Fluctuation scaling of quotation activities in the foreign exchange market

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Abstract

We study the scaling behavior of quotation activities for various currency pairs in the foreign exchange market. The components’ centrality is estimated from multiple time series and visualized as a currency pair network. The power-law relationship between a mean of quotation activity and its standard deviation for each currency pair is found. The scaling exponent α and the ratio between common and specific fluctuations η increase with the length of the observation time window Δt. The result means that although for Δt=1(min), the market dynamics are governed by specific processes, and at a longer time scale Δt>100(min) the common information flow becomes more important. We point out that quotation activities are not independently Poissonian for Δt=1(min), and temporally or mutually correlated activities of quotations can happen even at this time scale. A stochastic model for the foreign exchange market based on a bipartite graph representation is proposed.

Introduction

The complexity of economic and social systems has attracted a lot of attention from physicists recently [1], [2], [3], [4], [5], [6], [7], [8]. Collective behavior among interacting agents shows different properties from particles governed by Newtonian laws. However, intriguing universal properties could be found and mathematical models should be considered. This movement, called socio/econo-physics, is expected to bridge a gap between physics and our societies [9].

Financial markets are complex systems which consist of many interacting agents. The progress of understanding information flows among agents sheds light on the states of financial markets, i.e. the states of market participants. The recent accumulation of a massive amount of data about financial markets due to both the development and spread of information and communication technology allows us to quantify the states of financial markets in detail [10], [11]. In fact, the correlation structure of high-frequency financial time series is exhaustively and quantitatively investigated [12], [13]; however, the further the dimension of multiple time series increases, the more difficult it becomes to compute cross-correlations and to recognize them.

On the other hand, several studies in both socio/econo-physics and engineering were focused on the structure of corresponding complex networks, their internal dynamics and the flows of the constituents on them [14], [15], [16], [17]. Menezes and Barabási studied the scaling behavior of constituents’ flows on several constructions such as river networks (water flows), transportation systems (car flows), and computer networks (information flows) [18], [19]. As a result, scaling properties are found for flow fluctuations in such systems. This relationship is known as a fluctuation scaling or Taylor’s power law [20], [21]. Taylor’s power law is known as the scaling relationship between the mean of populations and their standard deviation in ecological systems. The ubiquity of Taylor’s power-law slopes between 1/2 and 1 suggests that there exists an underlying fundamental mechanism affecting the transportation of constituents.

Eisler and Kertész found that there is such a power-law relationship between a mean of traded volumes of stocks and their standard deviation on the New York Stock Exchange, and that the power-law exponent takes nontrivial values between 1/2 and 1 [22]. Jian et al. investigated trade volumes of stocks in the Chinese stock market [23]. They also found a non-universal scaling exponent of different fluctuations from 1/2 and 1. We think that their results provide a method to quantify the states of agents with multiple time series in the financial markets. This method is also useful to quantify the states of market participants from the viewpoint of information flows in financial markets.

The aim of this paper is to investigate information flows in the foreign exchange market (FX market) by means of quotation activities, measured as arrival rates of quotations on brokerage systems. We investigate the statistical properties of quotation activities in the FX market and quantify the total states of the market participants through fluctuation scaling.

The organization of this paper is as follows. Section 2 is a short overview of high-frequency financial data taken for our studies from the FX market. Section 3 is a brief summary of the power-law relationship (Taylor scaling) between a mean of constituents’ flow on a graph and their standard deviations. In Section 4 an empirical analysis of quotation activities is performed. In Section 5 the dependence of scaling exponents on a time window length is examined, and the relationship between the states of market participants in the FX market and the scaling exponents is discussed. Section 6 is devoted to concluding remarks, and addresses possible future studies.

Section snippets

Foreign exchange market

The foreign exchange market is the largest financial market in the world. It is a network consisting of brokers, bank traders, and investors. Recent developments in Information and Communication Technology have led to the spread of electronic trading systems all over the world. As a result, many market participants can directly access the FX market by using computer terminals. Moreover, trading activities are recorded in the computer servers, which perform matching operations among quotations

Fluctuation scaling

It is known [18], [19], [20], [22], [21], [27] that for complex systems consisting of many sites i, the mean values of their internal activities Xi,Δt(t)Xi,Δt=1Qj=0Q1Xi,Δt(jΔt), and their standard deviations σi,Δt=1Qj=0Q1(Xi,Δt(jΔt)Xi,Δt)2, can follow a power-law relationship σi,Δt=CXi,Δtα, where C is a positive constant, and α(1/2α1) denotes a scaling exponent. This power-law relationship is referred to as fluctuation scaling, or Taylors’ law [20], [21].

The limiting values α=1/2

Data analysis

For each currency pair, quotation activities are extracted from a T&S database. We calculate a mean of the quotation activities and their standard deviation for each currency pair following Eqs. (1), (2). As shown in Fig. 2, a power-law relationship between a mean and standard deviation in log–log plots exists. In order to estimate C and α, we used the least-squares method for log10Xi,Δt and log10σi,Δt. From a logarithmic form of Eq. (3), the fitting function is set as log10σi,Δt=αlog10Xi,Δt

Discussion

The nontrivial values of α presented in Fig. 3 may imply that the market participants are affected by information both from different origins relating to microscopic dynamics and from common sources relating to macroscopic dynamics, or may imply that market participants have strong interactions. In other words the arrival rates of quotations on the brokerage systems are not independently Poissonian.

The market participants communicate via electronic brokerage systems and perceive both endogenous

Conclusions

We found the power-law relationship between means of quotation activities for currency pairs at the FX and their standard deviation. The scaling exponents α take values in a range from 0.8 to 0.9; they increase with the time window Δt and vary in time depending on observation days. The nontrivial value of α implies that market participants may be affected by both endogenous and exogenous factors, or that they behave with a strong temporal correlation. The dependence α can be explained by the

Acknowledgements

This work is partially supported by the Kyoto University GCOE Program “Informatics Education and Research for Knowledge-Circulating Society” (A.-H. Sato and M. Nishimura) and the Initiative for Attractive Graduate School Education (M. Nishimura) from the Japanese Ministry of Education, Culture, Sports, Science and Technology, by the EU Grant MMCOMNET, No. 012999 (A.-H. Sato and J.A. Hołyst), and by the Polish Ministry of Science and Higher Education, Grants No. 1P03B04727 and

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