):154.67. Takaishi T. Recent scaling properties of Bitcoin price returns. arXiv:200906874.2020;.68. Avramov D, Chordia T, Goyal A. The impact of trades on daily volatility. TheReview of Financial Studies. 2006;19(4):1241–1277.69. Amihud Y. Illiquidity and stock returns: cross-section and time-series effects.Journal of financial markets. 2002;5(1):31–56.70. Roll R. A critique of the asset pricing theory’s tests Part I: On past and potentialtestability of the theory. Journal of financial economics. 1977;4(2):129–176.71. Taylor SJ. Financial Returns Modelled by the Product of Two StochasticProcesses, a Study of Daily Sugar Prices 1961-79. North-Holland, Amsterdam;1982.72. Taylor SJ. Modelling Financial Time Series. John Wiley & New jersy; 1986.73. Asai M, McAleer M, Yu J. Multivariate stochastic volatility: a review.Econometric Reviews. 2006;25(2-3):145–175.February 16, 2021 25/274. Gatheral J, Jaisson T, Rosenbaum M. Volatility is rough. Quantitative Finance.2018;18(6):933–949.75. Bennedsen M, Lunde A, Pakkanen MS. Decoupling the short-and long-termbehavior of stochastic volatility. arXiv:161000332. 2016;.76. Livieri G, Mouti S, Pallavicini A, Rosenbaum M. Rough volatility: evidence fromoption prices. IISE Transactions. 2018;50(9):767–776.77. Bouchaud JP, Matacz A, Potters M. Leverage effect in financial markets: Theretarded volatility model. Physical Review Letters. 2001;87(22):228701.78. Qiu T, Zheng B, Ren F, Trimper S. Return-volatility correlation in financialdynamics. Physical Review E. 2006;73(6):065103.79. Chen JJ, Zheng B, Tan L. Agent-based model with asymmetric trading andherding for complex financial systems. PloS one. 2013;8(11):e79531.February 16, 2021 26/27 h ( ) -0.2 -0.1 0 0.1 0.200.511.5 ∆ α ( ) -0.2 -0.1 0 0.1 0.2 γ ku r t o s i s (a)(b)(c) Fig 11.
Relationships between γ and (a) h (2), (b) ∆ αα