SHANNON ENTROPY AN ECONOPHYSICAL APPROACH TO CRYPTOCURRENCY PORTFOLIOS Noé Rodriguez-Rodriguez

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SHANNON ENTROPY:AN ECONOPHYSICAL APPROACH TO

CRYPTOCURRENCY PORTFOLIOS

Noé Rodriguez-Rodriguez

Facultad de Ciencias

Universidad Nacional Autónoma de México

Ciudad de México

noe.rdz@ciencias.unam.mx

Octavio Miramontes

Departamento de Sistemas Complejos

Instituto de Física

Universidad Nacional Autónoma de México

Ciudad de México

octavio@fisica.unam.mx

ABSTRACT

Cryptocurrency markets have attracted many interest for global investors because of their novelty,

wide online availability, increasing capitalization and potential proﬁts. In the econophysics tradition

we show that many of the most available cryptocurrencies have return statistics that do not follow

Gaussian distributions but heavy–tailed distributions instead. Entropy measures are also applied

showing that portfolio diversiﬁcation is a reasonable practice for decreasing return uncertainty.

Keywords Cryptocurrencies ·econophysics ·entropy ·portfolio uncertainty ·heavy-tailed ·distributions

1 Introduction

Financial mathematics and ﬁnancial economics are two scientiﬁc disciplines that together are the backbone of modern

ﬁnancial theory. Both disciplines make extensive use of their own models and theories coming from mathematics and

traditional economics and are aimed mostly to the predictive analysis of markets. By the other hand, physics historically

had an important inﬂuence on economical theory when the formalism of thermal equilibrium was inspirational in the

development of the theory of economic general equilibrium[

]. In recent years there is a renewed interest in the view

that economic phenomena share many aspects of physical systems and so they are susceptible for being studied under the

science of complex systems[

] and statistical physics[

], given rise to novel research ﬁelds such as econophysics[

One of the ﬁrst successful econophysical approaches was the discovery that stock market ﬂuctuations are not Gaussian

but heavy–tailed. Mandelbrot came to this conclusion when investigating cotton prices[

] and later Mantegna when

characterizing the Milan stock market[

]. This discovery forced a serious rethinking that the view introduced by Louis

Bachelier in the early XX Century stating that price variations are random and statistically independent[

], while

historically important, is not entirely appropriate. These prices are the outcome of the concurrent non-linear action of

many economic agents[

] and so the ﬂuctuations are usually correlated, meaning that the process is not a random

Brownian walk at all. In this article we explore ﬂuctuations in the cryptocurrency market and show for the ﬁrst time

that all of the most common coins analyzed fail a Shapiro–Wilk normality test and are best explained by heavy–tailed

statistical models.

By the other hand, the entropy is a fundamental key quantity in physics both in thermodynamics and information

theory. It is related to disorder and diversity and has been previously used in ﬁnances, speciﬁcally in portfolio selection

theory[

]. Acceptance of entropy in economic theory has historically some reticence, for instance the one expressed by

Paul Samuelson, a seminal ﬁgure in XX Century economics[

]. However it is gaining wide acceptance lately[

]

and specially with the development of non-equilibrium thermodynamics and complex systems theory[

]. It

is well known that portfolio diversiﬁcation is a good strategy to minimize speciﬁc risks and so we will explore the use

of entropy in the cryptocurrency market as a measure of return uncertainty and risk minimization [11, 19, 17].

arXiv:2210.02633v1 [physics.soc-ph] 6 Oct 2022

2 Methods, data and analysis

In this article we used the historical daily prices of 18 cryptocurrencies that are presented in CryptoDataDowload [

]

spanning at least 3 years. These are: Basic Attention Token (BAT), Bitcoin Cash (BCH), Bitcoin (BTC), Dai (DAI),

Eidoo (EDO), Eos (EOS), Ethereum Classic (ETC), Ethereum (ETH), Metaverse ETP (ETP), Litecoin (LTC), Neo

(NEO), OMG Network (OMG), Tron (TRX), Stellar (XLM), Monero (XMR), Verge (XVG), Ripple (XRP), and Zcash

(ZEC)[21, 22].

The opening and closing prices of the cryptocurrencies, quoted in dollars, from

10/16/2018

12/31/2021

were used,

giving a total of 1172 observations per cryptocurrency, to calculate daily returns. First, the distributions of the daily

returns of each cryptocurrency were statistically characterized through normality tests and parametric adjustments of

heavy–tailed distributions. Subsequently, an analysis where entropy functions are derived similarly as in Dionisio

et al. (2006), [

], Ormos and Zibriezky (2014) [

] and Mahmoud and Naoui (2017) [

]. From the set of 18

cryptocurrencies, the assets were randomly selected to compose investment portfolios, where the only premise used was

that the proportion invested in each asset is

, being

the number of assets in the portfolio. To compare entropy to

standard deviation consistently, normal entropy was used since it is a function of variance.

In the following subsections we will review some useful concepts regarding entropy functions: the discrete entropy

function, the continuous entropy function, the entropy as a measure of uncertainty, the comparison between entropy and

variance and investment portfolios.

2.1 Discrete entropy function

Let

be a discrete random variable,

{A1, A2, A3, ...An}

be a set of possible events and the corresponding probabilities

pX(xi) = P r(X=Ai)

, with

pX(xi)≥0

and

i=1 pX(xi)=1

. The generalized discrete entropy function or Rényi

entropy [24, 17] for the variable Xis deﬁned as

Hα(X) = 1

1−αlog n

i=1

pX(xi)α!(1)

where

is the order of the entropy and

α≥0

. This order can be considered as a bias parameter, where

α≤1

favors

rare events and where α≥1favors common events [25]. The base of the logarithm is 2.

When

α= 1

is a special case of the generalized entropy that assumes ergodicity and independence, which the

generalized case does not. However, substituting into 1 results in division by zero. By means of L’Hôpital’s rule, it can

be shown that when αtends to 1, we have the Shannon entropy

H1(X) = −

i=1

pX(xi)log(pX(xi)) (2)

Shannon entropy produces exponential equilibrium distributions, while generalized entropy produces power law

distributions.

2.2 Continuous entropy function

Let

be a continuous random variable that takes values of

and

pX(x)

be the density function of the random variable.

Continuous entropy is deﬁned as

Hα(X) = 1

1−αln ZpX(x)α(3)

Note that the logarithm base of 1 and 3 are different. Although the entropy depends on the base, it can be shown that

the value of the entropy changes only by a constant coefﬁcient for different bases.

When α= 1, we have the Shannon entropy for the continuous case

H1(X) = −ZpX(x)ln(pX(x))dx (4)

The properties of discrete and differential entropy are similar. The differences are that the discrete entropy is invariant

under variable changes and the continuous entropy is not necessarily so, furthermore the continuous entropy can take

negative values.

2.3 Entropy as a measure of uncertainty

According to Shannon (1948) [26], an uncertainty measure H(pX(x1), pX(x2), ..., pX(xn)) must satisfy:

1. Hmust be continuous on pX(xi), with i= 1, ..., n.

2. If pX(xi) = 1

n,Hmust be monotone increasing as a function of n.

If an option is split into two successive options, the original

must be the weighted sum of the individual

values of H.

Shannon showed that a measure that satisﬁes all these properties is 2 multiplied by any positive constant (the constant

just sets the unit of measure). Among the properties that make it a good uncertainty choice are

1. H(X)=0if and only if all but one of pX(xi)are zero.

When

pX(xi) = 1

i.e. when the discrete probability distribution is constant,

H(X)

is maximum and equal to

log(n).

3. H(X, Y )≤H(X) + H(Y)

, where equality holds if and only if

and

are statistically independent i.e.

p(xi, yj) = p(xi)p(yj).

4. Any change towards the equalization of the probabilities pX(xi), increases H.

5. H(X, Y ) = H(X) + H(Y|X)

. So the uncertainty of the joint event

(Y|X)

is the uncertainty of

plus the

uncertainty of Ywhen Xis known.

6. H(Y)≥H(Y|X)

which implies that the uncertainty of

is never increased by knowledge of

. Decreases,

unless Xand Yare independent, in which case it doesn’t change.

2.4 Comparison between Entropy and Variance

Ebrahimi et al. (1999) [

] showed that entropy can be related to higher order moments of a distribution, thus it can

offer a better characterization of

pX(x)

because it uses more information about the probability distribution than the

variance (which is only related to the second moment of a probability distribution).

The entropy measures the disparity of the density

pX(x)

of the uniform distribution. That is, it measures uncertainty in

the sense of using

pX(x)

instead of the uniform distribution [

]. While the variance measures an average of distances

from the mean of the probability distribution. According to Ebrahimi et al. [

], both measures reﬂect concentration,

but use different metrics. The variance measures the concentration around the mean and the entropy measures the

density diffusion regardless of the location of the concentration. Statistically speaking, entropy is not a mean-centered

measure, but takes into account the entire empirical distribution without concentrating on a speciﬁc moment. This

way you can take into account the entire distribution of returns without focusing on one particular [

]. The discrete

entropy is positive and invariant under transformations, but the variance is not. In the continuous case neither the

entropy nor the variance are invariant under one-to-one transformations [

] [

]. According to Pele et al. (2017) [

]

entropy is strongly related to the tails of the distribution, this feature is important for distributions with heavy tails or

with an inﬁnite second-order moment, where the variance is obsolete. Furthermore, the entropy can be estimated for

any distribution, without prior knowledge of its functional form. These authors found that heavy–tailed distributions

generate low entropy levels, while light–tailed distributions generate high entropy values.

2.5 Investment Portfolios

A portfolio or investment portfolio is simply a collection of assets. They are characterized by the value invested in each

asset. Let wibe the fraction invested in asset iwith i= 1,2, ..., n, the required constraint is that

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SHANNONENTROPY:ANECONOPHYSICALAPPROACHTOCRYPTOCURRENCYPORTFOLIOSNoéRodriguez-RodriguezFacultaddeCienciasUniversidadNacionalAutónomadeMéxicoCiudaddeMéxiconoe.rdz@ciencias.unam.mxOctavioMiramontesDepartamentodeSistemasComplejosInstitutodeFísicaUniversidadNacionalAutónomadeMéxicoCiudaddeMéxicooctavio@f...

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SHANNON ENTROPY AN ECONOPHYSICAL APPROACH TO CRYPTOCURRENCY PORTFOLIOS Noé Rodriguez-Rodriguez

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