Thesis Econophysics on Interactions of Markets Yukihiro Aiba Department of Physics, Graduate School of Science, University of Tokyo December 2005 Acknowledgments I would like to express my sincere gratitude to Professor Naomichi Hatano, for his guidance, useful discussions, and encouragements. Without his cordial support, this work has never been performed. I also would like to express my sincere gratitude to Dr. Hideki Takayasu for his guidance and fruitful discussions. I am grateful to Prof. Hajime Takayama, Prof. Miki Wadati, Prof. Kazuyuki Aihara, Prof. Shinichi Sasa and Prof. Naoki Kawashima for their critical reading of the manuscript and useful comments. I would like to thank all members of Hatano Laboratory for stimulating discussions and encourage- ments. In particular, I would like to thank Mr. Tetsuro Murai, Ms. Junko Ya- masaki, Mr. Masahiro Kawakami, Mr. Kouhei Oikawa, Mr. Kenji Kawamura, Dr. Manabu Machida, Dr. Shunji Tsuchiya, Dr. Akinori Nishino, Dr. Keita Sasada, Dr. Yuichi Nakamura, Mr. Masashi Fujinaga and Mr. Naoya Sato for stimulating discussions and their encouragements. I acknowledge the ¯nancial support from University of Tokyo 21st Cen- tury COE Program \Quantum Extreme System and Their Symmetries." Finally, I thank my family and all of my friends for their continual en- couragement and support. i Contents 1 Introduction: What is Econophysics? 1 1.1 Economic systems as strongly correlated many-body systems . 1 1.2 Scaling properties of ¯nancial prices . 2 1.3 Econophysics of wealth distributions . 3 1.4 An example of modeling ¯nancial fluctuations using concepts of statistical physics . 6 1.4.1 Sznajd model . 6 1.4.2 Sato and Takayasu's dealer model . 10 1.5 Summary . 14 1.6 The contents of the thesis . 18 2 Triangular Arbitrage as an Interaction among Foreign Ex- change Rates 19 2.1 Introduction . 19 2.2 Existence of triangular arbitrage opportunities . 20 2.3 Feasibility of the triangular arbitrage transaction . 21 3 A Macroscopic Model of Triangular Arbitrage Transaction 29 3.1 Macroscopic model of triangular arbitrage . 29 3.1.1 Basic time evolution . 30 3.1.2 Estimation of parameters . 32 3.1.3 Analytical approach . 36 3.2 Negative auto-correlation of the foreign exchange rates in a short time scale . 38 3.3 What makes the rate product converge . 40 4 A Microscopic Model of Triangular Arbitrage Transaction 43 4.1 Introduction . 43 iii iv 4.2 Microscopic model of triangular arbitrage . 44 4.2.1 Microscopic model of triangular arbitrage: interacting two systems of the ST model . 44 4.3 The microscopic parameters and the macroscopic spring constant 48 5 Summary 55 Chapter 1 Introduction: What is Econophysics? 1.1 Economic systems as strongly correlated many-body systems Systems consisting of many interacting units such as strongly correlated many-body systems are of great interest of statistical physics. In such sys- tems, exotic phenomena like phase transitions occur, but we cannot see them emerging if we look at each unit separately. Statistical physics treats the interacting units as a whole and thereby have successfully elucidated the mechanism of the phenomena. Economic systems obviously consist of a large number of interacting units. Thus one may expect it possible that methods and concepts developed in the study of strongly correlated systems may yield new results in economics. In fact, some empirical laws are founded and models aiming to reproduce such phenomena are constructed, using the methods and the concepts developed in statistical physics. Econophysics is a word used to describe work being done by physicists in which ¯nancial and economic systems are treated as complex systems [1, 2]. Many physicists have contributed to quantifying and modeling economic fluctuations in recent years. The content of this chapter is in preparation for submission. 1 2 Chapter 1 1.2 Scaling properties of ¯nancial prices Mandelbrot, who is famous as the advocator of the concept of fractal, origi- nally found a self-similar structure by analyzing the fluctuations of the cotton price in a commodity market [3]. Recently, Mantegna and Stanley [4, 5, 6, 7] found a scaling law in the fluctuations of a stock index. The stock index is a weighted average of the stock prices. Speci¯cally, Mantegna and Stanley used a stock index called the S&P 500. They analyzed the price fluctuation of the S&P 500 as follows. Let G(T ) be the logarithm of the price change in a time step T [min]: G(T ) = ln Y (t) ¡ ln Y (t ¡ T ); (1.1) where Y (t) is the price at time t. The value G is often called `return.' Mantegna and Stanley drew the histograms P (G) for the time steps T =1, 3, 10, 32, 100, 316, 1000 [min] (Fig. 1.1). The shape of the histogram of course depends on T ; it spreads as T increases. However, the histograms for various values of T collapsed onto one curve by scaling G G~ ´ (1.2) T 1=¯ and P (G~) P~(G~) ´ ; (1.3) T ¡1=¯ where ¯ = 1:4. This fact means that the price fluctuations have a self-similar structure often found in critical phenomena in physical systems. Gopikrishnan et al. [8, 9] later analyzed a database documenting each and every trade in the three major US stock markets, the New York Stock Exchange (NYSE), the American Stock Exchange (AMEX), and the National Association of Securities Dealers Automated Quotation (NASDAQ) for the entire two-year period. They thereby extracted a sample of approximately 4 million data points, which is much larger than the 500 thousand data points analyzed by Mantegna and Stanley, and the 2000 data points studied by Mandelbrot. Gopikrishnan et al. found an asymptotic power-law behavior with an exponent ¯ ' 3 for the cumulative distribution (Fig. 1.2). They refer to this phenomenon as an `inverse cubic law' [10]. The power-law behavior was also found in the foreign exchange markets [11, 12, 13]. These results motivated many physicists to analyze ¯nancial fluctuations and to ¯nd `universality' in the economic system in recent years. Section 3 3 (a) (b) ) ) ~ G G ( ( ~ P P log log ~ G G Figure 1.1: (a) The probability density functions of price changes measured at di®erent time horizons T =1, 3, 10, 32, 100, 316, 1000 minutes. The distributions spread with increasing T . (b) The same data as in (a), but plotted in scaled units. The distributions collapse well onto the distribution for T =1 [min]. Both graphs are adapted from [4]. 1.3 Econophysics of wealth distributions Another important topic in econophysics is a power-law behavior of wealth distributions [14]. Here, the `wealth' means the income of individuals, the size of business ¯rms or the GDP of countries. The fact that wealth distributions have power-law tails has been recog- nized for over 100 years. Pareto [15] investigated the statistics of the wealth of individuals by modeling them as a scale-invariant distribution f(x) » x¡γ; (1.4) where f(x) denotes the number of people having income equal to or greater than x, and γ is an exponent that Pareto estimated to be 1:5. Nowadays, many works have analyzed the data of personal income and modeled them [16]{[20]. The size distributions of business ¯rms also obey the power law. Okuyama et al. [21] analyzed the income of business ¯rms in Japan and Italy. Figure 1.3 is a logarithmic plot of the distributions of the income of Japanese and Italian companies. The data for the Japanese ¯rms can be approximated by 4 Chapter 1 β Figure 1.2: The cumulative distribution of normalized daily price changes. The price change is often called `return.' This graph is adapted from [10]. Section 3 5 Figure 1.3: The cumulative distribution of the income of Japanese companies (the bold line with x a million yen) and Italian ¯rms (the dashed line with x a hundred thousand lira). The two straight lines show the power law with the exponent ¡1, namely Zipf's law. This graph is adapted from [21]. a straight line with slope ¡1 in the range of income less than 105; this means that the distribution follows a power law with an exponent very close to ¡1, namely Zipf's law. The data for the Italian ¯rms are roughly on a straight line with the same slope ¡1, but the Italian data deviate from the straight line in comparison to the Japanese data. Okuyama et al. concluded that this was because of the lack of data of smaller companies. Furthermore, M.H.R. Stanley et al. [22, 23] calculated histograms of how the ¯rm size changes from one year to the next. They made 15 histograms for each of 15 bins of the ¯rm size. The largest ¯rms have very narrow distributions of growth rates. This means that the percentage of the size 6 Chapter 1 change from year to year for the largest ¯rms cannot be so great. A tiny ¯rm, on the other hand, can radically increase or decrease in size from year to year. These 15 histograms thus have widths that depend on the ¯rm size. The width showed a power law of the ¯rm size with an exponent ¸ ' 1=6 over 8 orders of magnitude, from the tiniest ¯rm to the largest ¯rm [22, 23]. The growth rate therefore can be normalized and the data collapse on a single curve. This scaling property can be extended to the growth rate of countries by analyzing the GDP. Lee et al. [24] found that the histograms of the country size in the GDP behave the same way as the histograms of the ¯rm size. They analyzed the annual growth rate R ´ ln[g(t + 1)=g(t)], where g(t) is the GDP of a country in the year t.