Computer Physics Communications 121–122 (1999) 145–152 www.elsevier.nl/locate/cpc Econophysics: can statistical physics contribute to the science of economics? L.A.N. Amaral, P. Cizeau, P. Gopikrishnan, Y. Liu, M. Meyer, C.-K. Peng, H.E. Stanley 1 Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215, USA Abstract We address a current question in econophysics: Are fluctuations in economic indices correlated? To this end, we analyze 1-minute data on a stock index, the Standard and Poor index of the 500 largest stocks. We extend the 6-year data base studied by Mantegna and Stanley by including the 13 years 1984–1996 inclusive, with a recording frequency of 15 seconds. The total number of data points in this 13 years period exceed 4.5 million, which allows for a very detailed statistical analysis. We find that the fluctuations in the volatility are correlated, and that the correlations are well described by a power law. We also briefly describe some recent scaling results in economics, specifically some surprising features that appear to be common to the growth rates of business firms, countries, research budgets, and bird populations. 1999 Elsevier Science B.V. All rights reserved. 1. Introduction we have at our disposal the tools of computational physics and the computing power to carry out any We begin by pointing out that economics, as a number of approaches, this abundance of data is to our science, is not well understood. This seems to be great advantage. For us, studying the economy means almost generally accepted; in fact, the cover article studying a wealth of data on a well-defined complex of the 23 August 1997 issue of The Economist was system. Today we are going to look at some examples actually entitled “The Puzzling Failure of Economics”. of scale-invariant correlations that are of interest to What does this new field of econophysics have to social scientists. offer to any solution to this puzzle? We begin with At one time, it was imagined that the “scale-free” an assumption that I and my coworkers first encoun- phenomena are relevant to only a fairly narrow slice tered years ago when we worked on critical phenom- of physical phenomena [1]. However, the range of ena: “Everything depends on everything else”. A care- systems that apparently display power law and hence ful analysis of any system involves studying the prop- scale-invariant correlations has increased dramatically agation of correlations from one unit of the system to in recent years, ranging from base pair correlations the next. We learned that these correlations propagate in noncoding DNA [2,3], lung inflation [4,5] and in- both directly and indirectly. terbeat intervals of the human heart [6–9] to com- Our approach is to begin empirically, with real data plex systems involving large numbers of interact- that we can analyze in some detail. In economics, we ing subunits that display “free will”, such as city have available to us a great deal of real data and, since growth [10–12], animal behaviour [14–19], and even economics [20,21]. In particular, economic time se- 1 E-mail: [email protected]. ries, as, e.g., stock market indices or currency ex- 0010-4655/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S0010-4655(99)00301-X 146 L.A.N. Amaral et al. / Computer Physics Communications 121–122 (1999) 145–152 change rates depend on the evolution of a large number 2. Quantification of correlations in S&P 500 of strongly interacting systems far from equilibrium, and belong to the class of a complex evolving systems. 2.1. Data description and detrending Thus, the statistical properties of ®nancial markets The S&P 500 index, an index of the New York have attracted the interests of many physicists [22±33]. Stock Exchange, consists of the 500 largest companies Methods originating in statistical physics have been in the US. It is a market-value weighted index (stock proven useful in analyzing ®nancial indices. They price times number of shares outstanding), with each are also used to construct new models for the pric- stock's weight in the index proportionate to its market ing of derivatives and the assessment of the involved value. The S&P 500 index is one of the most widely risk [25]. used benchmarks of US equity performance. Our data The recent availability of very high frequency data cover 13 years (from Jan. 1984 to Dec. 1996) with a allows to study economic time series with a high ac- recording frequency of 15 seconds interval. The total curacy on a wide range of time scales varying from number of data points in this 13 years period exceed less than 1 minute up to more than 10 years. Conse- 4.5 million, which allows for a very detailed statistical quently, a large number of methods known from sta- analysis. tistical physics have been applied to characterize the The S&P 500 index Z.t/ from 1984 to 1996 tends to time evolution of stock prices and foreign exchange increase constantly on a semi-log graph except during rates [22±26,30±33]. It turns out that the distributions crashes, e.g., October 1987 and May 1990. Since the standard deviation of Z.t C1t/− Z.t/ is proportional of the increments of economic time series, both in to the price level, we take the logarithmic of the index stock market indices and foreign currency exchange as everyone does. We de®ne the forward change rates, are nearly symmetric and have strong ªleptokur- ticº wings [34,26]. Index increments as a function of G.t/ ≡ loge Z.t C 1t/ − loge Z.t/; (1) time show only weak correlations on short time scales where 1t is the time-lag (set to 1 minute in the below 10 minutes [34,35], which seemingly makes correlation study). them fundamentally different from well known exam- We only count the number of minutes during the ples of complex dynamic systems in physics such as, opening hours of the stock market, and remove the e.g., turbulent ¯ow where power law correlations on nights, weekends and holidays from the data set, i.e., long time scales are commonly observed [36]. the closing and the next opening of the market is The situation is different for the volatility, i.e., continuous. the market ¯uctuations averaged on a suitable time The absolute value of G.t/ describes the amplitude interval. There is long time persistence much larger of the ¯uctuation. jG.t/j is, by de®nition, always than the correlation time in volatility [37]. Volatility positive, and there are no obvious global trends visible, which is due to the logarithmic difference, i.e., the is the key input of virtually all option pricing models, relative increment on the original index Z.t/.The including the classic Black and Scholes [38] and Cox, large values of Z.t/ correspond to the crashes and Ross, and Rubinstein [39] binomial models that are big rallies of the index. It is known in the ®nancial based on estimates of the asset's volatility over the literature that the volatility varies in time [40], as remaining life of the option. So to understand the expected the jG.t/j quantity also ¯uctuates in time. dynamics of the volatility has very important practical It is known that there exits intra-day patterns in reason. NYSE and S&P 500 index data, one simple explana- Here, we quantify long range power law correla- tion is that there are many information traders active tions in the volatility of the S&P 500 stock index and near the open and many liquidity traders active near report an occurrence of a cross-over phenomena of this the close [41]. We ®nd the similar intra-day pattern in long range correlation. Furthermore, we discuss the our S&P 500 index data set. The intra-day pattern distribution of the volatility, and show that it can be N jG.ti;same/j A.t/ ≡ PiD1 ; (2) ®tted very well by a log-normal distribution. N L.A.N. Amaral et al. / Computer Physics Communications 121–122 (1999) 145–152 147 where N is the total trading days over the 13-year A detailed description of the DFA algorithm appears period and tsame is the same time of each day (N D elsewhere [43,7]. Brie¯y, the jg.t/j time series (with 3309 in our study). In order to avoid the arti®cial N data) is ®rst integrated, correlation caused by this daily oscillation, G.t/ signal t is normalized by the intra-day pattern y.t/ ≡ jg.i/j: (5) X iD1 g.t/ ≡ G.t/=A.t/; (3) Next the integrated time series is divided into boxes of equal length, n. In each box of length n, a least squares i.e., each data point divided by the intra-day pattern of line is ®t to the data (representing the trend in that its corresponding time during the day. box). The y coordinate of the straight line segments is denoted by yn.t/.Nextwedetrendtheintegrated time series, y.t/, by subtracting the local trend, y .t/, 2.2. Methods to calculate correlations n in each box. The root-mean-square ¯uctuation of this integrated and detrended time series is calculated by We have three methods to quantify the correlations. N The direct method to study the correlation property is v 1 2 F.n/Du y.t/−yn.t/ : (6) the correlation function estimation, which is de®ned as uN X t tD1 hG.t/G.t C τ)i−hG.t/i2 This computation is repeated over all time scales C.τ/ ≡ ; (4) (box sizes) to provide a relationship between F.n/, hG2.t/i−hG.t/i2 theaverage ¯uctuation as a function of box size.