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Chapter 1

Inﬂuence Networks in the Foreign Exchange

Market

Arthur M.Y.R. Sousa, Hideki Takayasu, and Misako Takayasu

Abstract The Foreign Exchange Market is a market for the trade of currencies and

it deﬁnes their relative values. The study of the interdependence and correlation

between price ﬂuctuations of currencies is important to understand this market. For

this purpose, in this work we search for the dependence between the time series

of prices for pairs of currencies using a mutual information approach. By applying

time shifts we are able to detect time delay in the dependence, what enable us to

construct a directed network showing the inﬂuence structure of the market. Finally,

we obtain a dynamic description of this structure by analyzing the time evolution of

the network. Since the period of analysis includes the great earthquake in Japan in

2011, we can observe how such big events affect the network.

1.1 Introduction

The Foreign Exchange Market is a market in which currencies are traded; it is

continuously open during the weekdays and it has the largest transaction volume

among the ﬁnancial markets (average of $5.3 trillion/day in April 2013 [1]). The

importance of this market is that it deﬁnes the relative values of currencies and

affects other markets, such as the stock markets [2].

In this market, traders can make orders for buying and selling which are

organized in the order book according to their corresponding prices. The highest

price of the buy orders in a given time is called best bid and the lowest price of the

sell orders, best ask, and their average deﬁnes the mid-quote; a deal occurs when

the best bid meets the best ask.

A.M.Y.R. Sousa () • M. Takayasu

Department of Computational Intelligence and Systems Science, Interdisciplinary Graduate

School of Science and Engineering, Tokyo Institute of Technology, G3-52 4259 Nagatuta-cho,

Yokohama 226-8502, Japan

e-mail: yamashita.a.ai@m.titech.ac.jp

H. Takayasu

Sony Computer Science Laboratories Inc., 3-14-13 Higashigotanda, Shinagawa-ku, Tokyo

141-0022, Japan

© The Author(s) 2015

H. Takayasu et al. (eds.), Proceedings of the International Conference on Social

Modeling and Simulation, plus Econophysics Colloquium 2014, Springer

Proceedings in Complexity, DOI 10.1007/978-3-319-20591-5_1

3

4 A.M.Y.R. Sousa et al.

Information about dependence between price ﬂuctuations of currencies is impor-

tant to understand the foreign exchange market. Several studies try to model this

market and access those dependences [3–5]. However there are no studies on the

inﬂuence structure in this market and the time evolution of the dependences. To

contribute to ﬁll this gap, we analyse the dependences in foreign exchange data

during a period of 3 weeks using the mutual information, a non-linear dependence

measure from the information theory [6,7]. By doing a time shift analysis we can

infer temporal dependence between markets making possible the construction of

directed networks that show the inﬂuence structure of the foreign exchange market.

1.2 Data and Method

We analyze the foreign exchange data of the Electronic Broking Services (EBS)

by ICAP. This data contains the orders for pairs of currencies in a resolution

of 0.1 s. Here we use the 6 currencies with the largest transaction volume: USD

(United States dollar), EUR (Euro), JPY (Japanese yen), GBP (Pound sterling),

AUD (Australian dollar) and CHF (Swiss franc) in the period between 2011, March,

07th and 2011, March, 25th, each day from 22:00:00 to 21:59:59 GMT. The chosen

period is a special one because it includes the great earthquake in Japan on 2011,

March, 11th and the announcement of the intervention in the foreign exchange

market as a response to the effects of the earthquake on 2011, March, 17th [8].

For this data we deﬁne the price P(t) as the last mid-quote, where t is the real time

in intervals of 0.1 s. As an example of the data, Fig. 1.1 shows the price P(t) for the

market USD/JPY on 2011, March, 09th, before the great earthquake in Japan.

We work with the sign of the difference of price P(t) [9]:

S.t/DsignŒP.t/P.t1/; (1.1)

so that we obtain a time series for each pair of currencies with the symbols C

(price increasing), (price decreasing) and 0 (price unchanged). By comparing

two of these time series, we can identify 4 states not containing 0: (C,C), (C,),

(,C)and(,). The removal of the states with 0, e.g. (C, 0), is an important

step because then we compare the series only when there is activity in both of

them, avoiding issues regarding the volume difference and the time zone difference.

Table 1.1 illustrates the number of occurrence of each state when comparing the

EUR/USD with other markets on 2011, March, 07th (time series of each market

with 863,999 points).

Studies in ﬁnancial markets commonly use the Pearson correlation coefﬁcient as

a measure to infer dependence [5,10]. But the correlation coefﬁcient detects only

linear correlation between two variables, not having information about the depen-

dence. The mutual information on the other hand deals direct with the probability

distributions being a measure not only for linear and non-linear correlations, but also

for dependence. The mutual information is zero if and only if the random variables

1 Inﬂuence Networks in the Foreign Exchange Market 5

Fig. 1.1 Price P(t) for the market USD/JPY on 2011, March, 09th. Here we work with the sign of

the difference of the price P(t)

Tab l e 1 . 1 Number of states

for EUR/USD and other

markets on 2011, March, 07th

(no time shift)

Market (C,C) (C,) (,C) (,) 0a

AUD/JPY 3256 2904 2941 3303 851;595

AUD/USD 2425 1707 1591 2332 855;944

CHF/JPY 125 129 184 184 863;377

EUR/AUD 55 59 66 48 863;771

EUR/CHF 3817 3061 3160 3895 850;066

EUR/GBP 3956 3305 3272 4086 849;380

EUR/JPY 5351 3918 3956 5202 845;572

GBP/AUD 53 47 45 53 863;801

GBP/CHF 43 47 56 52 863;801

GBP/JPY 4791 4431 4238 4807 845;732

GBP/USD 3088 2359 2533 3134 852;885

USD/CHF 2874 3656 3689 3032 850;748

USD/JPY 5822 7131 7081 5743 838;222

a(C,0),(, 0), (0, 0), (0, ), (0, C)

are independent. There are evidences that mutual information can reveal aspects

ignored by the correlation coefﬁcient and studies comparing both measures [11–

13]. Another reason for using mutual information in this work is that we are dealing

with symbolic series, in which the numerical values that are taken in account for the

correlation coefﬁcient have no meaning.

The mutual information I(X;Y) between two random variables Xand Y:

I.XIY/DX

x

X

y

p.x;y/log p.x;y/

p.x/p.y/;(1.2)

6 A.M.Y.R. Sousa et al.

which can also be expressed in term of the entropies H:

I.XIY/DH.X/H.XjY/(1.3)

or

I.XIY/DH.Y/H.YjX/: (1.4)

H(X) is the entropy of the random variable Xand can be understood as a

measure of its uncertainty. Similarly, H(X jY) can be seen as the uncertainty of X

given Y. Thus, one interpretation for the mutual information is the reduction in the

uncertainty of a random variable given the knowledge of the other. If the variables

are independent, the knowledge of one variable does not give information about the

other and then the mutual information is zero.

The ﬁnal dependence measure we use is the global coefﬁcient:

.XIY/Dp1e2I.XIY/;(1.5)

This quantity has desired characteristics for a dependence measure, as taking

value zero for independent variables and being in the range [0;1] [14], and has been

used in ﬁnancial data [12].

In order to compute the global coefﬁcient of the ﬁnancial series, we estimate the

probability of each state using the relative frequency in a time window of 1day.

We also determine a signiﬁcance level to decide if the computed coefﬁcient is

signiﬁcantly different from the one of a random series; we randomize the analysed

series and calculate the global coefﬁcient until it reaches a stationary value which

corresponds to the coefﬁcient for the corresponding random series and we take this

value as the signiﬁcance level.

1.3 Results and Discussion

For each two pairs of currencies we compute the global coefﬁcient for their sign time

series as function of the time shift between them. For this data, we ﬁnd four general

types of structures according to the presence of peaks that represent dependence

between the markets, as illustrated in Fig. 1.2.

• No peak: no dependence between markets.

• Peak at time shift zero: both markets are synchronized. External inﬂuences (e.g.

economic news) make the markets to have similar behaviour, the change in the

price occurs simultaneously in both markets.

1 Inﬂuence Networks in the Foreign Exchange Market 7

Fig. 1.2 Examples of results for the time shift cross-analysis. (a) GBP/JPY and USD/CHF on

2011, March, 09th: no dependence between the markets, same result for random time series. (b)

EUR/JPY and GBP/USD on 2011, March, 09th: dependence at time shift 0. (c) AUD/JPY and

USD/JPY on 2011, March, 09th: dependence when the USD/JPY series is shifted 0.1 s forward in

relation to the AUD/JPY series. (d) EUR/CHF and USD/CHF on 2011, March, 09th: dependence

at t ime shift 0.1 s in both directi ons. Dotted lines indicate the signiﬁcance level

8 A.M.Y.R. Sousa et al.

• Peak at a time shift different of zero: one market inﬂuences the other, i.e., there is

an internal inﬂuence. This means that the past of one market affects the present

of the other market, which could be interpreted as an information ﬂow.

• Two peaks at time shifts in both directions: there are also internal inﬂuences, but

in this case both markets affect each other during the analysed period.

We can build an inﬂuence network deﬁning the pairs of currencies as nodes and

adding the links according to the time shift cross-analysis between the markets that

correspond to the nodes: (a) no peak: no link; (b) peak at time shift zero: undirected

link; (c) peak at a time shift different from zero: directed link from the market that

inﬂuences the other one, i.e., the market that goes ahead, whose past values affects

the present values of the other market; (d) two peaks at time shifts in both directions:

extraverted link.

We proceed with this analysis for all weekdays from 2011, March, 07 to 2011,

March, 25. In this period two important events took place: the great earthquake in

Japan on March, 11 and the announcement intervention in the foreign exchange

market on March, 17. Figures 1.3,1.4 and 1.5 show the time evolution of the

inﬂuence network with day resolution during those 3weeks. Figure 1.6 shows the

time evolution of the different types of links in the inﬂuence network.

We observe that the structure does not present major changes within the ﬁrst

week from March, 07th to March, 10th, before the earthquake in Japan. Some

characteristic features are: (a) EUR/USD and USD/JPY are the nodes with higher

out-degree, meaning those are the markets that always go ahead being followed

by the others, and (b) almost no extraverted links (with exception of link between

USD/CHF and EUR/CHF, which is always present), i.e., information ﬂows only in

one direction, creating a hierarchy of importance between the markets.

From March, 11th (ﬁrst week) to March, 17th (second week), which corresponds

to the period between the earthquake in Japan and the intervention, we notice that

the inﬂuence network changes compared to the structure in the ﬁrst week. An

important change is the increase in the number of directed and extraverted links,

suggesting the interdependence between markets becomes stronger (not only due

external inﬂuences, but internal ones). The new extraverted links that appeared

involve the nodes EUR/USD and USD/JPY, that continue being the most important

nodes (highest out degree), but now they are also inﬂuenced by other markets. One

possible interpretation is that the players of these important markets are now being

more careful, waiting for the information of other markets to decide to change the

price.

After the announcement of the intervention on March, 17th, we observe another

change in the structure, specially the disappearance of the extraverted link between

EUR/USD and USD/JPY. Gradually the inﬂuence network returns to a structure

similar to the one of the ﬁrst week (before the earthquake).

Those results suggest that the event of the earthquake affected the dependence

between markets and the event of the announcement of the intervention contributed

for the return of the market to a state previous the earthquake, i.e., it was efﬁcient in

the sense of reversing the changes caused by the earthquake in the foreign exchange

1 Inﬂuence Networks in the Foreign Exchange Market 9

Fig. 1.3 Inﬂuence Networks of the Foreign Exchange Market for the currencies USD, EUR, JPY,

GBP, AUD and CHF from 2011, March, 07th to 2011, March, 11th. The Great Earthquake in

Japan took place on 2011, March, 11th. In this network nodes represent the pairs of currencies and

there are three types of links according to the time shift cross-analysis: (i) undirected link (gray)

corresponding to peak at time shift zero; (ii) directed link (black), peak at a time shift different

from zero, in this case 0.1 s, from the market that inﬂuences the other one; (iii) extraverted link

(red), two peaks at time shifts, also 0.1 s, in both directions

10 A.M.Y.R. Sousa et al.

Fig. 1.4 Inﬂuence Networks of the Foreign Exchange Market for the currencies USD, EUR, JPY,

GBP, AUD and CHF from 2011, March, 14th to 2011, March, 18th. The Intervation in the Foreign

Exchange Market was announced in the end of 2011, March, 17th

1 Inﬂuence Networks in the Foreign Exchange Market 11

Fig. 1.5 Inﬂuence Networks of the Foreign Exchange Market for the currencies USD, EUR, JPY,

GBP, AUD and CHF from 2011, March, 21st to 2011, March, 25th

12 A.M.Y.R. Sousa et al.

Fig. 1.6 Time evolution of the number of the different types of links in the inﬂuence network from

2011, March, 07th to 2011, March, 25th. Dotted lines indicate the number of links on 2011, March,

07th

market. It is possible that other factors besides the intervention contributed to the

stabilization of the market; to discuss this aspect, it would be necessary the analysis

of other periods where stability was reached with no intervention.

1.4 Final Remarks

In this paper we used a non-linear dependence measure based on the mutual

information to access the dependence between pairs of currencies of the foreign

exchange market. We analysed the sign of price difference of these markets from

2011, March, 07th to 2011, March, 25th, a period that includes the great earthquake

in Japan and the intervention. By applying a time shift between the sign series

we obtained different dependence structures between markets and then constructed

an inﬂuence network based on them. The analysis of the inﬂuence network and

its time evolution showed that the markets EUR/USD and USD/JPY are the most

important nodes, with the information ﬂowing from them to the other markets. It

also suggested that the event of the earthquake changed the inﬂuence structure of

the network, intensifying the interdependence between markets and changing the

dynamics of the markets EUR/USD and USD/JPY; and the announcement of the

intervention was effective in reverting the effects of the earthquake: changes could

be observed in the day rightafter the announcement and the network totally returned

to the state previous the earthquake in less than 1 week. The results represent a

contribution to understand how the foreign exchange market reacts to big events

and thus what can be done in periods of crisis. The analysis can also be useful to

1 Inﬂuence Networks in the Foreign Exchange Market 13

predict the behavior of one market based on the past behavior of another, if there is

an inﬂuence relationship between them.

One important observation is that in the time shift cross-analysis the typical time

shift is 0.1 s, i.e., when we have a market inﬂuencing another the time delay is 0.1 s.

This fact is possibly related to the resolution of the data, also 0.1 s. We analysed

the same data but with resolution 1s and could not detect time delay between

markets as we found for resolution 0.1 s. We still need to study if we can detect

the directionality between markets in other time resolution data or if the resolution

0.1 s is essential to detect such feature. Further researches also should include other

currencies, a larger period of analysis and the possibility of time windows smaller

than 1 day.

Open Access This book is distributed under the terms of the Creative Commons Attribution Non-

commercial License which permits any noncommercial use, distribution, and reproduction in any

medium, provided the original author(s) and source are credited.

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