Hideki Takayasu Nobuyasu Ito Itsuki Noda - Proceedings of the International Conference on Social Modeling and Simulation, plus Econophysics Colloquium 2014

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Part I
Financial Market

The Author(s) 2015

Hideki Takayasu , Nobuyasu Ito , Itsuki Noda and Misako Takayasu (eds.) Proceedings of the International Conference on Social Modeling and Simulation, plus Econophysics Colloquium 2014 Springer Proceedings in Complexity 10.1007/978-3-319-20591-5_1

1. Influence Networks in the Foreign Exchange Market

Arthur M. Y. R. Sousa 1 , Hideki Takayasu 2 and Misako Takayasu 1

(1)

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

(2)

Sony Computer Science Laboratories Inc., 3-14-13 Higashigotanda, Shinagawa-ku, Tokyo 141-0022, Japan

Arthur M. Y. R. Sousa

Email:

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 [].

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.

Information about dependence between price fluctuations of currencies is important to understand the foreign exchange market. Several studies try to model this market and access those dependences []. 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 0.1s. 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 [ shows the price P(t) for the market USD/JPY on 2011, March, 09th, before the great earthquake in Japan.

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)

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

(1.1)

so that we obtain a time series for each pair of currencies with the symbols + (price increasing), (price decreasing) and 0 (price unchanged). By comparing two of these time series, we can identify 4 states not containing 0: (+, +), (+, ), (, +) and (, ). The removal of the states with 0, e.g. (+, 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 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).

Table 1.1

Number of states for EUR/USD and other markets on 2011, March, 07th (no time shift)

Market	(+, +)	(+, )	(, +)	(, )	0a
AUD/JPY	3256	2904	2941	3303	851,595
AUD/USD	2425	1707	1591	2332	855,944
CHF/JPY					863,377
EUR/AUD					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					863,801
GBP/CHF					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(+,0), (, 0), (0,0), (0, ),(0, +)

Studies in financial markets commonly use the Pearson correlation coefficient as a measure to infer dependence []. 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 X and Y :

(1.2)

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

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