Vol.:(0123456789)
Annals of Operations Research (2022) 313:47–75
https://doi.org/10.1007/s10479-020-03680-y
1 3
S.I. : FINANCIAL MODELLING AND RISK MANAGEMENT OF ENERGY 
AND ENVIRONMENTAL INSTRUMENTS AND DERIVATIVES
Financial modelling, risk management of energy instruments 
and the role of cryptocurrencies
Toan Luu Duc Huynh1,2  · Muhammad Shahbaz3,4 · Muhammad Ali Nasir1,5 · 
Subhan Ullah6
Published online: 2 July 2020 
© The Author(s) 2021
Abstract
This paper empirically investigates whether cryptocurrencies might have a useful role in 
financial modelling and risk management in the energy markets. To do so, the causal rela-
tionship between movements on the energy markets (specifically the price of crude oil) and 
the value of cryptocurrencies is analysed by drawing on daily data from April 2013 to April 
2019. We find that shocks to the US and European crude oil indices are strongly connected 
to the movements of most cryptocurrencies. Applying a non-parametric statistic, Transfer-
ring Entropy (an econophysics technique measuring information flow), we find that some 
cryptocurrencies (XEM, DOGE, VTC, XLM, USDT, XRP) can be used for hedging and 
portfolio diversification. Furthermore, the results reveal that the European crude oil index 
is a source of shocks on the cryptocurrency market while the US oil index appears to be a 
receiver of shocks.
Keywords Energy markets · Risk management · Crude oil · Cryptocurrency · Transfer 
entropy · Financial instruments
JEL Classification O31 · G18 · O32 · O33
 * Toan Luu Duc Huynh 
 toanhld@ueh.edu.vn
1 School of Banking, University of Economics Ho Chi Minh City, Ho Chi Minh City, Vietnam
2 Chair of Behavioral Finance, WHU – Otto Beisheim School of Management, Burgplatz 2, 
56179 Vallendar, Germany
3 School of Management and Economics, Beijing Institute of Technology, Beijing, China
4 Department of Land Economy, University of Cambridge, Cambridge, UK
5 Leeds Business School, Leeds Beckett University, Leeds, UK
6 Division of Accounting, Nottingham University Business School, University of Nottingham, 
Nottingham, UK
48 Annals of Operations Research (2022) 313:47–75
1 3
1 Introduction
Climate change is one of the most crucial and hence widely debated issues over the last 
decade. The need to tackle it at the global scale provides a rationale to explore the role of 
energy and environmental instruments in reducing carbon emissions. The financial sector, 
with its crucial role in the economy and society, should play its part (Shahbaz et al. 2018; 
Nasir et al. 2019).
The oil price always conveys a large amount of information to researchers, investors 
and policymakers. Empirical research (Balke et al. 2002; Hamilton 2003; Barsky and Kil-
ian 2004; Chen and Chen 2007; Kilian and Park 2009; Elder and Serletis 2010; Jo 2014; 
Degiannakis and Filis 2017) has confirmed that the uncertain movement of oil prices can 
have different effects on different economies and financial assets. It is highly likely that 
major economic indicators strongly rely on the crude oil market, which reflects all ‘good-
or-bad’ information relating to global geopolitics. For example, the West Texas Intermedi-
ate (WTI) index peaked at around US$75 per barrel at the beginning of October 2018 and 
then fell to around US$40 per barrel at the beginning of January 2019. Keynes’s (1923) 
seminal work in this area encourages researchers to study the stochastic dynamics in oil 
and commodity markets. Although there is plenty of theoretical and empirical research on 
oil price modelling to capture the uncertain movements in the oil price, oil prices move-
ments still attract much scholarly attention, with more sophisticated empirical methods. 
As of July 2018, the cryptocurrency market accounted for over US$300 billion worth of 
market capitalization, with nearly 2000 ‘coins’ trading.
Crude oil shipments were worth $486.3 billion (from World’s Top Exports) whereas the 
market capitalization for cryptocurrencies was worth $171 billion (from coinmarketcap.
com, 26th April 2019). Given such a high level of investment in these two assets, we are 
interested in examining whether there are links between the crude oil and the cryptocur-
rency markets.
There are a number of other reasons to examine the causal relationship between crude 
oil indices and movements on cryptocurrency markets. First, recently, many countries 
have shown interest in issuing their own coins for oil trading, including Venezuela, Rus-
sia, and the United States. Nicolás Maduro, Venezuela’s President, stated: “In 2019, we 
have a schedule for [oil] to be sold for Petro (the name of cryptocurrency backed by crude 
oil prices) and in this way continue to free us from a currency that the elite of Washing-
ton uses”.1 Taking inspiration from the mechanisms of current cryptocurrency platforms 
(Bitcoin, Ethereum, Litecoin, etc.) many countries around the world are likely to introduce 
their own cryptocurrencies to trade oil and other commodities. Second, understanding of 
the co-movements is necessary in establishing the initial valuation of a new coin. Valua-
tion models provide a preliminary pricing mechanism for those interested in issuing new 
cryptocurrencies. Third, the phenomenal growth of the cryptocurrencies market has been 
attracting the attention of both individual and institutional investors, as cryptocurrencies 
provide an alternative investment platform for the hedging and diversification of investors’ 
portfolios. We believe the next decade will see substantial research outputs on the linkage 
between crude oil and cryptocurrencies in different markets around the globe.
Several governments, including those of Russia, Iran and Venezuela, have already taken 
measures to regulate the oil markets by issuing their own coins. For instance, Igor Yusufov, 
1 https:// www. ccn. com/ crypt ocurr ency- accep ted- venez uela- will- sell- oil- for- petro- maduro- says.
49Annals of Operations Research (2022) 313:47–75 
1 3
former energy minister of Russia, announced that his country is in the final stages of issu-
ing a cryptocurrency based on the fundamental of the petrodollar. Every cryptocurrency 
platform (Bitcoin, Ethereum, Litecoin, etc.) needs to possess the key characteristics of any 
strong fundamental coin. It is anticipated that the application of cryptocurrencies in trading 
crude oil will gradually become a normal method of payment for many governments and 
will be an enormously useful means to control their commodity prices. Many countries 
in Eastern Asia are currently facing conflicts, trade restrictions and economic sanctions 
from the United States, and these states are likely to adopt alternative payment platforms 
using cryptocurrencies by introducing their own petrol-coins.2 Hence, understanding the 
dynamic relationship between a coin and crude oil is one of the initial steps in assessing 
the possibility of launching an oil trading currency. The rapid growth of the cryptocur-
rency market is an emerging area of research in the field of energy economics and finance, 
and has recently attracted much scholarly attention. Cryptocurrency is not only a trend-
ing financial investment but is also considered to be ‘The decentralized transaction sys-
tem empire’ (Ammous 2018). The bitcoin standard: the decentralized alternative to central 
banking. John Wiley & Sons.Bitcoin was first introduced in 2009 with a market price of 
a few cents. Its price had climbed to over $14,000 by December 2017. More recently, the 
market capitalization of Bitcoin has surpassed US$200 billion, while around six million 
users have virtual wallets and nearly one billion confirmed transactions occur every day 
(Hileman and Rauchs 2017).
Investors who choose cryptocurrencies as alternative investment platforms are likely to 
consider the linkage between these two markets. Why? To answer this, we need to empiri-
cally establish the nature of directional causality to understand how crude oil prices influ-
ence cryptocurrencies and vice versa. Investors need to ascertain the determinants of this 
causal relationship to make wise investment decisions. A deep understanding of this con-
nection between crude oil and the cryptocurrency market is of great importance in several 
ways. First, oil and cryptocurrencies are not homogenous assets and, currently, there are 
several kinds of internationally traded crude oils as well as a range of cryptocurrencies. 
The different varieties of oil commodities and cryptocurrencies offer a good opportunity 
to investigate the underlying relationship between these complex payment instruments and 
crude oil prices. Second, the movement of oil prices depends on macroscopic economic 
factors, which leads to the market experiencing many external exogenous shocks. Although 
the cryptocurrency market is a fairly new addition to the global financial system, and is 
relatively independent of macroeconomic factors in these early stages, over time prices of 
cryptocurrencies are likely to become increasingly connected with the global energy mar-
ket. This implies that investors will tend to use cryptocurrencies to hedge against volatility 
in the oil markets. Thus, the investigation of this causal relationship will guide investments 
in financial instruments. Third, Laherrere (2006) showed the role of benchmark crude oil 
indices, such as Brent for Europe and WTI for the American market. These indices convey 
many prospective indicators used for economic and financial forecasting. While the rapid 
expansion of blockchain and cryptocurrency can be practically applied in oil trading, it is 
possible to examine the connections among these highly volatile assets. Fourth, the crypto-
currency market has similar features to the oil market, such as market risk (Koutmos 2019), 
the network correlation with the other conventional markets (Giudici and Polinesi 2019), 
2 Petrol-coins refers to the kind of cryptocurrency which is called ‘Petrol’ backed by crude oil.
50 Annals of Operations Research (2022) 313:47–75
1 3
technical trading and returns (Hudson and Urquhart 2019). Therefore, it is necessary to 
consider the link between cryptocurrencies and energy markets.
Finally, this study adds to the existing and growing body of energy-finance literature 
relating to cryptocurrencies. The theoretical and empirical literature on cryptocurrencies 
has developed in various areas. A few studies have focused on the regulatory aspect of Bit-
coin. For instance, Stokes (2012), Raymaekers (2015), Böhme et al. (2015), Vandezande 
(2017), Pieters and Vivanco (2017) evaluated the regulatory framework of Bitcoin as a 
financial asset being traded on real markets. Other researchers discuss whether Bitcoin is 
likely to be treated as a speculative asset (Baur et al. 2018; Blau 2017; Bouri et al. 2017; 
Dyhrberg 2016; Glaser et al. 2014; Yermack 2015). Their results indicate that Bitcoin is a 
suitable investment for diversification, and has the potential to be used as a safe haven for 
hedging purposes. Bouri et al. (2017) further examine whether Bitcoin can be considered 
a safe haven, hedge or a diversifier against volatility in a commodity index. Their findings 
indicate that Bitcoin is a safe haven and a strong hedge for energy commodities. Recent 
studies have also measured the relationship between Bitcoin volatility and its returns in 
the comparative international context (Balcilar et  al. 2017; Katsiampa 2017; Peng et  al. 
2018). In addition, a few recent studies have examined the contagion risk and spillover 
effects among cryptocurrencies (for example, Ciaian and Rajcaniova 2018; Corbet et  al. 
2018; Symitsi and Chalvatzis 2018; Huynh et al. 2018; Huynh 2019). Recently, research-
ers have also raised concerns about the efficiency of the cryptocurrency market (Urquhart 
2016; Nadarajah and Chu 2017; Bariviera 2017; Sensoy 2019). However, few studies have 
investigated the influence of cryptocurrencies on crude oil markets or vice versa, and this 
study attempts to fill this gap in the energy-finance literature.
In this context, the aim of this paper is to examine information spillovers between a 
commodity asset (specifically oil) and a digital cryptocurrency (e.g., Bitcoin). Our data 
cover 80% of the market capitalization of cryptocurrency alongside two main oil indexes 
(US crude oil and European crude oil). With regard to methodology, we use an entropy 
approach to capture the direction of spillovers within the information network, from a static 
perspective. In contrast, previous studies have focused on lagged terms, for example within 
a Vector-Auto-Regression model. We employ different indicators to identify the impor-
tance of each cryptocurrency being studied within the global energy market.
The remainder of this paper is organized as follows. Section 2 reviews the literature and 
highlights our contribution. The methodology and data sources are explained in Sect. 3. 
Section 4 presents the preliminary data analytics and empirical findings. Finally, Sect. 5 
concludes the paper with policy implications.
2  Literature review
Regulatory bodies and investors, in particular, are concerned about excessive volatility and 
abnormal returns in the cryptocurrency market. Higher abnormal returns in these markets 
make investment in cryptocurrency more appealing to the investor community. Recent 
research on cryptocurrencies shows that investors can achieve substantial returns by diver-
sifying across different cryptocurrencies (Liu 2018).
A few studies have attempted to test the assumptions of market efficiency for crypto-
currencies. Examining the seasonality effect for cryptocurrencies, Kaiser (2018) could not 
observe robust and consistent calendar effects on cryptocurrency returns. He nevertheless 
found that the volatility and spreads, as well as trading volume, were generally lower on 
51Annals of Operations Research (2022) 313:47–75 
1 3
weekends and in January. Behavioural finance researchers have also raised concerns about 
the volatility of the cryptocurrency market. Investigating the widespread noise and senti-
ment-driven trading in the cryptocurrency market, Kallinterakis (2019) found a significant 
herding effect. He argued that the extreme noise and volatility in these markets had a desta-
bilizing effect on associated financial markets.
In the energy research literature, Krugman’s (1983) seminal work provides a solid foun-
dation for understanding the economic perspectives on and implications of oil prices. He 
argued that volatility in oil prices can cause a reciprocal change in the foreign exchange 
markets in terms of multilateral relationships. A number of other studies have also con-
tributed to the literature on oil prices and currencies. For instance, Lizardo and Mollick 
(2010) asserted that there is a significant relationship between positive oil shocks and sub-
sequent depreciation of the US dollar. This result is similar to the work of Adler and Sosa 
(2011), Aloui et al. (2013), Reboredo (2012), Reboredo et al. (2014), Coudert and Mignon 
(2016), who all reported a strong relationship between oil prices and exchange rate volatil-
ity. Kunkler and MacDonald (2019) attempted to exclude the numéraire effect (see German 
et al. 1995 for the numéraire effect) in evaluating the co-movements between oil prices and 
G-10 countries’ currencies. They examined the spillover and causal interference from the 
oil market to those currencies. Using intraday data, many recent studies (such as Andersen 
et al. 2007; Degiannakis 2008; Corsi et al. 2010; Bekaert and Hoerova 2014; Duong and 
Swanson 2015; Bollerslev et al. 2016; Wang et al. 2016a, b; Tian et al. 2017; Gong and Lin 
2018a) have found there is a causal relationship between oil prices and currency markets. 
Research on the return and volatility spillovers among 18 cryptocurrencies indicated that 
Bitcoin is a key contributor to return-volatility spillover across all cryptocurrencies, which 
implies a high degree of interconnectedness and interdependence amongst cryptocurren-
cies (Koutmos 2018).
Another strand of literature investigates the relationship between oil prices and product 
prices. Serletis (1994) and Gjolberg and Johnsen (1999) reported the existence of a long-
run equilibrium, while Adrangi et al. (2001) and Asche et al. (2003) emphasized the role 
of a unidirectional causal relationship. Generally, crude oil and product prices have strong 
dynamics. Product prices have played an important role in explaining short-run and long-
run petroleum markets. This effect is stronger in the European market than in the US mar-
ket (Lanza et al. 2005). Using intraday oil prices, previous studies (see in Sévi 2014; Hau-
gom et al. 2014; Prokopczuk et al. 2016; Phan et al. 2016; Wen et al. 2016; Gong and Lin 
2017; Degiannakis and Filis 2017; Liu et al. 2017; Ma et al. 2017; Gong and Lin 2018b; 
Ma et al. 2019) sought to model oil’s stochastic dynamics (including price and volatility).
When it comes to the connections between the different oil and other markets, several 
papers have contributed to the literature. For example, Lin and Tamvakis (2001) indicated 
the connection between NYMEX and International Petroleum Exchange (IPE) crude oil. 
They emphasized the spillover effects particularly when two markets are trading at the 
same time. Sadorsky (2012) proved that the MGARCH model would better than the con-
ventional GARCH to detect contagion volatility and asymmetric effects on international 
oil markets. Similarly, Liu and Tu (2012) suggest the diversification from defining the cor-
relation3 among five types of petroleum future contracts. This means that understanding 
the connection of these markets is important for the purpose of valuation of these assets. 
Moreover, Furió and Chuliá (2012) found dependency between Brent crude oil (known 
3 The investors are advised to minimize the variance by adding the opposite direction of correlation indi-
ces.
52 Annals of Operations Research (2022) 313:47–75
1 3
in Europe) and Zeebrugge natural gas forward prices and also reported a causal relation-
ship in both markets. Furthermore, Jiao et al. (2007) and Chen et al. (2009) explored the 
economic transmission from the Chinese oil and international oil markets, which is consid-
ered to be of vital importance in understanding the interconnectedness of two different oil 
markets.
Indeed, there is a large amount of empirical evidence on the role of oil in economic 
activities. For example, Pindyck (2003), Brown and Yucel (2008), Zamani (2016), Jadidza-
deh and Serletis (2017), Serletis and Rangel-Ruiz (2004), Bachmeier and Griffin (2006), 
Ramberg and Parsons (2010), Brigida (2014) and Atil et al. (2014) investigated the rela-
tionship between crude oil prices and natural gas prices. Although there are inconsistent 
results in terms of strong evidence or weak evidence, these studies suggest crude oil can be 
a market dominant for natural gas.
The relationship between crude oil and other commodities and currencies is well estab-
lished in the energy literature. With the emergence of the new cryptocurrencies, a hand-
ful of studies have investigated the relationship between cryptocurrencies and oil prices. 
Van Wijk (2013) found that Bitcoin had a negative relationship to oil prices. Employing 
the Vector Error Correction Model, this author also examined the relationship of Bitcoin 
to other types of financial assets. However, these relationships were explored for only 
that particular cryptocurrency, ignoring other prominent cryptocurrencies. Most cryp-
tocurrencies have been developed using the first version of Bitcoin or using a decentral-
ized version (version 2.0) such as Ethereum (ETH) or Ripple (XRP) or Litecoin (LTC). 
Therefore, an examination of causal relationships is necessary to explain the spillover or 
contagion effects across these markets. Wang et  al. (2016a) found that trading activities 
on oil markets influenced Bitcoin returns and volatilities. However, Wang et  al. (2016b) 
focused on only one kind of cryptocurrency and their methodology was mainly based on 
linear assumptions. Guesmi et  al. (2018) argue that Bitcoin can be used for hedging. If 
investors have the option to add assets to their portfolio of gold, oil and equities, Bitcoin 
will be an optimal option to minimize the overall risk of the portfolio. Using quantile-and-
quantile regression and CoVaR, Selmi et al. (2018) argued that, by using Bitcoin, investors 
may earn higher returns in the hedging process than they can with oil and gold. Recently, 
Gajardo et al. (2018) employed the Multifractal Asymmetric Detrended Cross-correlation 
analysis method to examine the relationship between Bitcoin and stock (through DJIA), 
gold and several major currencies. The results showed that Bitcoin has an asymmetric cor-
relation with other financial assets; however, this phenomenon is evident only in the short 
run. Symitsi and Chalvatzis (2019) presented new insights relating to the economic value 
of Bitcoin. They found that an oil portfolio consisting of Bitcoin generates higher risk-
adjusted returns.
With the growing momentum of research on cryptocurrencies and energy products, 
we will briefly summarize here the current literature on the linkage between crypto-
currency and crude oil. First, Gronwald (2019) considered whether Bitcoin could be 
considered a commodity asset or not. If so, a co-movement with the commodity market 
would be expected. Using jump-GARCH and short-run supply–demand shocks, Gron-
wald drew the conclusion that Bitcoin has the characteristic of an exhaustible commod-
ity resource, similar to crude oil. This could result in the co-movement of these two 
kinds of assets. Although Giudici and Abu-Hashish (2019) found that oil has no link to 
Bitcoin prices, several studies have reported a relationship between these two assets, for 
example Gajardo et al. (2018), using cross-correlation dependence. Similarly, Jin et al. 
(2019) concluded that Bitcoin prices are susceptible to price fluctuations from oil by 
using MF-DCCA and MV-GARCH. Therefore, we can conclude that the evidence on 
53Annals of Operations Research (2022) 313:47–75 
1 3
the relationship between Bitcoin and crude oil is inconclusive. In addition, this paper 
not only examines the directional impact of Bitcoin but also considers the other main 
cryptocurrencies.
In brief, a vast amount of empirical work has examined the relationship between oil 
prices and other kinds of financial/non-financial assets. Nevertheless, the relationship 
between cryptocurrencies (as monetary currencies) and crude oil has received lim-
ited academic attention. Using a more sophisticated estimation (‘Entropy Transfer’), 
this paper contributes to the embryonic literature on the transmission channel from oil 
shocks to the cryptocurrency market.
3  Empirical methodology and data
In this section we present our main estimation approach. Because of the complexity of 
data or stochastic processes, methods previously used to calculate information trans-
fer have been mainly derived from assumptions of specific subjects (or sometimes the 
process-built restrictions). An important challenge faced by energy researchers is to 
quantify the informational flow, as the Vector Auto Regression (VAR) approach cannot 
estimate this relationship. Therefore, we employ ‘Entropy Transfer’, which is considered 
a more robust method (Schreiber 2000) to overcome the methodological concerns raised 
over previous approaches. In addition, Granger (1988) causality underperform for struc-
tural models. Thus, Transfer Entropy is used here to measure the large amount of asym-
metric information and large networks in non-parametrically directed processes.
For example, if we consider two processes, I and J, the Transfer Entropy is defined 
as:
Then, Shannon (1948) entropy was defined as the independently encoded drawing:
in which, J is a discrete random variable with probability distribution p(j) and j stands for 
the different outcomes that this variable can take. Therefore, H is considered the optimal 
value. However, Shannon’s formula works well under uncertainty. Combined with the 
theoretical framework of Kullback and Leibler (1951), the Shannon process was further 
expanded into two processes, I and J, with the strict assumption that the time series should 
be stationary, for marginal probability distribution p(i), p(j) and joint distribution p(i,j). 
Following this, the average number of encoded observations for (t + 1) and value (k):
where i(k)t =
(
it,… , it−k+1
)
⋅ hJ(l) is considered analogously for process J. Then, with the 
generalized Markov process p
(
it+1|i
(k)
t
)
= p
(
it+1|i
(k)
t , j
(k)
t
)
 from Kullback and Leibler 
(1951). We then have the Shannon (1948) transfer entropy given as:
(1)
Transfer entropy from J to I =
(
Information for future observation I(t+1) absorbed from historical values of I and J
)
−
(
Information for future observation I(t+1) absorbed from the historical value of only I
)
(2)HJ = −
∑
j
p(j) ⋅ log (p(j))
(3)hI(k) = −
∑
i
p
(
it+1, i
(k)
t
)
⋅ log
(
p
(
it+1|i
(k)
t
))
54 Annals of Operations Research (2022) 313:47–75
1 3
where TJ→I evaluates the information flow from J to I.
Applying the ‘Transfer Entropy’ process, we generalize information flows from variable Y 
to variable X in the following formula:
in which  in is the nth component of the time series of variable X and  jn is component n of 
the time series of variable Y.
The joint probability of X and Y is defined as:
Then, Erdös and Rényi (1970) introduced a new approach based on this with weighting 
parameter q > 0 for individual probabilities p(j).
As q → 1, Rényi entropy will converge to Shannon entropy while 0 < q < 1 means low prob-
ability with more weight. Otherwise, q > 1 means a higher initial probability for preference of 
outcome j. Endowing the escort distribution, Beck and Schlögl (1993) indicated the parameter 
휙q(j) =
pq(j)
∑
j p
q(j)
 , with q > 0 in order to normalize the weighted distribution. Jizba et al. (2012) 
developed the Rényi entropy, which we employed as follows:
The process of Shannon and Rényi Transfer Entropy is based on discrete random data. 
Then, the process is based on the Markov block bootstrap (Dimpfl and Peter 2013) and 
repeated by bootstrap. Finally, the null hypothesis for transferring entropy states that no infor-
mation is transferred. We also generalize the Rényi Transfer Entropy:
(4)TJ→I(k, l) =
�
i,j
p
�
it+1, i
(k)
t j
(l)
t
�
⋅ log
⎛
⎜
⎜
⎜
⎝
�
it+1�i
(k)
t , j
(k)
t
�
�
it+1�i
(k)
t
�
⎞
⎟
⎟
⎟
⎠
(5)
TEY→X(k, l) =
∑
in+1,i
k
n
,jl
n
p
(
in+1i
k
n
, jl
n
)
log2 p(in+1|i
k
n
, jl
n
)
−
∑
in+1,i
k
n
,jl
n
p
(
in+1i
k
n
, jl
n
)
log2 p(in+1|i
k
n
)
=
∑
in+1,i
k
n
,jl
n
p
(
in+1i
k
n
, jl
n
)
log2
p(in+1|ikn, j
l
n
)
p(in+1|ikn)
(6)p
(
in+1, i
(k)
n
, j(l)
n
)
= p
(
in+1, in,… , in−k+1, jn,… , jn−l−1
)
(7)HqJ =
1
1 − q
log
(
∑
j
pq(j)
)
(8)RTJ→I(k, l) =
1
1 − q
log
⎛
⎜
⎜
⎜
⎝
∑
i 휙q
�
i
(k)
t
�
pq
�
it+1�i
(k)
t
�
∑
i,j 휙q
�
i
(k)
t , j
(k)
t
�
pq
�
it+1�i
(k)
t , j
(k)
t
�
⎞
⎟
⎟
⎟
⎠
(9)RTEY→X(k, l) =
1
1 − q
log2
∑
in+1,i
k
n
휑q
�
ik
n
�
pq(in+1�ikn)
∑
in+1,i
k
n
휑q
�
ik
n
, jl
n
�
pq(in+1�ikn, j
l
n
)
55Annals of Operations Research (2022) 313:47–75 
1 3
where 휑q is the escort distribution given by:
The methodology of ‘Transferring Entropy’ is not only used by computer science 
researchers (e.g. Lizier et al. 2008; Lizier and Mahoney 2013), but has also been applied 
in many other interdisciplinary fields, for example in: neurosciences to detect neural cortex 
(Sumioka et al. 2007; Faes et al. 2013), social network and behaviors (Ver Steeg and Gal-
styan 2012); mathematical statistics (Barnett and Bossomaier 2012; Liu et al. 2014); and, 
especially, in thermodynamics (Prokopenko et al. 2013).
The applications of ‘Transferring Entropy’ in economics and finance is an emerging 
methodological development. Next, we briefly evaluate the application of this method in 
economics and finance. Marschinski and Kantz (2002) introduced this method to evalu-
ate the information flow from the S&P500 Index (USA)DAX (Germany). They found it 
provided more efficient and consistent estimations compared with other approaches as it 
subtracts the effects of noise as well as high volatility in the time series. Baek et al. (2005) 
employed this method to investigate the impact of energy industry returns on the over-
all health of an economy. Their sample covered 135 firms listed on the New York Stock 
Exchange over the period 1983 to 2003, and they found that energy companies signifi-
cantly influence the national economy. Interestingly, these results are in line with the paper 
by Kwon and Yang (2008) which reported on information flows from the US and European 
markets to the Asia Pacific region. Furthermore, Jizba et al. (2012) reexamined the work 
of Marschinski and Kantz (2002) with the integration of two methodological approaches 
(Shannon’s entropy and Rényi’s entropy). They also investigated the non-parametric dis-
tribution in terms of information transfer. Similarly, Dimpfl and Peter (2013) and Peter 
et al. (2011) investigated the information flow from the CDS (Credit Default Swap) mar-
ket to the corporate bond market by applying the ‘Transfer Entropy’ approach. The results 
reveal a dynamic relationship of market risk (measured by CBOE Volatility Index—VIX) 
among different financial assets. Kim et  al. (2013) analysed a set of variables including 
the equity index, consumer price index, real exchange rate, trade balance and industrial 
production index, representing macroeconomic determinants, from 18 economies over the 
period 1990–2000. They studied the interconnectedness among their samples and showed 
the US market was a net sender of information flows, whereas Europe appeared to be a 
passive recipient. Li et al. (2013) adapted this new perspective of ‘Transferring Entropy’ 
to investigate the contagion risks among Chinese banks by simulating the spillover risks 
of networks, and concluded that Chinese banks are likely to be stable under market risks.
In brief, several previous studies have applied Transfer Entropy to investigate causal 
relationships in different institutional and market settings. However, we are not aware of 
any study that has applied Transfer Entropy to investigate connections between the crude 
oil markets and the cryptocurrency markets. Therefore, our paper extends the application 
of this quantitative technique to investigate the causal relationship between the US and 
European crude oil returns and 14 different cryptocurrencies.
Our daily returns data are sorted into two main groups: First, oil returns data for the 
US index (DCOILWTICO) and European index (DCOILBRENTEU) are collected from 
Federal Reserve Economic Data (https:// fred. stlou isfed. org/). Noticeably, we focus on the 
developed regions only, specifically, the US and EU crude oil markets due to the large 
proportion of these developed economies in global crude oil market (Chuliá et al. 2019). 
Furthermore, Jin et al. (2012) also indicated that Dubai crude oil has similar properties to 
(10)휑q(i) =
pq(i)
∑
i p
q
i
56 Annals of Operations Research (2022) 313:47–75
1 3
Brent. Most importantly, WTI crude oil which has been traded in the New York Mercantile 
Exchange (NYME), the biggest commodity futures. This is the most crucial pricing bench-
marks in the global crude oil which could be considered as an important criterion for inves-
tors. In a recent study, Liu and Gong (2020) confirmed that WTI plays a prominent role in 
the international crude oil markets. Concomitantly, examining the relationship between the 
two largest crude oil markets and an inclusive sample of cryptocurrency coins by employ-
ing new econometrics models would provide us with deeper insights into the spill-over 
effects.
Second, data on 14 cryptocurrencies are collected from coinmarketcap.com. These cur-
rencies are: BTC (Bitcoin), LTC (Litecoin), ETH (Ethereum), XEM (Nem), DASH (Dash), 
DOGE (Dogecoin), XMR (Monero), VTC (Vertcoin), XVG (Verge), DGB (Digibyte), 
XLM (Stellar), USDT (Tether), MAID (MaidSafeCoin), XRP (Ripple). Our sample period 
covers data from 29th April 2013 to 06th April 2019, though not all cryptocurrencies were 
in existence over the whole period. We chose a long horizon to cover as much information 
as possible. In addition, one of Transfer Entropy’s advantages is that is allows unbalanced 
datasets to be used to estimate causal relationships. From the available pool of all crypto-
currencies, we selected those that met three main criteria: (i) continuous trading; (ii) having 
an equal number of or more observations than DCOILWTICO and DCOILBRENTEU; and 
(iii) accounted for at least 80% market capitalization of the cryptocurrency market. Fur-
ther information regarding market capitalization, circulating supply and date of issue are 
provided in “Appendix” (ii). In Table 1, we report the descriptive statistics for our sample.
Table 1  Summary statistics
Transfer Entropy allows us to estimate the causal relationship based on the unbalanced data. Our sample 
covers the period from 29th April 2013 to 06th April 2019 with different observations for our variables
Variable Obs. Mean SD Min Max Skewness Kurtosis
DCOILWTICO 1168 0.0005 0.0221 − 0.0905 0.1128 0.2535 5.8424
DCOILBRENTEU 1168 0.00004 0.0200 − 0.0690 0.0989 0.4333 5.7238
BTC 2169 0.0016 0.0433 − 0.2674 0.3614 − 0.1108 11.0616
LTC 2169 0.0013 0.0668 − 0.5192 0.8245 1.7000 27.7714
ETH 1338 0.0030 0.0763 − 1.3739 0.4034 − 4.0516 83.2739
XEM 1466 0.0035 0.0876 − 0.5025 1.0684 1.9692 23.8385
DASH 1877 0.0034 0.0777 − 0.4204 1.0668 2.5294 31.6831
DOGE 1938 0.0009 0.0808 − 0.6257 1.0625 1.5506 29.0534
XMR 1781 0.0018 0.0740 − 0.4404 0.5676 0.5846 9.01816
VTC 1902 0.0010 0.1118 − 0.6141 1.0923 2.2348 19.9729
XVG 1624 0.0044 0.1722 − 0.9162 1.9169 1.1696 16.0909
DBG 1885 0.0012 0.1024 − 0.6035 1.1522 1.8588 20.8781
XLM 1705 0.0021 0.0793 − 0.3334 0.7040 1.9281 17.8684
USDT 1501 − 0.0001 0.0120 − 0.1822 0.1511 − 2.3979 110.783
MAID 1804 0.0012 0.0694 − 0.4020 0.4643 0.17116 7.03972
XRP 2071 0.0019 0.0758 − 0.6011 1.0109 2.0258 30.2785
57Annals of Operations Research (2022) 313:47–75 
1 3
Following Miller and Scholes (1972),4 we continued data processing to calculate the 
index return by logarithm return.5 Table 1 presents the characteristics of WTI and BRENT 
EU indices returns as well as 14 cryptocurrencies in terms of daily trading activities. Over-
all, the mean daily return is approximately zero. Except for Bitcoin, Ethereum and USDT, 
the other variables show positive skewness, which indicates that these coins have positive 
extreme values in the datasets. Bitcoin, Ethereum and USDT are likely to have negative 
extreme values. With regard to kurtosis analysis, all variables have a heavy tail distribu-
tion, with the probability of extreme values in both tails. Surprisingly, Ethereum has the 
highest kurtosis value (83.27397) whereas the oil prices (in the US and Europe) have the 
lowest values (5.842435 and 5.723816, respectively). We therefore reject all the assump-
tions relating to normal distribution, implying there is an interconnectedness in the abnor-
mal distribution of variables. Therefore, we employ the entropy approach to investigate the 
causal relationship between crude oil markets and cryptocurrencies.
To define the data structures as well as their characteristics, plots are combined into 
a plot matrix. We then need to employ Transfer Entropy for further quantitative testing. 
Hence, Transfer Entropy is employed to capture the informational flows among these mar-
kets, assuming that the vast majority of information will have interacted. These results are 
likely to be robust, owing to the correction of selecting information by crude oil markets 
to cryptocurrencies through tensor of blockchains of these cryptocurrencies. Clearly, the 
anomalous distributions among our variables raises concerns that we need to employ the 
econphysics technique to estimate the informational flows to eliminate the outliers, and that 
we need to employ data transformation in order for our results to be robust (Figs. 1 and 2).
Fig. 1  Data Distribution and Correlation between the US Crude Oil Index and each Cryptocurrency. Note 
The correlation data have non-normal distribution (not linear). Furthermore, some dataset structures are 
skewed and heavy-tailed. Interestingly, there are some outliers correlated with the US crude oil index, 
which means that the dots are at a distance from the sample plot
4 r
t
= ln
(
P
t
P
t−1
)
 in which  Pt is the index at time t.
5 Many recent papers have applied this methodology for Bitcoin returns, but see in particular Jiang et al. 
(2018) and Sensoy (2019).
58 Annals of Operations Research (2022) 313:47–75
1 3
4  Empirical findings and discussion
4.1  Linear correlation analysis
First, we employed the linear correlation using a Bonferroni-adjusted significance level and 
the empirical results are reported in Table 2. Table 2 shows the empirical linear correla-
tions between DCOILWTICO, DCOILBRENTEU and the 14 cryptocurrencies.
Fig. 2  Data Distribution and Correlation between the European Crude Oil Index and each Cryptocurrency. 
Note The correlation data have a non-normal distribution (not linear). Furthermore, some dataset structures 
are skewed and heavy-tailed. Interestingly, there are some outliers correlated with the European crude oil 
index, which means that the dots are at a distance from the sample plot
Table 2  Linear correlation 
analysis
None of the correlations is statistically significant even at the 10% 
level
Cryptocurrencies DCOILWTICO DCOILBRENTEU
BTC 0.0041 − 0.0262
LTC − 0.0012 − 0.0333
ETH − 0.0019 − 0.0258
XEM − 0.0032 − 0.0084
DASH 0.0516 0.0252
DOGE 0.0446 0.0411
XMR 0.0120 0.0494
VTC 0.0319 0.0183
XVG − 0.0158 0.0323
DBG 0.0202 0.0073
XLM − 0.0020 0.0331
USDT − 0.0283 − 0.0179
MAID − 0.0312 − 0.0281
XRP − 0.0193 − 0.0221
59Annals of Operations Research (2022) 313:47–75 
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There is no correlation between crude oil returns and cryptocurrency values. The cor-
relation values are around zero and statistically insignificant. The results from linear cor-
relation therefore suggest there is no connection between crude oil and cryptocurrencies. 
However, in fact, the transmission of information still happens. Therefore, we need to fur-
ther investigate the connections among these relationships (including itself and between 
oil(s) and cryptocurrencies).
4.2  Unit root analysis
Before employing further estimation techniques, we need to test the stationary of all the 
variables. Table 3 gives a summary of stationarity tests such as Augmented Dickey–Fuller 
and Phillips–Perron. The purpose of having the stationary time series datasets is to ensure 
the results are unbiased and not spurious.
All variables are stationary at the 1% significance level from the original series. This 
shows that crude oil and cryptocurrency returns are integrated at I(0).
4.3  Transfer entropy analysis
In Table  4, we present the results of Renyi and Shannon (threshold) Transfer Entropy 
between the US crude oil index (DCOILWTICO) index returns and the 14 cryptocurrencies.
Interestingly, only six cryptocurrencies have a causal relationship with DCOILWTICO. 
Only two cryptocurrencies have a bilateral relationship, with shocks from US crude oil to 
DASH and XVG, and vice versa. DASH is a bitcoin-based currency featuring instant trans-
actions, decentralized governance and budgeting, and private transactions. Consequently, 
DASH has similar characteristics as Bitcoin but comes with more advanced features such as 
Table 3  Unit root analysis
The symbols *, **, and *** denote the level of significance at the 
10%, 5%, and 1% levels, respectively
Variables Augmented Dickey–
Fuller
Phillips–Perron
DCOILWTICO − 32.920*** − 32.796***
DCOILBRENTEU − 28.557*** − 28.580***
BTC − 46.438*** − 46.466***
LTC − 45.736*** − 45.798***
ETH − 34.781*** − 34.743***
XEM − 42.069*** − 41.977***
DASH − 43.387*** − 43.393***
DOGE − 40.463*** − 40.378***
XMR − 42.506*** − 42.511***
VTC − 41.902*** − 41.898***
XVG − 53.220*** − 52.886***
DBG − 43.755*** − 43.780***
XLM − 38.836*** − 38.884***
USDT − 26.252*** − 24.939***
MAID − 47.360*** − 47.374***
XRP − 42.666*** − 42.933***
60 Annals of Operations Research (2022) 313:47–75
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Table 4  Renyi and Shannon 
transfer entropy results 
from DCOILWTICO to 
Cryptocurrency Markets
Causality Renyi transfer entropy Shannon 
transfer 
entropy
DCOILWTICO → BTC 0.0083
[0.0027]
0.0080
BTC → DCOILWTICO 0.0081
[0.0025]
DCOILWTICO → LTC 0.0117*
[0.0028]
0.0116
LTC → DCOILWTICO 0.0042
[0.0024]
DCOILWTICO → ETH 0.0268***
[0.0043]
0.0267
ETH → DCOILWTICO 0.0055
[0.0042]
DCOILWTICO → XEM 0.0145
[0.0040]
0.0144
XEM → DCOILWTICO 0.0089
[0.0037]
DCOILWTICO → DASH 0.0171**
[0.0032]
0.0170
DASH → DCOILWTICO 0.0140*
[0.0033]
DCOILWTICO → DOGE 0.0109
[0.0028]
0.0108
DOGE → DCOILWTICO 0.0102
[0.0030]
DCOILWTICO → XMR 0.0136*
[0.0032]
0.0136
XMR → DCOILWTICO 0.0052
[0.0036]
DCOILWTICO → VTC 0.0076
[0.0031]
0.0076
VTC → DCOILWTICO 0.0045
[0.0031]
DCOILWTICO → XVG 0.0132*
[0.0033]
0.0131
XVG → DCOILWTICO 0.0184**
[0.0035]
DCOILWTICO → DGB 0.0104
[0.0032]
0.0103
DGB → DCOILWTICO 0.0065
[0.0031]
XLM → DCOILWTICO 0.0070
[0.0036]
0.0069
DCOILWTICO → XLM 0.0103
[0.0032]
DCOILWTICO → USDT 0.0122
[0.0036]
0.0122
USDT → DCOILWTICO 0.0075
[0.0038]
61Annals of Operations Research (2022) 313:47–75 
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private information and transactions. Therefore, it is cogent to use it as a payment method. 
Furthermore, the XVG is a technological platform based on the Tor Project.6 Interestingly, 
the Tor project was received a lot of offers from the United States National Agencies for 
developing their platforms; for example, The National Security Agency (NSA) (Guardian 
2013). Therefore, the effects of XVG on the US crude oil could be explained by the broad 
interest of technological and advanced platform which is designed.
In terms of transmission from the US crude oil market returns on cryptocurrency 
returns, the findings of Shannon transfer entropy estimation indicate a significant informa-
tion flow from the US crude oil to six of the cryptocurrencies (LTC, ETH, DASH, XMR, 
XVG and MAID). This is evidence of information flows from the US crude oil market 
returns to cryptocurrency markets. It supports the proposition that the macroscopic eco-
nomic value of the US crude oil has an effect on the cryptocurrency markets.
The process of transferring entropy from the US crude oil market to the cryptocurrency 
market lies in the range 1.1% (weakest at Litecoin) to 2.6% (strongest at Ethereum). Taking 
the Rényi’s transfer entropy through these graphs (in Fig. 3), we can further investigate this 
Table 4  (continued) Causality Renyi transfer entropy Shannon 
transfer 
entropy
DCOILWTICO → MAID 0.0107*
[0.0031]
0.0106
MAID → DCOILWTICO 0.0056
[0.0035]
DCOILWTICO → XRP 0.0094
[0.0027]
0.0093
XRP → DCOILWTICO 0.0103
[0.0027]
The symbols *, **, and *** denote significance at the 10%, 5%, and 
1% levels, respectively
Fig. 3  Rényi transfer entropy with different values of q from the US Crude Oil Market
6 Please see more at https:// www. torpr oject. org/.
62 Annals of Operations Research (2022) 313:47–75
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Table 5  Renyi and Shannon 
transfer entropy results 
from DCOILBRENTEU to 
Cryptocurrency Market
Causality Renyi transfer entropy Shannon 
transfer 
entropy
DCOILBRENTEU → BTC 0.0113*
[0.0025]
0.0112
BTC → DCOILBRENTEU 0.0106*
[0.0026]
DCOILBRENTEU → LTC 0.0116*
[0.0028]
0.0115
LTC → DCOILBRENTEU 0.0075
[0.0027]
DCOILBRENTEU → ETH 0.0177**
[0.0041]
0.0177
ETH → DCOILBRENTEU 0.0138
[0.0041]
DCOILBRENTEU → XEM 0.0088
[0.0038]
0.0087
XEM → DCOILBRENTEU 0.0080
[0.0038]
DCOILBRENTEU → DASH 0.0118
[0.0028]
0.0117
DASH → DCOILBRENTEU 0.0108
[0.0029]
DCOILBRENTEU → DOGE 0.0089
[0.0031]
0.0089
DOGE → DCOILBRENTEU 0.0058
[0.0031]
DCOILBRENTEU → XMR 0.0128*
[0.0032]
0.0127
XMR → DCOILBRENTEU 0.0042
[0.0035]
DCOILBRENTEU → VTC 0.0112
[0.0031]
0.0111
VTC → DCOILBRENTEU 0.0052
[0.0029]
DCOILBRENTEU → XVG 0.0093
[0.0034]
0.0093
XVG → DCOILBRENTEU 0.0056
[0.0034]
DCOILBRENTEU → DGB 0.0076
[0.0031]
0.0076
DGB → DCOILBRENTEU 0.0152**
[0.0031]
XLM → DCOILBRENTEU 0.0063
[0.0033]
0.0062
DCOILBRENTEU → XLM 0.0063
[0.0036]
DCOILBRENTEU → USDT 0.0137
[0.0037]
0.0137
USDT → DCOILBRENTEU 0.0077
[0.0037]
63Annals of Operations Research (2022) 313:47–75 
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relationship. From the results presented in Fig. 3, there is a possibility to reweight the prob-
abilities under Erdös and Rényi’s (1970) methodology (mentioned in the previous section). 
The value of q will converge to 1 (q → 1). As can be seen, the value of transfer entropy of 
Litecoin and Ethereum is highest at the smallest value of (q) and decreases as q increases. 
Finally, it will converge in Shannon’s TE.
Table  5 demonstrates the results of Renyi and Shannon (threshold) Transfer Entropy 
between DCOILBRENTEU index returns and the 14 cryptocurrencies.
Among the 14 cryptocurrencies, six have interconnectedness with DCOILBRENTEU. 
They are BTC, LTC, ETH, XMR, DGB, and MAID. Among them, only Bitcoin has a bilat-
eral causal relationship with the European crude oil market, which means that there is a 
strong connection from Bitcoin to DCOILBRENTEU, and vice versa.
Generally, all the shocks from the European crude oil market lead to shocks on the cryp-
tocurrency market, as evidenced by the single direction from DCOILBRENTEU to specific 
coins. It makes sense to say that DCOILBRENTEU is an indicator of the economic situa-
tion, and so offers investors to carefully watch these indices for their movements.
Four coins (LTC, ETH, XMR, MAID) are considered the receiver from the two mar-
kets. There are some common characteristics of these coins, including: (i) a decentraliza-
tion system, (ii) providing a platform for petrol-dollars in the future, and (iii) automatic 
Table 5  (continued) Causality Renyi transfer entropy Shannon 
transfer 
entropy
DCOILBRENTEU → MAID 0.0120*
[0.0032]
0.0120
MAID → DCOILBRENTEU 0.0065
[0.0033]
DCOILBRENTEU → XRP 0.0061
[0.0028]
0.0061
XRP → DCOILBRENTEU 0.0081
[0.0030]
The symbols *, **, and *** denote the significance at the 10%, 5%, 
and 1% levels, respectively
Fig. 4  Rényi transfer entropy with different values of q from the European Crude Oil Market
64 Annals of Operations Research (2022) 313:47–75
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confirmation of transactions. We also observe that the process of transferring entropy from 
the European crude oil market to the cryptocurrency market lies in the range of 0.76% 
(weakest at DGB) and 1.7% (strongest at Ethereum). Similarly, Ethereum also receives the 
highest transfer entropy from the US crude oil shocks. Therefore, we can conclude that 
Ethereum is likely to be the most passive receiver in the two crude oil markets. Concur-
rently, Ethereum is evaluated as the most innovative coin, as it allows many networks to be 
built on it, such as petrol coins or smart contracts in oil trading. Hence, there is likely to be 
greater information transfer from these two crude oil markets for Ethereum.
Figure 4 demonstrates the process of Rényi transfer entropy with different values of q 
from the European crude oil market.
Table 6  Renyi and Shannon 
transfer entropy results 
from DCOILWTICO to 
DCOILBRENTEU
The symbols *, **, and *** denote the significance at the 10%, 5%, 
and 1% levels, respectively
Causality Renyi transfer 
entropy
Shannon transfer entropy
DCOILWTICO → 
DCOILBRENTEU
0.0214***
[0.0029]
0.02138667
DCOILWTICO → 
DCOILBRENTEU
0.0126**
[0.0027]
Fig. 5  Rényi transfer entropy with different values of q from the DCOILWTICO to DCOILBRENTEU
65Annals of Operations Research (2022) 313:47–75 
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These graphs are similar to those for the transfer entropy process from the US crude 
oil market. At the highest q value, the transfer entropy value is lower. The highest transfer 
entropy values are at the lowest values of q.
The value of q will converge to 1 (q → 1). In Fig. 4, the transfer entropy value of DGB 
and Ethereum is highest at the smallest values of q and decreases as q increases. Finally, it 
will converge in Shannon’s TE.
4.4  The causal relationship between US and European Crude Oil Markets
We find strong evidence of a causal relationship between the US and European crude oil 
prices. These results show bilateral relationships, interconnectedness and spillover effects 
among these markets (Table 6; Fig. 5).
The highest transfer value is at the initial quantile. The lowest value for transfer is in the 
median quantile. Finally, it converges into Shannon’s TE at quantile (q = 1). This implies 
that the transfer entropy between the US and European crude oil markets follows a quad-
ratic curve. This bell shape indicates that the informational shock on either the US or the 
European crude oil index will result in a higher risk exposure on the other oil market.
Why do we perform the causal relationship analysis between the US and European 
crude oil markets? We would like to deeply investigate the multidimensional effects among 
Fig. 6  The causal relationship network between the US and European Crude Oil and Cryptocurrency Mar-
kets. Note Three coins have bilateral relationships whereas the remaining coins have a unilateral relation-
ship to a specific crude oil market. The interconnectedness between US and European crude oil markets is 
one of the key factors enhancing causal relationships among the cryptocurrencies
66 Annals of Operations Research (2022) 313:47–75
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these markets to explain the mechanism for transferring entropy. By illustrating the net-
work of causal relationships from these results, we illustrate what we achieve by employing 
‘Transfer Entropy’ (Fig. 6).
4.5  What we have learned from transfer entropy
We report empirical evidence on ‘Transferring entropy’ process from the US and European 
crude oil markets to the cryptocurrency markets. We identify what particular types of coins 
are senders or receivers in the transmission process. Our analysis reveals that 8 of the 14 
cryptocurrencies (over 50%) have a strong connection with crude oil returns. We also find 
that the European crude oil market seems to be a sender whereas the US crude oil market 
is likely to be a receiver. This conclusion is mainly based on the bilateral relationships of 
the Transfer Entropy process, as it is difficult to explain how many different cryptocur-
rencies have casual relationships with US and European crude oil markets. Nevertheless, 
our findings do indicate which coins have a directional impact and the exact relationships 
with crude oil markets. Applying an econophysics approach such as Transferring Entropy 
is one of the new methodological contributions in establishing the relationships between 
the crude oil and cryptocurrency markets.
There are already some oil-based cryptocurrencies in the exchange markets (such as 
Petro for Venezuela or Neft-coin for Russia). Hence, understanding the transfer entropy 
process is important to explain how the co-movements of these financial assets (crude oil 
and cryptocurrencies) work. For instance, the shocks in crude oil markets might cause a 
decline in oil-based coin and depreciate the oil-cryptocurrencies issued by the government. 
This could be considered a contagion effect on the linked markets.
Consistent with the findings of Guesmi et al. (2018) and Symitsi and Chalvatzis (2019), 
our results are crucial to investigate pairs of cryptocurrencies. We also argue that it is 
important to study the shocks from the US and European crude oil markets, as a number 
of countries have indicated their intention to develop their own cryptocurrencies (using the 
fundamental platform of Bitcoin, Ethereum and Litecoin). Examples of these new crypto-
currencies include Bilur Energy (UK), OilCoin and PetroDollar in the USA.
It is possible to draw the conclusion that ETH is the receiver from two markets. White 
et  al. (2020) also consider that many investors consider ETH as record transactions. In 
addition, ETH should be known as the oil-based characteristics to run the operations of 
many other cryptocurrencies. Finally, it is evident that there is no bilateral relationship 
between ETH and the crude oil markets. Our paper, therefore, contributes to the evidence 
on contagion networks as well as the precise directions from the US-European oil to cryp-
tocurrency markets. As expected, the fundamental cryptocurrencies (such as Bitcoin, 
Ethereum and Litcoin) have a strong relationship with the crude oil markets. Our paper is 
therefore a contribution to the embryonic literature on the linkage between cryptocurrency 
and crude oil markets.
4.6  Robustness check
In order to make our results robust and consistent, we employ another methodology, 
namely frequency connectedness, as proposed by Baruník and Kocenda (2019), to capture 
the interconnectedness as well as the spillover effect for significant Transfer Entropy values 
in Table 4 and Table 5. However, this method is based on balanced data; thus, we have the 
sub-sample from 2016 to 2019 for our variables. Two main conclusions can be drawn from 
67Annals of Operations Research (2022) 313:47–75 
1 3
the robustness check. First, our directional frequency interconnectedness is different from 
zero, which indicates the presence of a causal relationship in this sub-sample using a dif-
ferent methodology. Second, we also see from time to time increases in the connectedness 
among our variables.7
5  Conclusion and policy implications
Fintech, smart contracts, and blockchain, cryptocurrencies are becoming one of the alterna-
tive investments. Researchers and policymakers are interested in understanding the deter-
minants of their pricing mechanisms. Treating cryptocurrencies as another kind of mone-
tary currency, we can observe the influence of crude oil markets on them. Cryptocurrencies 
are currently used as a source of innovation within the financial system, to integrate the 
methods of settling individual/corporate/government payments, and to anonymously trans-
fer monetary funds. Our paper investigates the transmission channel measured by informa-
tional flows from the US and European crude oil markets to 14 different cryptocurrencies.
We found that over half of this sample of 14 cryptocurrencies are interconnected with 
the crude oil market (BTC (bilateral), LTC, ETH, XMR, DGB, DASH (bilateral), XVG 
(bilateral) and MAID). The US crude oil market is a receiver of informational flows from 
cryptocurrencies whereas the European crude oil is the sender in the transmission process. 
The features of these cryptocurrencies can be used to integrate them into a smart contract 
and create new coins for gasoline and crude oil trading.
Our results have important implications for investors and policymakers. First, by explor-
ing the relationship between crude oil indices and crypto returns, investors may choose 
independent coins (XEM, DOGE, VTC, XLM, USDT, XRP) for their portfolios for diver-
sification or hedging. In particular, investors can use these coins for hedging the move-
ment of US or European crude oil movements, as their return is quite different from that 
of the oil indices. In other words, these coins can be added to portfolios which already 
contain a selection of crude oil indices. Many cryptocurrencies use digital platforms and, 
as a result, the trading mechanism and hedging process are likely to cost-effective and less 
time-consuming.
Second, for policymakers who intend to use the fundamental platform to establish 
petrol-coins, these findings will be very useful in understanding the dynamics and pric-
ing mechanism of both assets. Understanding these directional movements, governments 
will be able to value their petrol coins and avoid the market risk or individual effects from 
these coins. Taking into consideration these risk factors, it is necessary for a country to use 
Bitcoin or Ethereum as its next-generation digital platform to build its own petrol coins, 
because the informational effects originating from these coins are strong.
Third, our paper adds to the theoretical literature on the economic valuation of cryp-
tocurrencies, which is an emerging area of research in the energy literature. We apply 
valuation models used in Physics and contribute to the empirical evidence in establish-
ing the transmission mechanisms of the economic value of crude oil and digital currency. 
Our findings thus add to the cryptocurrency puzzle and we identify alternative avenues of 
investments in this relatively underdeveloped area of research.
7 The results are available upon request.
68 Annals of Operations Research (2022) 313:47–75
1 3
Fourth, the time transfer calculated in our analysis supports the evidence in the lit-
erature that the cryptocurrency market is inefficient, which gives arbitrage opportunities 
to investors. However, it is advisable to consider the time discrepancy in implementing 
the most appropriate strategies for investors. Furthermore, US policymakers can also 
consider the implications of European shocks to make timely decisions to restrict or 
minimize the spillover effects.
However, our findings are subject to limitations. Our paper could not match the auto-
matically measured information flows from crude oil markets to cryptocurrencies. With 
the development of machine learning and deep learning, we can overcome this gap by 
using reinforcement learning with different approaches to solving this puzzle. We, how-
ever, leave this to future studies. Furthermore, our study only focuses on two of the 
biggest crude oil types such as WTI and BRENT while there is a potential avenue for 
examining the relationship between crude oil markets and cryptocurrencies in the devel-
oping economies.
We do identify avenues for future research. As cryptocurrency is an emerging line of 
research and we are observing inefficiency in the trading process, it will be interesting 
to investigate investors’ choices to use one cryptocurrency rather than another. Under-
standing this may not be possible using traditional economics and finance models, and 
hence we suggest a behavioural economic perspective in understanding investors’ pref-
erences in these complex new digital markets.
Appendix robustness check
WTI crude oil robustness check
Causality Full sample Frequency-
connected-
ness
DCOILWTICO → LTC 0.0117*
[0.0028]
0.74
DCOILWTICO → ETH 0.0268***
[0.0043]
0.01
DCOILWTICO → DASH 0.0171**
[0.0032]
1.15
DASH → DCOILWTICO 0.0140*
[0.0033]
0.01
DCOILWTICO → XMR 0.0136*
[0.0032]
0.66
DCOILWTICO → XVG 0.0132*
[0.0033]
13.83
XVG → DCOILWTICO 0.0184**
[0.0035]
7.50
DCOILWTICO → MAID 0.0107*
[0.0031]
2.68
69Annals of Operations Research (2022) 313:47–75 
1 3
Fig. 7  The connectedness between cryptocurrency and crude oil markets
See Fig. 7.
70 Annals of Operations Research (2022) 313:47–75
1 3
BRENT crude oil
Causality Renyi transfer entropy Frequency-
connected-
ness
DCOILBRENTEU → BTC 0.0113*
[0.0025]
0.01
BTC → DCOILBRENTEU 0.0106*
[0.0026]
0.11
DCOILBRENTEU → LTC 0.0116*
[0.0028]
0.09
DCOILBRENTEU → ETH 0.0177**
[0.0041]
0.16
DCOILBRENTEU → XMR 0.0128*
[0.0032]
0.11
DGB → DCOILBRENTEU 0.0152**
[0.0031]
0.02
DCOILBRENTEU → MAID 0.0120*
[0.0032]
0.02
Information on the variable’s full name
This appendix provides the insights of each cryptocurrency in terms of name, market capi-
talization, total supply and all time high in the exchange.
Symbol Name Market capitalization (USD) Circulating supply (unit) Date of issue
BTC Bitcoin 174,918,379,274 18,236,475 01st May 2013
LTC Litecoin 4,769,521,120 64,159,150 01st May 2013
ETH Ethereum 28,589,119,108 109,844,644 8th Aug 2015
XEM NEM 21,240,638 8,999,999,999 3rd Apr 2015
DASH Dash 938,226,050 9,344,247 15th Feb 2014
DOGE Dogecoin 315,532,005 123,461,542,248 19th Dec 2013
XMR Monero 1,398,868,850 17,458,076 21st May 2014
VTC Vertcoin 16,413,384 53,593,747 20th Jan 2014
XVG Verge 69,366,673 16,187,838,743 26th Oct 2014
DGB DigiByte 84,789,677 12,859,912,607 07th Feb 2014
XLM Stellar 1,370,120,659 20,205,010,713 05th Aug 2014
USDT Tether 4,653,741,622 4,642,367,414 07th March 2015
MAID MaidSafeCoin 43,986,645 452,552,412 28th Apr 2014
XRP Ripple 11,642,567,056 43,749,413,421 07th Aug 2013
Our database is updated to 25th February 2020 and retreived from coinmarketcap.com
71Annals of Operations Research (2022) 313:47–75 
1 3
Funding Open Access funding enabled and organized by Projekt DEAL.
Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, 
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