Recent scaling properties of Bitcoin price returns
RRecent scaling properties of Bitcoin price returns
T Takaishi
Hiroshima University of Economics, Hiroshima 731-0192, JAPANE-mail: [email protected]
Abstract.
While relevant stylized facts are observed for Bitcoin markets, we find a distinctproperty for the scaling behavior of the cumulative return distribution. For various assets, thetail index µ of the cumulative return distribution exhibits µ ≈
3, which is referred to as ”theinverse cubic law.” On the other hand, that of the Bitcoin return is claimed to be µ ≈ µ ≈
3, which is consistent with the inversecubic law. This suggests that some properties of the Bitcoin market could vary over time.We also investigate the autocorrelation of absolute returns and find that it is described by apower-law with two scaling exponents. By analyzing the absolute returns standardized by therealized volatility, we verify that the Bitcoin return time series is consistent with normal randomvariables with time-varying volatility.
1. Introduction
In 2008, Nakamoto[1] proposed Bitcoin as the first practical cryptocurrency based on theblockchain technology. His proposal was quickly accepted, and in 2009, the Bitcoin network waslaunched as a peer-to-peer payment network. Since then, a number of digitized cryptocurrencieshave been proposed and created , and growing cryptocurrency markets have a strong impacton contemporary financial markets. Bitcoin has recently attracted the interest of researchersand various properties and aspects of Bitcoin have been investigated. For example, volatilityanalysis [2, 3], price clustering[4, 5], adaptive market hypothesis [6], transaction activity [7],multifractality[8], liquidity and efficiency[9], Taylor effect[10], structural breaks [11], longmemory effects[12], rough volatility[13], power-law cross-correlation[14], asymmetric multifractalanalysis[15], and so forth.Universal properties across various financial time series are classified as stylized facts[16],which include (i) fat-tailed distributions, (ii) volatility clustering, and (iii) slow decay ofautocorrelation in absolute returns, among others. While stylized facts are also observed inBitcoin, for example, [8, 17], a distinct property has been observed for the tail index of returndistributions. For other assets such as stocks, it is known that the tail of the cumulative returndistribution is described by a power-law function and its tail index µ is obtained as µ ≈ µ ≈
2, which differs from the known stylized fact for other assets and is referredto as ”the inverse square law”[22]. Similar tail indices have also been reported in Ref.[23]. Sincethis is a newly emerging market, market properties of Bitcoin could change as trading activityincreases over time. In fact, the market efficiency measured by the Hurst exponent of the return e.g., see https://coinmarketcap.com/ for the current market capitalizations of cryptocurrencies. a r X i v : . [ q -f i n . S T ] S e p able 1. Results of tail index.(I) 2011-2013 (II) 2015-2020 2011-2020Fitting region [2,20] [2,20] [1,10]positive tail index µ -2.06 -3.33 -2.35negative tail index µ -2.08 -3.23 -2.38time series varies over time and at the early stage of the Bitcoin market, the Hurst exponentis observed to be less than 1/2, which indicates that the time series is anti-persistent[24]. Thisanti-persistent behavior could be related to the illiquidity of the Bitcoin market[9, 25]. As theliquidity of the Bitcoin market improves, the Hurst exponent moves to 1/2[24, 25].We investigate the possible change in scaling properties of the return distribution for therecent Bitcoin market. We also investigate the cumulative distributions and autocorrelation ofthe absolute returns and the return standardized by the realized volatility.
2. Data set
The data are Bitcoin tick data (in dollars) traded on Bitstamp from September 11, 2011 to June21, 2020 and downloaded from Bitcoincharts . We construct 1-min price data from the Bitcointick data and then calculate 1-min returns R t , t = 1 , , ..., N as R t = log p t − log p t − , (1)where p t is the 1-min price at t (min). We further standardize the returns data using¯ R t = R t − νσ . (2)where ν and σ represent the average and standard deviation of returns R t , respectively. We selectthe following two data periods from the data, which separate low and high liquidity periods: I)September 11, 2011-December 31, 2013 and II) January 1, 2015-June 21, 2020. Dataset I (II)contains low (high) liquidity data[25].
3. Results
First, we show the cumulative distributions of returns in Figure 1(a): positive tail and (b):negative tail, in a log-log plot, and recognize that the tail behavior is described by a powerlaw. While there is no significant difference between positive and negative tails, a considerabledifference is seen between data periods, that is, a heavier tail is observed for the cumulativedistribution of dataset (II), which contains high liquidity data. We obtain tail indices of thecumulative distributions by a regression fit to a power-law function of κ ¯ R − µ , where κ and µ arefitting parameters. The results of the tail indices are reported in Table 1. The tail index fordataset I is µ ≈
2, which is consistent with the previous results obtained for earlier periods[22].On the other hand, the tail index for II is µ ≈
3, which is consistent with the tail index obtainedfor other assets such as stocks. For the entire data set, we find a tail index between I andII. Our findings suggest that the scaling properties of the return distributions of the Bitcoinmarket could change over time, For the recent liquidity of the Bitcoin market, the tail index isin agreement with the well-known stylized facts for other assets. Due to a hacking incident, no data are available from January 4, 2015 to January 9, 2015. For these missingdata, we treat them as the price is unchanged. http://api.bitcoincharts.com/v1/csv/ Standardized R t C u m u l a ti v e d i s t r i bu ti on (a) Standardized R t C u mm l a ti v e d i s t r i bu ti on (b) Figure 1. (a) Cumulative distribution for positive tail. (b) Cumulative distribution for negativetail.Next, we show the autocorrelation function (ACF) of the absolute returns. The ACF of atime series x i : i = 1 , ..., N is defined by ACF ( t ) = (cid:104) ( x i − ρ )( x i + j − ρ ) (cid:105) σ x . (3)where ρ and σ x represent the average and variance of x i , respectively, and (cid:104) O i (cid:105) means takingan average over O i . Figure 2 displays the ACF of the absolute returns in the log-log plot andwe see that the ACF is also well described by a power law. A remarkable characteristic of theACF is that the power-law exponent seems to change at t ∼ t > t < able 2. Results of the power-law exponent.(I) 2011-2013 (II) 2015-2020 2011-2020fitting region [3,500] [3,500] [3,500]exponent µ -0.121 -0.116 -0.120fitting region [15000,10000] [1500,10000] [1500,10000]exponent µ -0.197 -0.235 -0.206
10 100 1000 10000 t A C F Figure 2.
ACF of absolute returns. The symbols at t=100 and 1000 show typical one-sigmaerrors calculated using the jackknife method.by R t ≡ σ t (cid:15) t , where σ t is the volatility at t and (cid:15) t is the standard normal random variable.Namely, the return R t is assumed to behave as a normal random variable with time-varyingvolatility. This assumption of returns automatically satisfies no autocorrelation in returns. Onthe other hand, the autocorrelation of absolute returns depends on the time structure of σ t . Ifwe standardize the (observed) returns R t by σ t , that is, R t /σ t ≡ ˜ R t , we expect to obtain thestandard normal random time series (cid:15) t . To test whether ˜ R t satisfies this expectation we needto estimate the volatility σ t of returns in the proper manner because volatility is not observablein the financial markets.In empirical finance, the standard technique to estimate volatility is to use volatility modelssuch as GARCH-type models[26–33]. The drawback of using such models is that volatilityestimates are model-dependent, and which estimate is better is not well defined. Recentavailability of high-frequency financial data enable us to obtain a model-free estimate knownas ”realized volatility (RV)”[34, 35] constructed by the sum of the square of intraday returns.The returns standardized by the RV are examined for the exchange rate[36, 37] and return[38],and the normality of the standardized return ˜ R t is established. However, detailed studies reportthe existence of deviation from the normality caused by the low sampling frequency in theRV[39–43]. Furthermore because ˜ R t is expected to be a random variable, we should observe noautocorrelation not only for ˜ R t but also for the absolute ˜ R t , that is, | ˜ R t | . The autocorrelation ofthe absolute ˜ R t is examined for Japanese stocks[44], and it is verified that no autocorrelation isobserved for the absolute ˜ R t . We examine the nonexistence of autocorrelation in the absolute˜ R t of Bitcoin. Figure 3 shows the autocorrelation of the absolute ˜ R t standardized by the RV at This behavior is also confirmed for returns simulated by the artificial spin model and standardized by GARCHvolatility[45].
10 20 30 40 50t00.20.40.60.8 A C F Figure 3.
ACF of the absolute returns standardized by RV. For the calculation of the ACF,we take the data from January 10, 2015 to June 21, 2020.a 5-min sampling frequency , and we confirm that there is no significant autocorrelation in theabsolute ˜ R t .
4. Conclusions
We investigate the scaling properties of the Bitcoin market and find that the tail index µ of thecumulative return distribution changes from µ ≈ µ ≈
3, which indicates that the recentBitcoin market exhibits the well-established scaling law, that is, the inverse cubic law. Ourfindings suggest that some properties of the Bitcoin market could change over time.We find that the autocorrelation of absolute returns shows a power law with two scalingexponents separated at approximately t = 1000min. The ACF of the absolute returnsstandardized by RV shows no autocorrelation, which indicates that the return time series isconsistent with a normal random variable with time-varying volatility. Acknowledgment
Numerical calculations for this work were carried out at the Yukawa Institute Computer Facilityand at the Institute of Statistical Mathematics. This work was supported by JSPS KAKENHIGrant Number JP18K01556.
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