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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.
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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 [35]. 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-
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... We perform a local analysis in the sign time series through a sliding window procedure: In each window of length w we assume the time series can be described by a stationary sign binary Markov process and compute the probability P of a symmetric process; we can then classify each window as corresponding to a symmetric process or not. Modeling the time series of the sign of price changes as Markov process is a consistent assumption from sampling time 0.4s: It has been shown that the autocorrelation function for the price difference in this market goes to zero after few ticks [40]; also in S5 Appendix we observe that the auto mutual information [41] from the mentioned sampling time has similar behavior for the present data set. Results shown here refer to sampling time 0.4s. ...
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