Physica A 569 (2021) 125770

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Physica A

journal homepage: www.elsevier.com/locate/physa

Quantifying the randomness of the forex market

Alfonso Delgado-Bonal

a,1,

∗

, Álvaro García López

b,1

a

Gentleman Scientist, 53C Crescent Road, Greenbelt, MD, USA

b

Universidad Rey Juan Carlos, Nonlinear Dynamics, Chaos and Complex System Group, Madrid, Spain

a r t i c l e i n f o

Article history:

Received 18 October 2020

Received in revised form 8 January 2021

Available online 18 January 2021

Keywords:

Complexity

Econophysics

Information Theory

Randomness

a b s t r a c t

Currency markets are international networks of participants opened all day during

weekdays without a supervisory entity. The precise value of an exchange pair is

determined by the decisions of the central banks and the behavior of the speculators,

whose actions can be determined on the spot or be related to previous decisions.

All those decisions affect the complexity and predictability of the system, which are

quantitatively analyzed in this paper. For this purpose, we compare the randomness

of the most traded currencies in the forex market using the Pincus Index. We extend

the development of this methodology to include multidimensionality in the embedding

dimension, to capture the influence of the past in current decisions and to analyze

different frequencies within the data with a multiscale approach. We show that, in

general, the forex market is more predictable using one hour ticks than using daily data

for the six major pairs, and present evidence suggesting that the variance is easier to

predict for longer time frames.

© 2021 Elsevier B.V. All rights reserved.

1. Introduction

The foreign exchange market (forex) is the largest market in the world and determines the exchange rates for every

currency. The participants in the market set the relative value of each currency pair by buying or selling positions, and

these actions are influenced by personal beliefs, trends or public announcements of central banks.

Unlike stock markets, forex is not being subject to an specific supervisory entity and is globally decentralized, open

to banks, commercial companies and private agents. The price of a currency pair at a given time is supposed to be a

reflection of economic factors, political conditions and the psychology of the participants.

The six most traded forex pairs analyzed in this paper are: EUR/USD (euro/US dollar); USD/JPY (US dollar/Japanese

yen); GBP/USD (British pound sterling/US dollar); AUD/USD (Australian dollar/US dollar); USD/CAD (US dollar/Canadian

dollar); USD/CNY (US dollar/Chinese renminbi).

The forex market has been the subject of intensive research for a long time, frequently focusing on the predictability

of its values but also on its variance. In this paper, we analyze the number of patterns within the data for the returns and

its variance in different timeframes, 1-hour (H1), 4-hours (H4) and daily.

Research on complexity includes a variety of algorithms and analysis techniques that usually come from the physical

or mathematical realm [1]. Complex systems are entangled by nonlinearly interacting elements and are found in different

fields such as the brain or financial markets. A review on the meaning of complexity and detailed analyses in those and

other processes can be found in [2].

∗

Corresponding author.

E-mail address: contact@adelgadobonal.com (A. Delgado-Bonal).

1

Both authors contributed equally to this work.

https://doi.org/10.1016/j.physa.2021.125770

0378-4371/© 2021 Elsevier B.V. All rights reserved.

A. Delgado-Bonal and Á.G. López Physica A 569 (2021) 125770

When dealing with the currency market, its movements can be separated into high-frequency variations [3] and slower

movements responsible for the trends [2]. This paper deals with the later, and explores its complexity relying on the

concept of entropy as defined in Information Theory and Kolmogorov complexity.

In the field of Information Theory, entropy is a magnitude that quantifies the uncertainty of a measure. On the

other hand, this paper follows the approach of Chaitin [4] and Kolmogorov [5] by defining complexity as in algorithmic

information theory, which takes into account the order of the points in a sequence. In this view, a chain is random if its

Kolmogorov complexity is at least equal to the length of the chain.

The benefit of the connection between information content and randomness is that it provides a way to quantify the

complexity of a dataset without relying on models or hypothesis about the process generating the data. By comparing the

entropy of our system with the maximum entropy rate possible, we can determine the degree of randomness of a series;

a complex (total random) process is defined as that process lacking pattern repetition.

The use of the entropy rate to study the complexity of a time series is not limited to stochastic processes. Sinai [6]

introduced the concept of entropy to describe the structural similarity between different dynamic systems that preserve

the measurements, giving a generalization of Shannon entropy for dynamic systems, known as Kolmogorov–Sinai entropy

(KS). Unfortunately, KS entropy is sometimes undefined for limited and noisy measurements of a signal represented in a

data series.

To overcome that limitation, Grassberger and Procaccia [7] used the Renyi entropy to define the correlation integral,

which in turn was used by Eckmann and Ruelle [8] to define the φ functions as a conditional probability. This ER entropy

is an exact estimation of the entropy of the system. Building upon those φ functions, Pincus [9] described the methodology

of ApEn, useful for limited and noisy data, providing a hierarchy of randomness based on the different patterns and their

repetitions.

ApEn measures the logarithmic probability that nearby pattern runs remain close in the next incremental comparison:

low ApEn values reflect that the system is very persistent, repetitive and predictive, with apparent patterns that repeat

themselves throughout of the series, while high values means complexity in the sense of independence between the data

and a low number of repeated patterns. The readers are encouraged to read a recent comprehensive tutorial on these

algorithms [10].

To use Approximate Entropy, it is necessary to specify two parameters, the embedding dimension (m) and the tolerance

of the measure (r), determined as a percentage of the standard deviation. Once the calculations have been performed, the

result of the algorithm is a positive real number, with higher values indicating more randomness. However, those values

are dependent on the characteristics of the dataset such as the influence of the past in the future prices or the volatility

of the prices.

In order to obtain a measure of randomness suitable for comparisons between evolving datasets, the Pincus Index (PI)

was introduced as a measure of the distance between a dataset and the maximum possible randomness of that system [11].

A value of PI equal to zero implies a totally ordered and completely predictable system, whereas a value equal to or greater

than one implies total randomness and unpredictability. The added benefit of the Pincus Index is that, unlike ApEn, it is

suitable for comparisons between different markets. This paper completes the development of that index by introducing

different kinds of multidimensionality in the measure. Thus, knowledge of the PI would be useful to fully understand the

concepts here presented and how the several levels of complexity of this measure are captured.

The Pincus Index was designed [11] to be independent on the parameter r by choosing the maximum value of

Approximate Entropy (MaxApEn), but the index is still dependent on the selection of the embedding dimension (m). This

parameter is related to the memory of the system and accounts for the length of the patterns compared in the sequence.

Techniques to determine the optimum value of the embedding dimension include the use the mutual information and false

nearest neighbor method [12–14], but since different markets may have different embedding dimensions, the comparisons

with a fixed m could be biased. To account for that possibility, we follow Bolea and coauthors [15] in the definition of

a Multidimensional index. Since such an index was based on MaxApEn, its extrapolation to a Multidimensional Pincus

Index is straightforward and provides a parameter-free index which allows for comparisons between evolving systems.

Besides multidimensionality in embedding dimension, dynamic systems may be composed of processes at different

frequencies with correlations at multiple time scales. Therefore, in the characterization of complexity, the comparison

of different frequencies may lead to incorrect conclusions. Costa and coauthors [16] proposed a multiscale procedure

to capture those correlations, showing its efficiency in distinguishing complexities in different dynamical regimes. To

describe the complexity of a time series at different levels, Costa and coauthors [17] generalized the multiscale procedure

to consider the complexity of higher statistical moments of time series. Here, we extend that methodology to create a new

Multiscale Pincus Index, showing how it is useful to correctly quantify the complexity of trading in different timeframes

and different statistical moments.

2. Methods and results

2.1. On the calculation of the Pincus Index

The Pincus Index (PI) captures the distance from a situation of total randomness for a given dataset, measured against

shuffled versions of the same data. To better quantify complexity and provide an index that is independent of the tolerance

2

A. Delgado-Bonal and Á.G. López Physica A 569 (2021) 125770

Fig. 1. ApEn, SampEn, and asymptotic lines depending on the alphabet for 50 pseudo-random binary (left) and decimal (right) sequences.

r, it is constructed based on the maximum value of Approximate Entropy (MaxApEn). The steps to compute the PI include

the determination of the MaxApEn of the original sequence and the MaxApEn of bootstrapped versions. Then, we use the

median value (50% percentile) of the empirical distribution of the bootstrapped versions to calculate the value of the

Pincus Index, and the 5% and 95% percentiles of the empirical cumulative distribution function to calculate the extremes

of the index. The rationale is simple: if the degree of randomness of the original sequence is similar to the shuffled

versions, the PI will be close to one, indicating randomness. If, on the other hand, the original sequence is ordered, the

PI will capture the distance from randomness as a fraction. For a detailed explanation of the methodology and several

examples of application, the reader is encouraged to see [10,11].

The Pincus Index is based on Approximate Entropy. When the number of data (N) is large, ApEn can be approximated

by Eq. (1). The error committed in this approximation is estimated to be smaller than 0.05 for N −m+1 > 90 and smaller

than 0.02 for N − m + 1 > 283 [18].

ApEn(m, r, N) ≃

−

1

N − m

N−m

i=1

log

N−m

j=1

[times that d[|x

m+1

(j) − x

m+1

(i)|] < r]

N−m

j=1

[times that d[|x

m

(j) − x

m

(i)|] < r]

.

(1)

where m is the length of the vectors being compared, and d measures the scalar distance between the vectors in a

component-wise way.

The Sample Entropy (SampEn) algorithm has been designed to avoid the self-bias included in ApEn [18], which is

mathematically formulated as [10]:

SampEn(m, r, N) =

− log

N−m

i=1

N−m

j=1,j=i

[times that d[|x

m+1

(j) − x

m+1

(i)|] < r]

N−m

i=1

N−m

j=1,j=i

[times that d[|x

m

(j) − x

m

(i)|] < r]

(2)

It is often said that SampEn is largely independent on the number of points because, unlike ApEn, it does not include

a prefactor

1

N−m

. However, it must be noticed that such independence is only true for homogeneous series, and it does

not hold for general situations [19]. In general, randomness depends for both algorithms on m and N [20].

In Fig. 1 we show the different behavior of ApEn and SampEn for 50 pseudo-random binary (left) and decimal (right)

chains using m = 2; we use r < 1 to make the analysis independent of this parameter, given their well-defined alphabet.

As it can be seen, the mean value of SampEn reaches the asymptotic limit of log k faster but with a larger standard

deviation than ApEn.

In the construction of the Pincus Index, we calculate the ratio MaxApEn

original

/MaxApEn

shuffled

. Since those quantities

are calculated using the same values of m and N, the ratio between them does not include the prefactor

1

N−m

appearing

in ApEn in Eq. (1). This fact makes the PI independent of the number of points in the same way that SampEn (i.e., for

white noise, or a homogeneously generated sequence, it captures the randomness independently of N).

Another reason for the construction of SampEn was the self-counting introduced in the calculation of ApEn: note that

the definition of SampEn explicitly avoids that situation by limiting j = i in Eq. (2). That bias can be as high as 20% or 30%

if the number of points is low [18]. In this regard, since the PI is constructed as a ratio and the bias in ApEn is present in

both the nominator and the denominator, the overall bias is modulated and severely corrected, providing a better measure

of complexity. It should be emphasized that the PI does not measure randomness specifically but how far away a series

is from total randomness.

3

A. Delgado-Bonal and Á.G. López Physica A 569 (2021) 125770

Fig. 2. Value of r for which MaxApEn is reached for the original series (red line) and the average of one hundred shuffled versions (black line). The

average of the standard deviations (bars) for the shuffled versions is 0.038 and increases with the embedding dimension.

2.2. The threshold: r

max

and MaxApEn

The use of an incorrect parameter selection when using ApEn or SampEn can lead to inaccurate estimations of the

complexity of datasets. By means of MaxApEn we can prevent the arbitrary selection of the threshold of r, which changes

depending on the complexity of the sequence.

Restrepo et al. [21] showed that the combined use of MaxApEn and r

max

can help to correctly characterize the

complexity. Using a dataset containing daily values of EURUSD from 2006 to 2010, we show in Fig. 2 that r

max

changes with

the embedding dimension selected. The distance between the maximum value of the threshold for the original (red line)

and the shuffled series (black line) shows that using a fixed common valued for the threshold would lead to misleading

results. Thus, even though the recommended range for r is commonly [0.1σ , 0.25σ ], that region does not guarantee to

capture the complexity correctly for all the values of the embedding dimension. It is advised to use a value equal to or

greater than the value of r

max

[10] to assure the relative consistency; the comparison with different values of r beyond

the maximum would lead to the same qualitative characterization of the order of the system.

As explained in [21], the differences in r

max

for the original and shuffled versions can be used as a mean to discern

between systems in noisy datasets with low number of samples N. Albeit in this work we focus on the development of

the Pincus Index, we take the opportunity to recall that, for some dynamical regimes, these combined techniques could

provide a better characterization of the systems, and show that the recommended range may not be adequate depending

on the embedding dimension m.

2.3. The embedding dimension: multidimensional analysis

In the methodology to calculate the Pincus Index, the tolerance r is automatically selected as the value which

maximizes Approximate Entropy [11]. However, the selection of the embedding dimension is a requirement for the

calculations. The embedding dimension determines the length of the patterns being compared, and it is related to how

much information from the past is used to determine the future values. In the search of a parameter-free application

of Approximate Entropy, Bolea et al. [15] proposed the use of MaxApEn combined with a multidimensional analysis by

adding the contribution of MaxApEn over a wide range of embedding dimensions to capture the influence of previous

values.

Since a priori the memory of the system is unknown and it may change in evolving datasets like the forex markets,

adopting the same methodology as Bolea and coauthors, it is straight forward to build a Multidimensional Pincus Index

(MPI) independent of both r and m, by defining:

MPI =

m

max

m

i

=1

MaxApEn

original

(m

i

)

m

max

m=1

MaxApEn

shuffled

(m

i

)

(3)

We illustrate the behavior of this new multidimensional index in Fig. 3 using the EURUSD exchange rate as an example.

Fig. 3 (left) shows the MaxApEn for different embedding dimensions for the original series (black line) and pseudo-

randomized versions of the same data (red) for a dataset containing daily values of EURUSD from 2006 to 2010. Based

on those values, we show how the MPI changes when we consider only the previous m values using Eq. (3). We observe

that, by adding the contribution of larger embedding dimensions, the MPI varies to capture the increased information

in the complexity of the series. The right side of Fig. 3 shows the MPI accounting for the contribution of the embedding

dimension up to 15 (MPI(m

max

= 15)) for rolling windows of four years of EURUSD daily exchange rate, i.e., approximately

N ∼ 1000 points.

4

A. Delgado-Bonal and Á.G. López Physica A 569 (2021) 125770

Fig. 3. Multidimensional Pincus Index.

The rationale for the inclusion of multidimensionality is its ability to capture complexity in a greater extent, as shown

by Bolea et al. [15]. Specifically for the forex or stock markets, or when drawing comparisons between different systems, it

is not guaranteed that the optimal value of the embedding dimension would be the same. In general, randomness depends

on m and N, as shown by Pincus and coauthors when they defined the maximum {m, N}-randomness [20,22]. The value

of the Pincus Index is its aptness to make comparisons between systems by measuring the distance of each series against

the maximum randomness of each alphabet. We shall remember that both ApEn and SampEn provide relative values, and

may be unsuitable for comparisons. By including multidimensionality in the definition of the MPI, we obtain an index

independent of preselected parameter values for both r and m which can be used with evolving datasets.

2.4. Sampling frequency: multiscale entropy

Another variable must be taken into account in order to capture complexity in all of its forms, which is the different

frequencies within the data. It is not uncommon for dynamical systems to be composed of subprocesses emerging at

different time scales. That situation is often observed in the markets when the trend at a certain frequency domain, let

us say 15 min, is not the same (or even the complete opposite) as the trend at 1 day data.

To account for that possibility, Costa et al. [16] designed the multiscale entropy (MSE) procedure based on the approach

proposed by Zang [23,24]. This measure is based on a weighted sum of scale dependent entropies, and it has been used

extensively since its appearance in the literature [25]. The main idea is the construction of coarse-grained time series

determined by a certain scale factor τ , averaging different time scales from the original time series. The coarse-graining

procedure reduces the length of the sequence by a scale factor τ , obtaining a coarse-grained time of length N/τ , with

N the original length. Thus, the larger the scale factor used, the shorter the resulting length of the coarse-grained time

series.

This procedure has become a prevailing method to quantify the complexity of data series and it has been vastly applied

in many different research fields, including finances [26]. After the creation of the coarse-grained sequences, the entropy

of each sequence is calculated and added up to obtain a multiscale entropy value. More detailed instructions of the

methodology can be found in Costa and coauthors works [16,27].

The MSE methodology has generally been applied in conjunction to Sample Entropy, given the above-mentioned fact

that is less dependent on the time series length since it does not include a prefactor in Eq. (2). However, similarly to

the comparison of different time series, different time scales may have different alphabets and the comparisons using

the same parameters may be biased. Some traded time frames will show higher variability, while the variations at

different frequencies may show lower changes and averaged values. Furthermore, Sample Entropy uses a fixed value

of the tolerance filter r which may not be adequate for all frequencies. This hinders the applicability of Sample Entropy

to characterize the randomness level appropriately.

As seen in the previous sections, the value of r which captures the maximum complexity is different for each sequence;

since the Pincus Index is based on MaxApEn, which automatically adapts to the maximum complexity of each frequency,

this index is able to capture the distance from total randomness of the different frequencies. As an example, Fig. 4 shows

the results of the Pincus Index for values traded Daily, at four hours (H4) and at one hour (H1) frequencies for m = 2.

We present the results for the six major traded pairs to display the evolution of the different frequencies.

In a previous communication we showed the effect of including more or less number of points (see Supplementary

Information in [11]). In this paper, our interest lies on characterizing the different frequencies knowing that approximately

the same number of points are used to draw comparisons about their complexity. To that end, we use rolling windows of

approximately N ∼ 1000 for all frequencies showed in Fig. 4, corresponding to four years in daily values, eight months in

5