An analysis of high-frequency cryptocurrencies prices dynamics using permutation-information-theory quantifiers
aa r X i v : . [ q -f i n . S T ] J u l An analysis of high-frequency cryptocurrenciesprices dynamics usingpermutation-information-theory quantifiers
Aurelio F. Bariviera
Universitat Rovira i Virgili,Department of Business.Av. Universitat 1, 43204 Reus SpainUniversidad del Pac´ıfico. Lima, Per´u [email protected]
Luciano Zunino
Centro de Investigaciones ´Opticas (CONICET La Plata - CIC)C.C. 3, 1897 Gonnet, ArgentinaDepartamento de Ciencias B´asicas, Facultad de Ingenier´ıaUniversidad Nacional de La Plata (UNLP), 1900 La Plata, Argentina [email protected]
Osvaldo A. Rosso
Departamento de Inform´atica en Salud, Hospital Italiano de Buenos Aires& CONICET, C1199ABB Ciudad Aut´onoma de Buenos Aires, Argentina.Instituto de F´ısica, Universidade Federal de Alagoas,Av. Lourival Melo Mota, s/n, 57072-970 Macei´o, AL, BrazilComplex Systems Group, Facultad de Ingenier´ıa y Ciencias Aplicadas,Universidad de los Andes, Las Condes, 12455 Santiago, Chile [email protected]
August 7, 2018
Abstract
This paper discusses the dynamics of intraday prices of twelve cryp-tocurrencies during last months’ boom and bust. The importance of thisstudy lies on the extended coverage of the cryptoworld, accounting formore than 90% of the total daily turnover. By using the complexity-entropy causality plane, we could discriminate three different dynamics inthe data set. Whereas most of the cryptocurrencies follow a similar pat-tern, there are two currencies (ETC and ETH) that exhibit a more per-sistent stochastic dynamics, and two other currencies (DASH and XEM)whose behavior is closer to a random walk. Consequently, similar financial ssets, using blockchain technology, are differentiated by market partici-pants. keywords: cryptocurrency; permutation entropy; permutation statisticalcomplexity; complexity-entropy causality plane; informational efficiency According to the traditional definition, a currency has three main properties:(i) it serves as a medium of exchange, (ii) it is used as a unit of account and(iii) it allows to store value. Traditional currencies are issued by central banks,on behalf of nation states, and their values are related to the confidence in thecentral bank policies and in the economy in which such currencies are based on.A few years ago, a new type of tradable asset, called broadly cryptocurrencies,emerged. The first and most world known is Bitcoin (BTC). It was createdfollowing the publication of a manuscript written by an unknown author underthe pseudonym “Nakamoto”[29]. Contrary to standard fiat money, its creationis not linked nor endorsed by any central bank and/or government. It is a fullyprivate creation of virtual money, whose value is not intrinsically based on anyprecious metal or any other underlying asset. Consequently, its intrinsic valueis zero[11]. Cryptocurrencies are based on a new technology called blockchain.Its main innovation is that transactions, instead of being validated by a centralauthority or clearing house, is done by several markets participants, who com-pete to validate them by solving complex cryptologic algorithms. In turn, thewinner in this validation quest is rewarded with some amount of the cryptocur-rency he/she is validating. This decentralized and encrypted transaction ledgermakes, according to those who are in favor of this technology, a more reliablevalidation than the centralized alternative.The ecosystem of cryptocurrencies has been growing at an increasing pace,and now there are around 1000 active and tradable cryptocurrencies, usingblockchain or similar protocols. Daily transactions are worth several millionsof dollars, and in recent times a growing literature is devoted to the study ofdifferent aspects of this new asset.The aim of this paper is to study the informational efficiency of the twelvemost important cryptocurrencies, using high-frequency data. All cryptocurren-cies rely in a similar blockchain technology, making them similar from a technicalpoint of view. However, none of them have any real or tangible asset in orderto price them. Consequently, the comparative analysis aims to test if these in-struments have different underlying (unobservable) dynamical structure. Thisarticle contributes to the literature in three important aspects. First, we expandthe empirical studies analyzing this new asset type. Second, we compare thedynamic behavior of the twelve major cryptocurrencies. Third, we describe thetemporal evolution of informational efficiency using high-frequency data. Therest of the paper is organized as follows: Section 2 describes the recent emergingliterature on Bitcoin and other cryptocurrencies, Sections 3 and 4 introduce themethodology used in the paper, Section 5 presents the set of data and discusses2he results of our empirical analysis, and, finally, Section 6 draws the mainconclusions.
The study of cryptocurrencies has different branches that spans law, computerscience and economics. The great innovation in Nakamoto’s paper[29] was prob-ably not the creation of BTC, but the development of an open-source, decen-tralized online payment system. In other words, financial transactions could bedone, at a very reduced fee[21], bypassing the established international bankingsystem. Even more, due to encryption, parties are note required to disclosetheir true identity. This feature could arise concerns within the law communityabout the use of BTC for illegal purposes. Computer science literature focus itsinterest on the technical design of the blockchain technology, security of cryp-tographic protocols, vulnerabilities, energy consumption, etc. Finally, financialand monetary economics focus mainly in either the economic determinants ofBTC price and in its informational efficiency. We will focus in this latter aspect.According to the classical definition by Fama[15], a market is informationallyefficient if prices convey all relevant information. In other words, and limitingthe information set to the series of prices of a given asset, we say that the marketfor that asset is efficient if the current price incorporates the information ofpast prices. As a corollary of such definition, the use of past prices for futureprices forecasting is futile. Samuelson[35] established that the time series ofprices of any given speculative asset should behave as a random walk (RW).The empirical literature in financial economics found several deviations fromthe RW hypothesis. In fact, Bariviera and coauthors have shown the presenceof time varying long-range dependence in the Thai Stock Market[5], studied theeffect of the 2008 financial crisis on the informational efficiency of Europeansovereign bonds[10], and they have also found an asymmetric response in thestochastic characteristics of European corporate and sovereign bonds[9]. Otherauthors have studied the relationship among predictability political crises andmarket crashes[20].Regarding the cryptocurrency markets, most of the literature concentratesits efforts in the analysis of BTC. However, the cryptocurrency ecosystem ispopulated by hundreds of competitors to BTC. Conmarketcap[12] gathers in-formation of around 1000 different active currencies. In this sense, our papergives a broader picture of this virtual market by analyzing other eleven cryp-tocurrencies in addition to the classical BTC.Cheah and Fry[11] found speculative bubbles in BTC market. Urquhart[39]reported informational inefficiency in the BTC market from 2013 until 2016.Similarly, Nadarajah and Chu[28] found that the time series behavior of BTCis not consistent with the Efficient Market Hypothesis (EMH), and Bariviera[6]has shown a reduced long-term memory effect in the period 2013-2016. Finally,Bariviera et al. [7] found that the long-term memory profile of BTC time seriesis similar at different time scales. It is also reported prices clustering at round3umbers (with 00 decimals)[40].
The departing point for many empirical studies in economics is a time series.Financial markets, and more precisely the growing cryptocurrency markets, pro-vide abundant material to process. Taking into account that each transaction isrecorded electronically, and that there are thousands of transactions per hour,the researcher can select data with different granularity. The abundance ofdata allows the introduction of more advanced techniques, mostly derived fromeconophysics, in order to shed light on economic phenomena.Information-theory-derived quantifiers could be very helpful to uncover in-formation conveyed by financial time series. The use of entropy quantifiers inthe financial literature can be traced back to the 1960s, with papers by Theiland Leenders[38], Fama[14], and Dryden[13]. These papers may be consideredisolated examples on the use of this technique, which was only recovered inrecent times, by the econophysics literature. In this line, Martina et al. [26]and Ortiz et al. [30] applied entropy and multiscale entropy analysis to assesscrude oil price efficiency. Alvarez-Ram´ırez et al. [2] also used entropy methodsto quantify the dynamics of the informational efficiency of the US stock marketover the last 70 years.Shannon entropy is a very natural and common way to measure the degreeof disorder in a system. According to Shannon and Weaver[36], given a discreteprobability distribution P = { p i ∈ R ; p i ≥ i = 1 , . . . , M } , with P Mi =1 p i = 1,Shannon entropy is defined as: S [ P ] = − M X i =1 p i ln p i . (1)This quantifier equals zero if the patterns are fully deterministic and reaches itsmaximum value for a uniform distribution.However, analyzing time series by means of Shannon entropy alone couldfall short. Feldman and Crutchfield[16] and Feldman et al. [17] advocate that anentropy measure does not quantify the degree of structure or patterns presentin a process, and that a measure of statistical complexity must be introducedinto the analysis in order to characterize the system’s organizational properties.Mart´ın et al. [24] and Lamberti et al. [22] have introduced a statistical complex-ity measure, based on the functional form developed by L´opez-Ruiz et al. [23],defined in the following way: C JS [ P, P e ] = H S [ P ] Q J [ P, P e ] (2)where H S [ P ] = S [ P ] /S max is the normalized Shannon entropy, P is the dis-crete probability distribution associated with the time series under analysis, P e is the uniform distribution and Q J [ P, P e ] is the so-called disequilibrium: Q J [ P, P e ] = Q { S [( P + P e ) / − S [ P ] / − S [ P e ] / } with Q a normalization4onstant. This disequilibrium is defined in terms of the Jensen-Shannon diver-gence, which quantifies the difference between two probability spaces. Mart´ın etal. [25] demonstrated the existence of upper and lower bounds for generalizedstatistical complexity measures such as C JS . Additionally, as highlighted in So-riano et al. [37], the statistical complexity is not a trivial function of the entropybecause it is based on two probability distributions.The planar representation of these two quantifiers, called the complexity-entropy plane, has been introduced in the econophysics literature for charac-terizing the informational efficiency of several markets. For example, to rankefficiency in stock markets[43, 45]; to rank efficiency in commodity markets[44];to link informational efficiency with sovereign bond ratings[42]; to assess theimpact of the establishment of a common currency and a deep and wide fi-nancial crisis in European sovereign bonds time series[10]; and to detect Libormanipulation[8, 4]. Many economic phenomena produce observable magnitudes, which are regis-tered at evenly distributed times. These observations, i.e. time series, are theraw materials used by quantitative analysts to model and scrutinize complexphenomena. This research area is broadly known as time series analysis. Oneof its goals is to describe the nature of the generating process. We can safelyassume that a straight departing point for this task is to find the appropriateprobability density function (PDF) associated with the time series. There areseveral competing methodologies for PDF estimation. Beyond traditional his-togram technique, and without attempting to be exhaustive, we can cite: binarysymbolic dynamics[27], Fourier analysis[31], wavelet transform[32], and ordinalpatterns[3]. The suitability of each method depends on the very own character-istics of the data. The methods for symbolic analysis of time series discretizeraw series and transform it into a series of symbols. These kind of methods arevery powerful because they are rarely affected by the presence of observationalnoise[18]. This property is specially important in the analysis of economic timeseries, where noise is a traditional feature. Among the symbolic-based tech-niques for PDF estimation, the Bandt and Pompe (BP) methodology[3] hasthe advantage of considering time causality in its estimation. This symbolicmethodology is robust to the presence of (observational) noise and requires no a priori model assumption, except weak stationarity. The starting point of thismethod is to consider the ordinal structure of D − dimensional partitions of thetime series. “Partitions” are devised by comparing the order of neighboringrelative values rather than by apportioning amplitudes according to differentlevels.Let consider a time series S ( t ) = { x t ; t = 1 , . . . , N } , an embedding dimension(pattern length) D > D ∈ N ), and an embedding delay (sampling frequency) τ ( τ ∈ N ), the BP-pattern of order D generated by s (cid:0) x s − ( D − τ , x s − ( D − τ , . . . , x s − τ , x s (cid:1) , (3)5s the one to be considered. To each time s , BP method assigns a D -dimensionalvector that results from the evaluation of the time series at times s − ( D − τ, s − ( D − τ, . . . , s − τ , and s . Clearly, the higher value of D , the more “timecausality” is incorporated into the ensuing vectors. By the ordinal pattern oforder D related to the time s , BP mean the permutation π = ( r , r , . . . , r D − )of (0 , , . . . , D −
1) defined by x s − r D − τ ≤ x s − r D − τ ≤ · · · ≤ x s − r τ ≤ x s − r τ . (4)In this way the vector defined by Eq. (3) is converted into a definite symbol π .So as to get a unique result, BP consider that r i < r i − if x s − r i τ = x s − r i − τ .This is justified if the values of x t have a continuous distribution so that equalvalues are very unusual.For all the D ! possible orderings (permutations) π i when embedding di-mension is D , their associated relative frequencies can be naturally computedaccording to the number of times this particular order sequence is found in thetime series, divided by the total number of sequences, p ( π i ) = ♯ { s | s ≤ N − ( D − τ ; ( s ) has type π i } N − ( D − τ . (5)In the last expression, the symbol ♯ stands for “number”. Thus, an ordinalpattern probability distribution P = { p ( π i ) , i = 1 , . . . , D ! } is obtained from thetime series.The ordinal pattern PDF is invariant with respect to nonlinear monotonoustransformations. Accordingly, nonlinear drifts or scalings artificially introducedby a measurement device will not modify the quantifiers’ estimation, a niceproperty if one deals with experimental data (see, e.g. , Saco et al. [34]). Theseadvantages make the BP approach more convenient than conventional methodsbased on range partitioning. Additional advantages of the method reside in itssimplicity (we need few parameters: the pattern length/embedding dimension D and the embedding delay τ ) and the extremely fast nature of the pertinentcalculation-process[19]. The BP methodology can be applied not only to timeseries representative of low dimensional dynamical systems but also to any typeof time series (regular, chaotic, noisy, or reality based)[3]. In fact, the existenceof an attractor in the D -dimensional phase space is not assumed. The onlycondition for the applicability of the BP method is a very weak stationaryassumption: for k ≤ D , the probability for x t < x t + k should not depend on t .For review of BP’s methodology and its multidisciplinary applications, pleasesee Zanin et al. [41] and references therein.In this work, the normalized Shannon entropy H S and the statistical com-plexity measures C JS (Eq. (2)), are estimated using the ordinal pattern probabil-ity distribution P = { p ( π i ) , i = 1 , . . . , D ! } . Defined in this way, these quantifiersare usually known as permutation entropy and permutation statistical complex-ity. They characterize the diversity and correlational structure, respectively, ofthe orderings present in the complex time series. The complexity-entropy causal-ity plane (CECP) is defined as the two-dimensional (2D) diagram obtained by6able 1: DataCryptocurrency Acronym Reuters Instrument Code (RIC)Bitcoin Cash BCH .MVBCHBitcoin BTC .MVBTCDash DASH .MVDASHEthereum Classic ETC .MVETCEthereum ETH .MVETHIOTA IOT .MVIOTLiteCoin LTC .MVLTCNEO NEO .MVNEONEM XEM .MVXEMMonero XMR .MVXMRRipple XRP .MVXRPZcash ZEC .MVZECplotting permutation statistical complexity (vertical axis) versus permutationentropy (horizontal axis) for a given system[33]. The term causality remem-bers the fact that temporal correlations between successive samples are takeninto account through the BP recipe used to estimate both information-theoryquantifiers. We use high-frequency price indices developed by MV Index Solutions (MVIS ® ).Data were obtained from Thomson Reuters Eikon terminal from one of the au-thors’ university. Data consist of 16,031 observations of price indices, for eachof the twelve cryptocurrencies detailed in Table 1. Data are equally spaced intime, being 5 minutes the time frame between each observation. The periodunder study spans from December 3, 2017 until February 14, 2018. This periodis very interesting since cryptocurrencies exhibited an unprecedented rise andsubsequent crash in their values. Consequently, it could be suitable to studythe co-movement of different currencies for testing if the underlying dynamicsof the different time series were the same.In spite of the fact that Bitcoin is, undoubtedly, the most famous cryp-tocurrency, there are several hundreds of tradable instruments using a similarblockchain technology. As can be seen in Table 2, the market is very concen-trated. Our twelve selected cryptocurrencies account for 88% of total marketcapitalization and 91% of 24 hours traded volume, among the 897 ones de-tailed in the website https://coinmarketcap.com/coins/views/all/ . Con-sequently, our study covers most of the cryptocurrency market.One feature of this market is that its dynamics is very similar for all theassets under study. Figure 1 shows how the permutation entropy varies across Accessed on 14/02/2018 https://coinmarketcap.com/coins/views/all/ .Market Capitalization Daily traded volumeAcronym USD % of USD % ofmillions cryptos millions cryptosBCH 22,931 5.5% 678 3.3%BTC 165,007 39.4% 9,128 44.0%DASH 5,355 1.3% 151 0.7%ETC 3,384 0.8% 765 3.7%ETH 90,727 21.7% 3,143 15.2%IOT 5,698 1.4% 68 0.3%LTC 12,580 3.0% 2,731 13.2%NEO 7,913 1.9% 265 1.3%XEM 5,049 1.2% 79 0.4%XMR 4,356 1.0% 123 0.6%XRP 44,039 10.5% 1,702 8.2%ZEC 1,566 0.4% 104 0.5%Total 368,606 88.1% 18,937 91.3%time. Sliding windows of size N = 360 data points and step δ = 60 havebeen implemented for the dynamical analysis. Behaviors are very similar forall cryptocurrencies. This could reflect coherent dynamics of the different timeseries.We can observe in Figure 2 that time series mostly exhibit persistent behav-ior, reflected in a location in the CECP compatible with fractional Brownianmotions (fBm) with Hurst exponents between 0.5 and 0.7. Previous studies onBTC time series reported an enhanced informational efficiency in the period2014-2016. Nevertheless, it seems that strong bull and bear markets could leadto more coordinated movements that reduce the informational efficiency.In order to verify if all cryptocurrencies follow the same stochastic process,we compute the sample mean and standard deviation of the information-theoryquantifiers for each currency. We depict results in Figure 3. We observe thatBTC occupies a central position among the other currencies. Additionally, thereare some other currencies more and less efficient than BTC.Taking into account that in our framework, informational efficiency is maxi-mal as H S [ P ] approaches 1 and C JS [ P ] approaches 0, we compute the Euclideandistance of the mean permutation entropy and permutation statistical complex-ity of each currency to ( H , C ) = (1 , D = 4, τ = 1, N = 360 and δ = 60. Hurst=0.5Hurst=0.6Hurst=0.7Hurst=0.8 Hurst=0.4
Hurst=0.3
Figure 2: Location of the cryptocurrencies in the CECP computed using slidingwindows with the following parameters: D = 4, τ = 1, N = 360 and δ =60. Black and red crosses are mean and standard deviation of 500 fractionalBrownian motion (fBm) simulations of 360 data points for the Hurst exponentsindicated in the figure. Dashed lines represent the upper and lower bounds ofthe quantifiers as computed by Mart´ın et al. [25].9igure 3: Mean and standard deviation of each cryptocurrency in the CECPduring the observation period. Quantifiers were calculated by implementingsliding windows with the following parameters: D = 4, τ = 1, N = 360 and δ = 60. Table 3: Informational efficiency ranking.Ranking Cryptocurrency efficiency measureposition d [( H , C ) − (1 , vis-`a-vis BTC. Results are displayedin Figures 4 and 5. We observe that there are seven cryptocurrencies (displayedin light gray in the figures), whose mean entropic and complexity behavior isindistinguishable form BTC (displayed in blue in the figures). However, wereject the null hypothesis of equal mean permutation entropy of BTC, withrespect to ETC, ETH, IOT and XEM (displayed in red in Figure 4). We alsoreject the null hypothesis of equal permutation statistical complexity of BTC,with respect to DASH, XEM, ETC and ETH (displayed in red in Figure 5). If weanalyze these results together with the graphical representation of mean valuesof Figure 3, we conclude that ETC and ETH are less efficient (more persistent)while DASH and XEM are more efficient than BTC. Actually, DASH and XEMdynamics are closer to a random walk behavior. One of the reasons for suchbehavior of ETC and ETH, could be found in the fact that this cryptocurrencieswere not created with the aim of substituting paypal-like systems. Ethereum’sgoal is using a blockchain for “smart contracts”, i.e. to replace internet thirdparties in order to validate trusted operations [Eth].Additionally, XEM and DASH appear as the most efficient cryptocurrencies.In this case, the reason could be found in the validation design. Both currenciesintroduced different ways of validating blocks. XEM introduced a proof-of-importance (POI) algorithm, and an Eigentrust++ reputation system in orderto check operations. Unlike BTC, DASH is comprised of three types of ’levels’,with specific roles and responsibilities on the network. In addition, from thebeginning the evolution, changes or upgrades in the currency can be proposedby anyone, establishing a decentralized governance by blockchain. This situationcould generate fairer transactions, which leads to a more efficient market.
We studied high-frequency data of the cryptocurrency market during a veryspecial period of boom and bust. Our paper reports detailed behaviors of thetwelve most important cryptocurrencies, which cover 88% of market capital-ization and over 91% of daily turnover. We detect that the majority of thecurrencies exhibit a similar behavior, compatible with some kind of persistentstochastic dynamics with Hurst exponents between 0.5 and 0.7. However, we11able 4: Anova analysis to test the equality of means among all cryptocurren-cies.ANOVA on permutation entropySource SS df MS F Prob > FCurrencies 0.2421 11 0.0220 39.8496 3.19E-81Error 1.7231 3120 0.0006Total 1.9651 3131ANOVA on statistical complexitySource SS df MS F Prob > FCurrencies 0.2039 11 0.0185 42.3817 2.12E-86Error 1.3647 3120 0.0004Total 1.5686 3131Figure 4: Anova analysis. Difference of mean permutation entropy for eachcryptocurrency with respect to BTC. Red lines indicate currencies whose meanpermutation entropy is different from BTC (at 1% significance for ETC, ETHand XEM, and 5% level for IOT). 12igure 5: Anova analysis. Difference of mean statistical complexity for eachcryptocurrency with respect to BTC. Red lines indicate currencies whose meanstatistical complexity is different from BTC (at 1% significance level).13an identify four cryptocurrencies whose behaviors are different from the rest.ETC and ETH exhibit more persistent behavior than the others, reflected insmaller mean permutation entropies and larger mean statistical complexities.On the contrary, DASH and XEM average behaviors are closer to a randomwalk. Our results uncover that, inside the cryptocurrency ecosystem, distinctbehaviors emerge. Even though the majority of the market follow the behav-ior of the leader (BTC), some alternative cryptocurrencies follow differentiateddynamics, which could indicate that these assets are not as homogeneous asexpected. The reason for such behavior could be found in the special charac-teristics of these currencies. Unlike BTC, the aim of ETC and ETH is to be avehicle for “smart contracts” rather than a virtual currency system. RegardingDASH and XEM, they introduced some innovations in the blockchain ecosystemand, consequently, investors could see them as more reliable assets.
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