Cryptocurrencies: Dust in the wind? Min Luo∗, Vasileios E. Kontosakos†, Athanasios A. Pantelous‡, Jian Zhou§ October 25, 2018
Abstract Analogous to the way wind blows single grains of sand and the subsequent settling back atop sand dunes, we find statistical evidence to claim that the prices of cryptocurrencies exhibit similar unpredicted patterns, characterized by positive or negative jumps. Motivated by extant evidence of asset returns’ non-normality, we capture distributional properties of the log-returns of the Bitcoin and the following three cryptocurrencies in terms of market capitalization (Ethereum, Ripple and Bitcoin cash). The total error induced by the fitted distribution is remarkably decreased when the generalized hyperbolic distribution is used, a finding further validated by a series of goodness-of-fit type statistical tests. A complementary analysis for the foreign exchange market is conducted with inherent similarities to that of cryptocurrencies. We reveal that the generalized hyperbolic distribution can also be used to model very widely traded currency pairs significantly more accurately than the log-normal. Keywords: Generalized Hyperbolic Distributions; Distribution Fitting; Cryptocurrency; Bitcoin; Foreign Exchange Market.
1
Introduction
The American progressive rock band Kansas in 1977 released their album “Point of Know Return”, their last-minute addition to the track lineup “Dust in the Wind” proved to be its greatest hit. Since 2017, the mania for trading cryptocurrencies winds up continuing debates in both financial circles and academia, as to whether the cryptocurrencies are arguably . . . “dust in the wind ”. Intuitively, our understanding so far about the cryptocurrencies generates similarities with the relatively quickly changing wind-driven dust displacement phenomenon. Without doubt, cryptocurrencies have boomed and busted intensively, especially in the last two years. Like when the wind blows and calms, it is very unlikely cryptocurrencies will not bubble and bust again, and most likely again and again (see Cheah and Fry, 2015; Corbet et al., 2018b; Fry, 2018). Although cryptocurrencies do not generate returns, as they are simply a fiat, distributed currency maintained and audited by networked hash-rate majority internet, they do ∗
School of Management, Shanghai University, Shanghai 200444, China. Department of Econometrics and Business Statistics, Monash Business School, Monash University, Clayton, Victoria 3800, Australia. ‡ Corresponding Author: Department of Econometrics and Business Statistics, Monash Business School, Monash University, Clayton, Victoria 3800, Australia: (
[email protected]). § School of Management, Shanghai University, Shanghai 200444, China. †
1
not have fundamentals (Fry and Cheah, 2016). In other words, their distinctive boundaries are infinitesimally small which affects scalable transaction - jumps (see Scaillet et al., 2018, among others). More precisely, their upward lift is triggered by the investors hope that the value of the cryptocurrency will increase in the future driven by collective market evaluation (Ciaian et al., 2016). Nevertheless, the sentiment towards cryptocurrencies has improved in recent days where prices have stabilised, which may lead to increased prices. In this paper, from our empirical analysis, we observe that the mass-size distribution of aeolian dust particles can provide an excellent candidate to capture intuitively as well as mathematically the relatively strong momentum returns in the turbulent cryptocurrency market exhibit. Taking lessons to learn from Nature, the founder of the physics of blown sands, Brigadier Ralph Alger Bagnold FRS, observes that “from the results of many analyses of sand samples it seems clear that it is difficult for someone to escape from the conclusion that sand grading is not a random phenomenon” (Bagnold, 1954). Similarly, the stochastic nature is an important ingredient for studying the behaviour of cryptocurrencies. In physics, the term saltation is defined as the motion of a solid which, while generally moving above a boundary, gets most of its upward lift by occasional contact with the boundary which results a succession of jumps. In this physical phenomenon, typically the particle of wind-driven dust will reach a height, and during its jump will obtain a horizontal velocity, which will be a considerable fraction of the wind speed at this height (Zheng, 2009). Eventually, when the particle hits the dust surface again, it will do so very violently, and as a consequence the particle will often either rebound or hurl one or more other particles up into the air flow. Therefore, once the saltation is initiated, a chain reaction is released, implying that the dust transport can be maintained by a weaker wind than the one which initiated it (see Barndorff-Nielsen, 1977; Barndorff-Nielsen et al., 1985, among others). On one hand, cryptocurrencies have neither an intrinsic value nor are capable of being valued according to fundamental analysis (Fry and Cheah, 2016), thus institutional investors are largely remaining away. Unavoidably, cryptocurrencies must rely on millions of individual investors worldwide. On the other hand, the downside is that individual investors are less likely buy-and-hold investors or have a very long investment horizon, something that would bring stability to the market. As the sentiment towards cryptocurrencies has improved, so that the popularity of cryptocurrencies among individual investors. However, whenever a large price jump can provide enough market momentum to persuade the smallest investors to trade, implying that cryptocurrencies can remain extremely volatile even when sustaining trading may result in substantially weaker reactions than the initial movement. The connection between the hyperbolic laws in aeolian process and the jumps observed in the price evolution of cryptocurrencies begs the liken to the Brownian motion (pedesis1 ) - asset price evolution analogue in modern finance. Once it was the botanist Brown (1828) who observed at a microscopic level unexpected movements of pollen grains immersed in water, but which remained unexplained until Thiele (1880), who described the mathematics behind the observed process. Later, Bachelier (1900) in his celebrated PhD thesis, independently further formalized the mathematics of Brownian motion, deriving a model applicable to stock returns.2 1
From ancient Greek: “leaping”. Bachelier modelled the fluctuation of stock prices by arguing that over short time–increments they should not depend of the absolute price level and they should also be independent of past observations (later formalized by Markov as the well known memoryless property of stochastic processes) while using Central Limit Theorem he also shows that Brownian increments are independent and identically 2
2
Despite cryptocurrencies’ increasing popularity over the past years as an alternative investment opportunity and the subsequent proliferation of the number of cryptocurrency-based transactions, very little work has been done in the direction of systematically examining its distributional properties and trading environment (e.g. Guo and Li, 2017; Fry, 2018; Gkillas and Katsiampa, 2018; Phillip et al., 2018). Recently, Elendner et al. (2017) study the general trading environment for cryptocurrencies by examining their cross section correlation along with their statistical properties, however, without attempting to model their underlying distribution. In our paper, we rigorously investigate and connect the distributional properties of the four largest cryptocurrencies – in terms of market capitalization – by fitting the generalized hyperbolic distribution to their daily log–returns and comparing the fit against that suggested by the normal distribution. Barndorff-Nielsen (1977) observed that the “energy cascade” might be considered in parallel with the “information cascade” effect, whereby as Bingham and Kiesel (2001) reported, “price-sensitive information originates in a global newsflash trickles down through national and local level to smaller and smaller units of the economic and social environment.” In finance, Eberlein and Keller (1995), Rydberg (1999) and Eberlein (2001) introduced the generalized hyperbolic (GH) distribution, which offers a significantly better fit compared to the normal distribution for modeling daily log–returns of widely traded cryptocurrencies. This affords markedly lower errors in comparison to their empirical density, as measured by Kolmogorov–Smirnov (KS) distance, Cramer–von Mises (CVM) statistic and Kullback–Leibler (KL) divergence. The benefits of GH over the normal distribution stems from the following two properties: its ability to approximate heavier tails, mainly in the left tail, as well as the larger density mass concentrated around zero. Furthermore, similar statistical analysis for the currency market, reveals that the GH distribution can better capture the distributional properties of widely traded foreign exchange pairs. The latter, combined with the gradual change of the trading environment (increased security, lower volatility, establishment of a futures market with underlying asset the Bitcoin, BTC3 ) provides us with evidence that, at the moment, cryptocurrencies cannot be directly considered equivalent market players to conventional currency pairs, largely due to their volatility (see Yermack, 2015; Li and Wang, 2017; Corbet et al., 2018a). However, continuing their current trend and especially their decreasing volatility it is more likely than not that they will indeed become equivalent to currency market players. The remainder of this paper is structured as follows: Section 2 provides the description of the data and a brief discussion about the main features of cryptocurrencies and blockchain. Section 3 discusses why the log–normal distribution is not a good fit for our data and introduces the class of hyperbolic distributions. Sections 4 and 5 present and analyse the results of the distribution fitting along with error interpretation and the statistical tests conducted. Finally, Sections 6 and 7 complete this study with the discussion on whether cryptocurrencies are directly comparable to currency, followed by concluding remarks.
2
Data
Before we describe the data used in this paper, let us first outline the main features of BTC and the cryptocurrencies in general as well as the blockchain technology. distributed (i.i.d). 3 Financial times: https://www.ft.com/content/9404475c-dd9d-11e7-a8a4-0a1e63a52f9c
3
This will provide us with the necessary background to grasp specific features inherent to cryptocurrencies, which also affect the statistical properties of the traded coins, thus largely contributing to the establishment of the current trading environment.
2.1
Cryptocurrencies and Blockchain
Cryptocurrencies, such as BTC, Ethereum (ETH), Ripple (XRP) among hundreds of others, provide a form of digital currency created around the idea of noncentralized banker (conventionally governments) and distribution protocol (as is the case with real money issued by central banks) (B¨ohme et al., 2015). Particularly, the BTC network is a form of peer–to–peer (P2P) communication system (Nakamoto, 2008), where each user can transact with all the other market participants directly without the need to be directed via a centralized intermediary (Li and Wang, 2017). Fundamentally, each user can be generator, sender or receiver of the cryptocurrency and subsequently the corresponding monetary amount in any of the available currencies. BTC can be infinitely divided into smaller units with limitations imposed only by its network architecture and the computing structure that it is built upon. The network generates new BTC units using a prespecified algorithm which increases the circulation at a decreasing rate until it reaches a predetermined limit (e.g. Zohar, 2015). Other cryptocurrencies have no or much wider limits as to the maximum number of coin units that can be generated. This results in them having a circulation many times larger than that of BTC, but of course their price is much lower as well. Moreover, a cryptocurrency user can generate a unique address based on an alpharithmetic string through which they can send and receive blocks of coins (usually referred to as tokens), while to securely store their coins they use another digital entity called a wallet, either an on-line or off-line one (e.g. Dryall, 2018). Each transaction conducted using cryptocurrency tokens is recorded on a decentralised ledger named blockchain – (Zyskind and Nathan, 2015, for an application of blockchain) – which is managed and maintained by a network of dedicated machines called miners (Tschorsch and Scheuermann, 2016, for a detailed study of decentralized digical currencies). Initially, this was with computer processing units from normal desktop computers, then with graphics processing units. Eventually, hardware known as an Application-Specific Integrated Circuit was designed specifically for mining BTC) Miners are typically very powerful specific hardware which guarantee the operability of the crypto network and are rewarded by the network with cryptocurrency tokens which increase the total coin circulation and are also partially indemnified by the transaction fees (e.g. White, 2015). The latter makes BTC the most desired traded cryptocurrency but does not necessarily apply to others, a primary distinguishing feature of offered coins. ETH for instance, is the main provider of blockchain applications with the vast majority of blockchain–based platforms built on ETH’s network4 while XRP’s goal is the construction of a consolidated decentralized payment system to facilitate payments and money transfers between users. Finally, Bitcoin cash (BCH) – a hard fork (a term borrowed from software engineering) of BTC5 – was created to alleviate and speed up the transac4
Commonwealth Bank of Australia (CBA), which is one of the largest commercial banks in Australia, has tested a new system based on the Ethereum blockchain to deliver world’s first blockchain bond. Read more (Media release on 10th August 2018): https://www.commbank.com.au/guidance/newsroom/ cba-picked-by-world-bank-to-deliver-world-s-first-standalone-blo0-201808.html. 5 A hard fork is in essence the result of a split of the initial blockchain into two smaller branches.
4
tion rate of the BTC network by using blocks of larger size compared to the initially designed ones.
2.2
Data description
We consider daily log returns for the four cryptocurrencies with the largest market capitalization6 , namely the BTC, ETH, XRP and the BCH.7 Summary statistics for the four cryptocurrencies can be found in panel A of Table 1. The sample periods for each cryptocurrency differ as their trading started at different times and the available datasets are not sufficiently large to allow us to use the same period for all of them and still have enough data in order for the parameter estimation to be reliable. In addition, we study whether the GH distribution can provide us with a better fitting compared to the log-normal distribution – the standard for modeling asset returns – for more conventional asset classes. Particularly, we find that the foreign exchange market offers numerous cases where log-normality is violated, and the return distribution of the traded pairs exhibits certain characteristics: mainly greater peakedness around zero and a larger amount of mass concentrated in the tails, particularly in the left tail (e.g. Schmitt et al., 1999). Both effects can be captured more adequately by the GH distribution and this leads us to also include four widely traded pairs from the foreign exchange market, the Euro/Japanese Yen (EUR/JPY), United States Dollar/Euro (USD/EUR), United States Dollar /Japanese Yen (USD/JPY) and United States Dollar/Singaporean Dollar (USD/SGD). The sample period for all the four pairs is the same, as we use daily data from January 2013 to August 2018. Detailed summary statistics can be found in panel B of Table 1.
3 Why is the log–normal distribution not a good fit? The starting point for our analysis is the detection of whether our data are normally distributed – and thus they can be represented by the log-normal distribution – or they can be more sufficiently modelled with another family of distributions. Extensive literature in the area of asset return modelling has revealed that asset returns frequently deviate from Gaussianity, exhibiting features impossible to be captured by the normal distribution. These include among others, heteroskedasticity, volatility clustering, serial correlation and more mass in the left tail, also accompanied by negative skewness and leptokurtosis (Lamoureux and Lastrapes, 1990; Nelson, 1991; Andersen et al., 2001; Cont, 2001; Carr et al., 2002, among numerous others). Subsequently, the detection of the underlying distribution has important implications in portfolio structuring, risk management8 and pricing and hedging of derivatives (Rachev et al., 2005). This analysis carries over into the market of cryptocurrencies which appears as a strong candidate to exhibit distributional similarities (Scaillet et al., 2018) to those for more conventional assets as studied in the aforementioned literature. 6
According to the web platform ”CoinMarketCap”. The time series for the cryptocurrencies were obtained by the Bitcoin exchange ”Bitstamp” and the open source cryptoasset analytics platform ”Coin Metrics”. 8 Bitcoin in this context can be specifically used for risk management purposes, providing large diversification benefits, given its low correlation with traditional tradable assets. (Baur et al., 2018) 7
5
Panel A Name
Symbol
No. of Obs.
Mean
Std.Dev.
Skewness
Kurtosis
Bitcoin
BTC
2,487
0.0013
0.0214
-1.8985
66.8440
Ethereum
ETH
1,084
0.0025
0.0329
0.1089
6.9720
Ripple
XRP
1,827
-0.0010
0.0341
-2.0274
30.5381
Bitcoin Cash
BCH
378
0.0006
0.0427
0.6191
7.4576
Panel B Symbol
No. of Obs.
Mean
Std.Dev.
Skewness
Kurtosis
EUR/JPY
1,482
0.0000
0.0028
-0.4281
11.6127
USD/EUR
1,482
0.0000
0.0023
-0.0883
5.3588
USD/JPY
1,482
0.0000
0.0029
-0.0621
7.7572
USD/SGD
1,482
0.0000
0.0014
-0.2300
4.7854
Table 1: Summary statistics. Panel A presents summary statistics for the four cryptocurrencies. Panel B presents summary statistics for the four traded foreign exchange pairs. The inconsistency in the number of observations for the cryptocurrencies in Panel A stems from the different times the trading of the cryptocurrencies started. Focusing not only on BTC, but also on all four cryptocurrencies with the largest market capitalization, it is straight forward to show both visually and statistically that their log–returns are not normally distributed. An easy way to show nonnormality is using the quantile-quantile (QQ) plots (Fig. 1). A QQ plot in essence sorts the input data (in our case log–returns) in ascending order, and subsequently plots them against quantiles formed from the theoretical distribution (in our case the normal). When the two distributions under comparison are very similar to each other the points of the QQ plot form a straight line (dashed– dotted line in Fig. 1), while any deviation from it signals that the two compared distributions are inherently different. Evidently, since the theoretical distribution considered for this exercise is the normal, any divergence from the dashed–dotted line of Fig. 1 provides us with strong evidence that the log–returns of the examined time–series are not normally distributed. As a result, we reject Gaussianity for the log–returns of all the four cryptocurrencies while, after a careful observation of the plots, we should expect more mass concentrated in the tails, compared to that assumed by the normal distribution. Finally, we reach the same conclusion by running a Jarque and Bera (1987) test which results in rejecting the null hypothesis of normality in the examined data (see Table 2).
4
Distribution fitting
Having introduced the basic features for BTC and the class of cryptocurrencies, our main goal for the fitting exercise is to detect a probability distribution which can capture more sufficiently the distributional characteristics of BTC’s log–returns and the other three examined cryptocurrencies. A class of distributions which has been used – but not as extensively as one would expect - in finance, is that of the family of hyperbolic distributions. In the emerging literature, we meet attempts to make use of the convenient distributional properties of the said family of distributions mainly
6
log return quantiles
0.2 0 -0.2 -0.4 -4
0.5
-2
0
2
4
standard normal quantiles XRP
log return quantiles
log return quantiles log return quantiles
BTC
0.4
0
-0.5 -4
-2
0
2
4
standard normal quantiles
ETH
0.2 0.1 0 -0.1 -0.2 -4
0.2
-2
0
2
4
standard normal quantiles BCH
0.1 0 -0.1 -0.2 -4
-2
0
2
4
standard normal quantiles
Figure 1: QQ plots of the log–returns of the four examined cryptocurrencies, i.e., Bitcoin (BTC), Ethereum (ETH), Ripple (XRP) and Bitcoin cash (BCH).
7
Panel A Cryptocurrencies BTC
ETH
XRP
BCH
λ (tail parameter)
-0.1144
0.5850
-0.2165
-0.0613
ζ (shape parameter)
0.1323
0.0726
0.1196
0.4024
ρ (skewness parameter)
0.0502
0.0934
-0.1603
0.1681
µ (location parameter)
0.0014
0.0025
-0.0010
0.0006
δ (scale parameter)
0.2233
0.0651
0.2321
0.4628
Skewness
0.3743
0.4894
-1.2849
0.8029
Kurtosis
15.4652
7.2593
17.3374
7.4844
7.0481e+03
2.3151e+03
4.2891e+03
707.4988
Log-likelihood
Panel B Foreign exchange pairs EUR/JPY
USD/EUR
USD/JPY
USD/SGD
λ (tail parameter)
-1.4926
-1.3591
-0.0180
0.0086
ζ (shape parameter)
0.4474
1.2239
0.5937
1.3573
ρ (skewness parameter)
-0.0451
0.0336
-0.0114
-0.0133
µ (location parameter)
0.0000
0.0000
0.0000
0.0000
δ (scale parameter)
1.1983
0.5953
0.5995
1.0093
Skewness
-0.1133
0.0714
-0.0463
-0.0365
Kurtosis
9.7052
5.4692
7.1023
5.0304
-1.9822e+03
-2.0339e+03
-3.5219e+03
-2.0647e+03
Log-likelihood
Table 2: Parameter estimation. Five-parameter estimation for the generalized hyperbolic (GH) distribution for the 4 cryptocoins with the largest market capitalization (Panel A) and the foreign exchange pairs (Panel B).
8
by Eberlein and Keller (1995), Prause (1997), K¨ uchler et al. (1999), Rydberg (1999), Bingham and Kiesel (2001), Eberlein (2001) and Bibby and Sørensen (2003). All the aforementioned studies examine in similar ways whether the GH distribution is a “good” fit for the returns of certain asset classes, where goodness-of-fit is measured by several statistical tests which we also adopt in the present study. Here, we carry over this analysis in the cryptocurrency market to figure out whether the GH distribution can provide us with a better fit, compared to the normal, where this is quantified in terms of the KS distance, the CVM statistic and the KL (relative entropy) divergence.
4.1
The class of generalized hyperbolic distributions
The classic empirical studies of Bagnold (1954) and Bagnold and Barndorff-Nielsen (1980) reveal that when log-density is plotted against log-size of a dust particle, a unimodal curve which approaches linear asymptotics at infinity is derived. Consequently, GH distributions are first introduced by Barndorff-Nielsen (1977) where they are used to model the aeolian process observed in dust dune movements which stems from certain patterns of dust particle trajectories caused by wind blowing (see Bagnold and Barndorff-Nielsen, 1980; Barndorff-Nielsen et al., 1985; Zheng, 2009). The family of GH distributions nests a number of relevant distributions, including the hyperbolic, the normal, the inverse Gaussian and the variance-gamma among others (see Eberlein and Hammerstein, 2004, for a detailed analysis). The probability distribution function of the GH is given as in Barndorff-Nielsen (1978) and Barndorff-Nielsen and Stelzer (2005) by fGH (x; λ, α, β, δ, µ) = a(λ, α, β, δ, µ)[δ 2 + (x − µ)2 ]λ/2−1/4 exp[β(x − µ)] p × Kλ−1/2 (α δ 2 + (x − µ)2 ), (1) where a(·) is the norming constant equal to a(λ, α, β, δ, µ) = √
(α2 − β 2 )λ/2 p . 2παλ−1/2 δ λ Kλ (δ α2 − β 2 )
(2)
In Eq. (1) α > 0 is the shape parameter, 0 ≤ |β| < α controls for skewness, µ ∈ < is the location parameter, usually set at zero, δ > 0 is a scaling parameter and λ ∈ < is the tail index parameter, which in essence defines the amount of mass concentrated in the tails of the distribution; in addition, K(·) is the modified Bessel function of the third kind (Watson, 1995). Introducing a second scaling parameter σ > 0 – so as to incorporate standard deviation in the functional for pdf of the GH distribution and use the standardized variable (x − µ)/σ– and using the second parameterization of Prause (1999) the pdf in Eq. (1) can be written as fGH (x; λ, ζ, ρ, δ, µ) = a(λ, ζ, ρ, δ, µ)
λ/2−1/4 ζρ x−µ +1 exp p δσ 1 − ρ2 s ! ζ x−µ 2 Kλ−1/2 p + 1 , (3) δσ 1 − ρ2
x−µ δσ
2
which is the expression also used in BenSa¨ıda and Slim (2016). In Eq. (3) p ζ = δ α2 − β 2 , β ρ= α
9
(4)
while the norming constant a(·) now takes the value √ a(λ, ζ, ρ, δ, µ) =
ζ(1 − ρ2 )λ/2−1/4 √ . 2πδσKλ (ζ)
(5)
We can easily show that the two expressions, Eqs. (1) and (3), are in essence equivalent and can be used interchangeably. For example, for the argument of the Bessel function in Eq. (1) we have that s α
x−µ σ
s s 2 x − µ x−µ 2 2 +δ =α δ + 1 = αδ +1 δσ δσ s ζ ζ x−µ 2 p = αp = √ +1 δσ α2 − β 2 ( α2 )−1 α2 − β 2 s s ζ ζ x−µ 2 x−µ 2 +1= p + 1, (6) =q 2 δσ δσ 1 − ρ2 1 − αβ
2
which is the argument of the Bessel function in the pdf of Eq. (3). Performing similar simple algebraic transformations one can validate that the two expressions are equivalent. To estimate the five parameters in Eq. (3), we use maximum likelihood where we maximize the L(θ; X) =
n Y
f (xi |θ),
(7)
i=1
which is equivalent to maximizing its logarithm l(θ; X) =
n X
log f (xi |θ),
(8)
i=1
where θ = (λ, ζ, ρ, µ, δ) is the set of parameters estimated by Eq. (8). For the estimation above we can use one of the proposed algorithms such as that in Jensen (1988), Blaesild and Sørensen (1992) or BenSa¨ıda and Slim (2016). We opt for the latter, written in MATLAB as it also derives the first four moments of the distribution (i.e., mean, standard deviation, skewness and kurtosis). Results of the estimation are reported in Table 2.
4.2
Higher moments
To capture higher moments (i.e., the third and fourth) of the GH distribution around the mean µ we use, as in Scott et al. (2011), the following moment generating function k X
Mk = b
l= k+1 2
c
2 λ+k Kλ+k (ζ) k! 2l−k δ β , ζ Kλ (ζ) (k − l)!(2l − k)!2k−l
(9)
where k is the order of the moment. The first four moments, derived using Eq. (9) for the moments about µ and the binomial theorem have as follows: ρδ M1 = µ + p Rλ,1 (ζ), 1 − ρ2
10
(10)
M2 = M3 =
δ 2 ρ2 δ2 2 (R (ζ) − R ) + Rλ,1 (ζ), λ,2 λ,1 1 − ρ2 ζ
(11)
3δ 3 ρ δ 3 ρ3 3 2 p (R (ζ)−3R (ζ)R (ζ)+2R (ζ))+ (Rλ,2 (ζ)−Rλ,1 (ζ)), λ,3 λ,2 λ,1 λ,1 (1 − ρ2 )3/2 ζ 1 − ρ2 (12)
M4 =
ρ2 3δ 4 ρ4 4 6δ 4 ρ4 3 2 (R (ζ) + R (ζ)R (ζ)) − R (ζ) λ,2 λ,1 1 − ρ2 λ,1 1 − ρ2 1 − ρ2 λ,1 δ 4 (3 − 6ρ2 + (ζ 2 + 3)ρ4 ) 2δ 4 ρ2 (λρ2 + 3) + R (ζ) + Rλ,3 (ζ) λ,2 ζ 2 (1 − ρ2 )2 ζ(1 − ρ2 ) 4δ 4 ρ2 3Rλ,2 (ζ)Rλ,1 (ζ) ρ2 Rλ,3 (ζ)Rλ,1 (ζ) , (13) − + 1 − ρ2 ζ 1 − ρ2
where Rλ,i (ζ) = Kλ+i (ζ)/Kλ (ζ), while for the derivation of the moments in Eqs. (10) to (13) the second parametrization in Eq. (4) is used.
5
Empirical Results
We first visualize the realised benefit by adopting the GH distribution in place of the normal by plotting the fitted pdf for each competing case. Fig. 2 shows the result from the distribution fitting of the normal (dashed line), GH (black solid line) and the kernel–smoothed GH (solid blue line) against the empirical density (circle markers) of the BTC log–returns. It is evident that using the GH instead of the log-normal distribution, we capture two effects observed in the distribution of BTC’s log returns: it is in fact more peaked as a large number of returns are concentrated around zero and it has a large number of observations concentrated in the tails of the distribution. An even more optimal fitting is obtained by kernel–smoothing the GH, using a weighting scheme which places more weight to adjacent observed points. The same benefit by using the GH is observed in all four cases of the examined cryptocurrencies (ETH, XRP, BCH) as Fig. 3 shows. GH offers a remarkably better fit compared to log-normal, further validated by plotting the error and the cumulative error in Figs. 4 and 5 . We observe that both the frequency error (Fig. 4) and the cumulative frequency error (Fig. 5) is lower when the GH (red cross markers and red line) is used to model log returns in place of the normal distribution (blue circles and blue line). Fig. 4 shows the error in the distribution fitting induced from the normal distribution (blue circle markers) and the GH (red cross markers) respectively. To correctly interpret this graph we should think of it as a graphical representation measuring the number of misclassifications of observations to the corresponding bins of empirical density’s histogram. In essence, we first compare the normal distribution against the empirical density (blue) and then the GH against the empirical density. It is evident that using the latter we minimize the number of misclassifications especially for BTC and XRP, while the approximation for ETH and BCH is also significantly improved. This finding is more emphasized and further visually corroborated by plotting the cumulative error - which will now follow the shape of a cumulative distribution function - in Fig. 5. One can see the benefit of using the GH distribution in place of the normal, as the total fitting error is remarkably lower when the former one is used (red line). Finally, we also plot the cdf for the
11
BTC
60
50
empirical pdf normal fitting hyperbolic fitting kernel-smoothed hyperbolic
frequency
40
30
20
10
0 -0.1
-0.08 -0.06 -0.04 -0.02
0
0.02
0.04
0.06
0.08
0.1
log return Figure 2: Bitcoin (BTC) distribution fit: This figure demonstrates the result from the distribution fitting of the normal (dashed line), geometric hyperbolic (GH) (black solid line) and the kernel–smoothed GH (dashed-dotted blue line) against the true empirical density (circle markers) of the BTC log–returns.
12
0.1 0.05 0 0 -0.15
5
10
15
-0.1
-0.05
log return
BCH 20
log return
0 -0.1
5
10
15
20
25
30
35
-0.05
0
XRP
0.05
0.1
frequency log return
0 -0.1
5
10
15
20
25
30
-0.05
0
ETH
0.05
0.1
frequency frequency
Figure 3: Distribution fit for Ethereum (ETH), Ripple (XRP) and Bitcoin cash (BCH), the three cryptocurrencies with the largest capitalization after bitcoin. The distribution fitting of the normal (dashed line), generalized hyperbolic (GH) (solid black line) and the kernel– smoothed GH (dashed-dotted blue line) against the empirical density (circle markers) of the ETH/XRP/BCH log–returns.
13
BTC
30 20 10 0
30 20 10 0
-0.2
0
0.2
-0.1
log return XRP
30 20 10 0
0
0.1
log return BCH
40
frequency error
40
frequency error
ETH
40
frequency error
frequency error
40
30 20 10 0
-0.4
-0.2
0
0.2
-0.1
log return
0
0.1
log return
Figure 4: Frequency error induced by the fitting of the generalized hyperbolic (red cross markers) on the cryptocurrencies Bitcoin (BTC), Ethereum (ETH), Ripple (XRP) and Bitcoin cash (BCH) log–returns and the normal distribution (blue circle markers) in comparison with the empirical density.
14
frequency total error
frequency total error
BTC
300 200 100 0
0
200 100 0 -0.4
-0.2
100 0 -0.1
log return XRP
300
200
0.2
frequency total error
frequency total error
-0.2
0
ETH
300
0.2
0.1
log return BCH
300 200 100 0 -0.1
log return
0
0
0.1
log return
Figure 5: Cryptocurrencies Bitcoin (BTC), Ethereum (ETH), Ripple (XRP) and Bitcoin cash (BCH): total error from distribution fitting. Total error from the distribution fitting between the normal and the empirical (solid blue line) and the generalised hyperbolic and the empirical (dashed–dotted red line) of the examined cryptocurrencies.
15
BTC
0.5
0
0.5
0 -0.2
0
0.2
-0.1
log return XRP
0.5
0
0
0.1
log return BCH
1
CDF
1
CDF
ETH
1
CDF
CDF
1
0.5
0 -0.4
-0.2
0
0.2
-0.1
log return
0
0.1
log return
Figure 6: Cumulative distribution function (CDF) for the normal (dashed-doted red line), generalised hyperbolic (dashed blue line) and the empirical density (solid black line) of the examined cryptocurrencies: Bitcoin (BTC), Ethereum (ETH), Ripple (XRP) and Bitcoin cash (BCH).
three distributions (empirical, normal, GH) to find that in all four cases the empirical (dashed line) is closely followed by GH (solid line), providing thus supportive evidence that the GH can capture the features observed in the examined datasets much more sufficiently than the normal. On top of the evidence obtained by graphically illustrating the goodness-of-fit for the examined distributions, we further measure how suitable the proposed GH distribution is for the considered datasets, by running a series of statistical tests, namely the KS, CVM, and the KL divergence between the proposed distribution and the empirical density. Using the KS test (Massey Jr, 1951), we quantify the distance of the examined distribution from the empirical density of the sample data, given by the following equation KS = sup |Fd (x) − F (x)| ,
(14)
x
where Fd (·) is the empirical density of the given dataset and F (·) is the cumulative distribution function under examination (i.e., the normal of the GH). From Eq. (14), it follows that the smaller the value of KS the closer the two distributions are and the stronger the evidence that the data follow the hypothesized distribution. Table 3 shows the exact values for the KS statistic for the normal and the GH distribution, according to which the null hypothesis that the data come from a normal
16
distribution is rejected at the one percent significance level. Obviously, this does not tell us anything about the actual distribution the sample data follow, but provides us with strong evidence that they are not normally distributed. Qualitatively, we obtain similar evidence by calculating the CVM statistic as follows: Z ∞ 2 [Fd (x) − F (x)]2 dF (x), (15) ω = −∞
where again Fd (·) is the empirical cumulative density and F (·) the hypothesized distribution we fit on the given data. The expression in Eq. (15) can be written as n X 1 2i − 1 CV M = nω = + − F (xi ) , 12n 2n 2
(16)
i=1
which is the value of the CVM statistic; the statistic rejects the null hypothesis whenever its value is larger than the critical (Csorgo and Faraway, 1996). Table 3 shows that the null hypothesis that data follow a specific distribution is rejected at the one percent significance level in the case of the normal distribution, while it is not rejected for the GH case. A final measure we adopt to quantify the difference between two distributions is the KL divergence (or relative entropy). Although KL divergence should not be directly interpreted as a measure of distance between two distributions, as it not a symmetric statistic (KL divergence of distribution Q from P is difference than that of P from Q), it still provides useful information on how close the two examined distributions are. KL divergence is defined as Z ∞ p(x) DKL (P ||Q) = p(x) log dx, (17) q(x) −∞ for distributions P and Q. The definition above can be written in its discrete form DKL (P ||Q) =
X
P (xi ) log
i
P (xi ) , Q(xi )
(18)
which is the expected value of the logarithmic difference between P and Q. The representation in Eq. (18) reveals the way KL divergence works: it quantifies the contribution of each individual sample i to the evidence we are trying to obtain. It follows that whenever P (xi ) is very close to Q(xi ) their ratio will be close to one and their logarithmic difference very close to zero, thus resulting in no further increasing the value of DKL . On the contrary, when the ratio P (xi )/Q(xi ) is much larger or smaller than one then DKL will increase with every additional sample giving us evidence to support that Q is not a good candidate distribution to represent P (P´erez-Cruz, 2008, for more details on the computation of KL divergence). Table 3 shows the KL divergence between the normal/GH and the empirical density; in all cases of the examined cryptocurrencies the KL statistic is significantly lower when the GH instead of the normal distribution is considered, while similar benefit is reported for all four examined currency pairs.
6
Is Bitcoin a crypto–currency?
In the present section, motivated by certain characteristics met in the foreign exchange market (i.e., currency is used as a medium of exchange, continuous 24–hour market operation, sensitivity to a wide range of factors that affect rates, strong
17
Panel A Cryptocurrencies BTC
ETH
XRP
BCH
4.23e+05***
714.7458***
5.89e+04***
337.0989***
Normal Distribution
0.3924
0.1621
0.3666
0.2039
GH Distribution
0.0524
0.0361
0.0476
0.1137
0.1538***a
0.1026***a
0.1609***a
0.1096***a
0.018
0.0275
0.1051***a
0.0289
2.1326***a
1.1878***a
2.3867***a
1.0575***a
0.3528
0.3409
0.3543
0.3433
Jarque–Bera test Kullback–Leibler divergence
Kologorov–Smirnov test Normal Distribution GH Distribution Cramer–von Mises test Normal Distribution GH Distribution
Panel B Foreign exchange pairs EUR/JPY
USD/EUR
USD/JPY
USD/SGD
4.6414e+03***
342.6935***
2.4902e+03***
210.6142***
Normal Distribution
0.1320
0.0989
0.1189
0.0911
GH Distribution
0.0572
0.0743
0.0440
0.0741
Normal Distribution
0.0654***a
0.0445***a
0.0629***a
0.0440***a
GH Distribution
0.0580***a
0.0382
0.0591***a
0.0402
2.0151***a
1.3328***a
1.7057***a
1.2239***a
0.3537
0.3524
0.3397
0.3527
Jarque–Bera test Kullback–Leibler divergence
Kologorov–Smirnov test
Cramer–von Mises test Normal Distribution GH Distribution
***: normality hypothesis rejected at one percent significance level. ***a: null hypothesis that data come from the given distribution rejected at one percent significance level.
Table 3: Statistical tests. Normality (Jarque–Bera test), Kullback-Leibler divergence (KL, also know as relative entropy), Kolmogorov-Smirnov (KS) test and Cramer–von Mises (CVM) statistic for the normal and the generalized hyperbolic distribution fit for the four examined cryptocurrencies (Panel A), Bitcoin (BTC), Ethereum (ETH), Ripple (XRP) and Bitcoin cash (BCH) and the four foreign exchange pairs (Panel B).
18
log return quantiles
log return quantiles
EUR/JPY
0.02 0 -0.02
0.04
-2
0
2
standard normal quantiles USD/JPY
0.02 0 -0.02 -4
-2
0
2
0.01 0 -0.01 -0.02 -4
4
log return quantiles
log return quantiles
-0.04 -4
USD/EUR
0.02
0.01
standard normal quantiles
0
2
4
standard normal quantiles USD/SGD
0.005 0 -0.005 -0.01 -4
4
-2
-2
0
2
4
standard normal quantiles
Figure 7: QQ plots for the foreign exchange pairs: EUR/JPY, USD/EUR, USD/JPY, and USD/SGD. The deviation of the empirical distribution from the dashed-dotted line shows evidence of non-normality.
trend following, geographic dispersion; see Chen, 2009) we investigate the possibility of being able to also capture distributional characteristics of the currency market, by modelling currency pairs’ returns via the the GH distribution. To this end, we attempt to detect distributional similarities between the Bitcoin and foreign exchange currencies, by fitting the GH distribution to four pairs, namely the Euro/Japanese Yen (EUR/JPY), United States Dollar/Euro (USD/EUR), United States Dollar /Japanese Yen (USD/JPY) and United States Dollar/Singapore Dollar (USD/SGD). In Figs. 7 to 11, one can see the results of performing similar analysis to that for the cryptocurrencies using the GH distribution. Not surprisingly, we find that the log–returns of currency pairs are not normally distributed, as both the QQ plots (Fig. 7) and the output of the Jarque–Bera normality provide supportive evidence in this direction. Moreover, in Fig. 8 we see the results of the fitting exercise to the selected currency pairs where one can directly observe that the GH distribution provides superior fit compared to the normal (blue solid line against dashed line in). The latter is further confirmed by observing the error plots in Figs. 9, 10 and 11. Again, the resulted error by the use of the GH distribution (red line and red cross markers) is measurably lower to that induced by the normal (blue line and
19
EUR/JPY
200 100 0 -0.01 -0.005
0
0.005
log return USD/JPY
100 0 -0.01 -0.005
0
0.005
200 100
400
200
USD/EUR
0 -0.01 -0.005
0.01
frequency
300
frequency
300
frequency
frequency
300
0.01
log return
0
0.005
0.01
log return USD/SGD
300 200 100 0 -5
0
log return
5 10
-3
Figure 8: Distribution fit for the foreign exchange pairs, EUR/JPY, USD/EUR, USD/JPY, and USD/SGD. The distribution fitting compares the empirical density (circle markers) against the normal distribution (dashed line) and the generalized hyperbolic (solid line) and the kernel– smoothed generalized hyperbolic (dashed-dotted line).
20
EUR/JPY
150 100 50 0
150 100 50 0
-0.02 -0.01
0
0.01
-0.01
log return USD/JPY
200
frequency error
200
frequency error
USD/EUR
200
frequency error
frequency error
200
150 100 50 0
0
0.01
log return USD/SGD
150 100 50 0
-0.01
0
0.01
0.02
-0.005
log return
0
0.005
log return
Figure 9: Frequency error calculated by the fitting of the generalized hyperbolic (red cross markers) on the four foreign exchange pairs EUR/JPY, USD/EUR, USD/JPY and USD/SGD, against that introduced by the normal distribution (blue circle markers).
21
frequency total error
frequency total error
EUR/JPY
4000 3000 2000 1000 0
0
3000 2000 1000 0 -0.01
0
2000 1000 0 -0.01
log return USD/JPY
4000
3000
0.01
frequency total error
frequency total error
-0.02 -0.01
0.01
USD/EUR
4000
0.02
4000
0
log return USD/SGD
3000 2000 1000 0 -0.005
log return
0.01
0
0.005
log return
Figure 10: Foreign exchange pairs: EUR/JPY, USD/EUR, USD/JPY and USD/SGD total error from distribution fitting. Total error from the distribution fitting between the normal and the empirical (blue solid line) and the generalised hyperbolic and the empirical (red dashed line) of the examined foreign exchange pairs.
blue circle marker), showcasing why the use of the former is preferable. The aforementioned findings support the conclusion that, distribution-wise, BTC and the cryptocurrencies overall exhibit similar distributional properties to the examined currency pairs. However, from a monetary point of view, this does not suffice to conclude that cryptocurrencies behave like a traditional currency pair. They still face certain limitations which makes their adoption as currencies uncertain (see Yermack, 2015; Corbet et al., 2018a, among others), even though it has undoubtedly made progress in becoming accepted as such. The main concerns are raised with respect to whether cryptocurrencies can be considered a monetary entity as economists’ consensus indicate that for the latter to be the case, three attributes should strictly be met: first, the entity to serve as a unit of account, second to be able to be used as a medium of exchange and third to be a store of value (Brunner and Meltzer, 1971). The requirement most sufficiently met by cryptocurrencies is that of serving as a medium of exchange. Indeed, even though limited acceptance, the adoption of them as a method of payment has been growing quickly. However, still a large majority of cryptocurrencies users will either keep them stored in their (e-)wallets, in anticipation of excessive returns, or they will perform very few transactions, including mainly selling them to somebody else (see Glaser et al., 2014; Baur et al., 2018). Nevertheless, the main feature of cryptocurrencies that impede them from
22
cumulative density function
0.5
0 -0.02 -0.01
0
0.01
log return USD/JPY
1
0.5
0 -0.01
0
0.01
0.02
log return
cumulative density function
cumulative density function cumulative density function
EUR/JPY
1
USD/EUR
1
0.5
0 -0.01
1
0
0.01
log return USD/SGD
0.5
0 -0.005
0
0.005
log return
Figure 11: Cumulative distribution function (CDF) for the normal (dashed-doted red line), generalized hyperbolic (dashed blue line) and the empirical density (solid black line) of the examined foreign exchange pairs EUR/JPY, USD/EUR, USD/JPY and USD/SGD.
23
BTC rolling annualized volatility 1.6
1.6
252-day rolling window
1.4
1.2
1.2
annualized volatility
annualized volatility
30-day rolling window
1.4
1 0.8 0.6
1 0.8 0.6
0.4
0.4
0.2
0.2
0
0 2012 2013 2014 2015 2016 2017 2018
2012 2013 2014 2015 2016 2017 2018
year
year
Figure 12: Annualized rolling volatility of Bitcoin (BTC)’s log returns using a 30-day rolling window (left) and a 252-day (trading year) rolling window (right).
serving as a unit of account is their high volatility (e.g. Corbet et al., 2018a). Generally speaking, for an entity to be used as a unit of account it means that it can be used as a benchmark (i.e., numeraire) against which prices can be compared in a meaningful way. As a result, cryptocurrencies’ high volatility does not allow for such a use as the purchasing power of an X amount of them today might be vastly different than that of a few days later. However, by examining BTC’s rolling volatility (Fig. 12) we observe that its annualized value has significantly decreased over the past years. In fact, from mid–2013 to 2017 volatility trends downwards taking values for most of this interval between 12% and 21%. These figures are still larger than those of the volatility of conventional traded currency pairs, but the decreased volatility is noticeable. This relatively quiet period is followed by one of increased volatility, unavoidably busted by the price surge of BTC which gained more than 1, 000% and peaks at the time the BTC futures are issued by the Chicago Mercantile Exchange (CME). Today, given its current, again downward trend, the scenario that it reaches comparable figures to that of widely traded currency pairs is not an unlikely event (see Fig. 13). Subsequently, if the volatility of cryptocurrencies continue to decrease they will possibly serve as a numeraire to facilitate meaningful comparisons with traded assets and services. BTC’s seemingly decreasing volatility can be the key component in the direction of gradually becoming a store of value.9 When the underlying volatility is high, investors will typically refrain from holding the asset in the longrun as its purchasing power by the end of holding period is prone to a large degree of uncertainty. This is the main reason investors resort to gold or the USD as it is commonly accepted that their intrinsic value will remain relatively stable even in 9
Commonly, an asset, commodity or currency constitutes a store of value when it can be stored for future use without its value depreciating in an unforeseen way.
24
USD/JPY rolling annualized volatility 0.1
0.1
30-day rolling window
0.09
0.09
0.08
annualized volatility
0.08
annualized volatility
252-day rolling window
0.07 0.06 0.05 0.04 0.03
0.07 0.06 0.05 0.04 0.03 0.02
0.02
0.01
0.01 2012 2013 2014 2015 2016 2017 2018
0 2012 2013 2014 2015 2016 2017 2018
year
year
Figure 13: Annualized volatility of USD/JPY pair’s log returns using a 30–day rolling window (left) and a 252-day (trading year) rolling window (right).
financial turmoil. There are of course then issues left to be resolved in order for the BTC (or other of the cryptocurrencies) to be considered a valid alternative investment to the currency market. A practical problem that arises is its high exchange rate to conventional currencies which renders it hard to use in daily life as it is not that intuitive to quote prices with several decimal places or negative exponents (e.g., 5×10−4 BTC for a cup of coffee). Of course a remedy to that would be the adoption of subdivisions of BTC as happens with tradition currencies. However, the main issue yet to be resolved is the frequent violation of the law of one price which states that under the same currency, identical goods listed at different locations should be sold for the same price. This is not the case with BTC as different exchanges offer different quotes which clearly violates the law of one price. Without a doubt, the main motive of BTC’s market participants so far has been speculation: the idea of spending very little for a hopefully unlimited upward potential. This behaviour is mainly driven by BTC’s yet unclear raison d’ ˆetre as well as its limited trading history. As every newly-introduced asset, BTC and the market of cryptocurrencies overall, are still growing and as yet far from mature assets. Once this is achieved, speculation won’t be the main incentive for investors to enter the market as such opportunities won’t be easily detectable or in case they are, the price of risk will be very high. Both the positive trading environment and the distributional properties of cryptocurrencies, and particularly for the most popular BTC which have changed over time, appear to contribute towards converting it into what its purpose and name dictates: a currency.
25
7
Concluding remarks
Drawing upon the natural phenomenon of the aeolian transport observed in sand dune movements where instantaneous wind bursts must first surpass inertial barriers then sand particle jumps and less energy can sustain continued change before eventually settling (Bagnold, 1954), we at least temporarily extend the statistical analysis conducted by Barndorff-Nielsen (1977) and Barndorff-Nielsen et al. (1985) into the market of cryptocurrencies. The latter represents a highly unregulated and news–driven trading environment where price jumps happen on a daily basis fuelled by major market participants’ speculative appetite and then unobserved latent variables affect the price level of the traded coins (see Wang and Vergne, 2017; Li and Wang, 2017, for relevant studies). Motivated by available literature and the preliminary finding that cryptocurrencies’ log returns are not normally distributed, the main research question we pose here is whether the underlying empirical density of the four major cryptocurrencies - in terms of market capitalization - can be captured by a theoretical probability distribution. The generalized hyperbolic distribution, used to model the abovementioned aeolian process, provides us with a remarkably better fit compared to the normal, as measured by several statistics and in-depth error analysis. Especially in the case of Bitcoin, we find that the generalized hyperbolic distribution can capture its distributional properties almost perfectly, minimizing the error against the empirical density. Further analysis of the foreign exchange market reveals that the said distribution can offer a better fit compared to the normal. This initiates the discussion over whether cryptocurrencies are actually what their name suggests. A closer look provides us with evidence that their behaviour and statistical properties have largely changed over the years, and although as yet cannot be strictly considered equivalent market choices to conventional currencies, undoubtedly their distributional properties and the trading environment may continue, over time, to make them progressively similar to the latter.
Acknowledgement: The authors would like to thank cordially Nicholas H. Bingham for his continuous encouragement, for giving us a good number of excellent references for the hyperbolic distributions, and for his kind feedback. We are grateful to Clare Livesey and Nicholas S. Spyrison for their comments that significantly improved the writing and presentation of this paper. The remaining errors are ours.
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