A model for correlations in stock markets

We propose a group model for correlations in stock markets. In the group model the markets are composed of several groups, within which the stock price fluctuations are correlated. The spectral properties of empirical correlation matrices reported in [Phys. Rev. Lett. {\bf 83}, 1467 (1999); Phys. Rev. Lett. {\bf 83}, 1471 (1999.)] are well understood from the model. It provides the connection between the spectral properties of the empirical correlation matrix and the structure of correlations in stock markets.Comment: two pages including one EPS file for a figur

Agents Play Mix-game

In mix-game which is an extension of minority game, there are two groups of agents; group1 plays the majority game, but the group2 plays the minority game. This paper studies the change of the average winnings of agents and volatilities vs. the change of mixture of agents in mix-game model. It finds that the correlations between the average winnings of agents and the mean of local volatilities are different with different combinations of agent memory length when the proportion of agents in group 1 increases. This study result suggests that memory length of agents in group1 be smaller than that of agent in group2 when mix-game model is used to simulate the financial markets.Comment: 8 pages, 6 figures, 3 table

Scaling of the distribution of fluctuations of financial market indices

We study the distribution of fluctuations over a time scale $\Delta t$ (i.e., the returns) of the S&P 500 index by analyzing three distinct databases. Database (i) contains approximately 1 million records sampled at 1 min intervals for the 13-year period 1984-1996, database (ii) contains 8686 daily records for the 35-year period 1962-1996, and database (iii) contains 852 monthly records for the 71-year period 1926-1996. We compute the probability distributions of returns over a time scale $\Delta t$, where $\Delta t$ varies approximately over a factor of 10^4 - from 1 min up to more than 1 month. We find that the distributions for $\Delta t \leq$ 4 days (1560 mins) are consistent with a power-law asymptotic behavior, characterized by an exponent $\alpha \approx 3$, well outside the stable L\'evy regime $0 < \alpha < 2$. To test the robustness of the S&P result, we perform a parallel analysis on two other financial market indices. Database (iv) contains 3560 daily records of the NIKKEI index for the 14-year period 1984-97, and database (v) contains 4649 daily records of the Hang-Seng index for the 18-year period 1980-97. We find estimates of $\alpha$ consistent with those describing the distribution of S&P 500 daily-returns. One possible reason for the scaling of these distributions is the long persistence of the autocorrelation function of the volatility. For time scales longer than $(\Delta t)_{\times} \approx 4$ days, our results are consistent with slow convergence to Gaussian behavior.Comment: 12 pages in multicol LaTeX format with 27 postscript figures (Submitted to PRE May 20, 1999). See http://polymer.bu.edu/~amaral/Professional.html for more of our work on this are

Universal and non-universal properties of cross-correlations in financial time series

We use methods of random matrix theory to analyze the cross-correlation matrix C of price changes of the largest 1000 US stocks for the 2-year period 1994-95. We find that the statistics of most of the eigenvalues in the spectrum of C agree with the predictions of random matrix theory, but there are deviations for a few of the largest eigenvalues. We find that C has the universal properties of the Gaussian orthogonal ensemble of random matrices. Furthermore, we analyze the eigenvectors of C through their inverse participation ratio and find eigenvectors with large inverse participation ratios at both edges of the eigenvalue spectrum--a situation reminiscent of results in localization theory.Comment: 14 pages, 3 figures, Revte

Statistical properties of absolute log-returns and a stochastic model of stock markets with heterogeneous agents

This paper is intended as an investigation of the statistical properties of {\it absolute log-returns}, defined as the absolute value of the logarithmic price change, for the Nikkei 225 index in the 28-year period from January 4, 1975 to December 30, 2002. We divided the time series of the Nikkei 225 index into two periods, an inflationary period and a deflationary period. We have previously [18] found that the distribution of absolute log-returns can be approximated by the power-law distribution in the inflationary period, while the distribution of absolute log-returns is well described by the exponential distribution in the deflationary period.\par To further explore these empirical findings, we have introduced a model of stock markets which was proposed in [19,20]. In this model, the stock market is composed of two groups of traders: {\it the fundamentalists}, who believe that the asset price will return to the fundamental price, and {\it the interacting traders}, who can be noise traders. We show through numerical simulation of the model that when the number of interacting traders is greater than the number of fundamentalists, the power-law distribution of absolute log-returns is generated by the interacting traders' herd behavior, and, inversely, when the number of fundamentalists is greater than the number of interacting traders, the exponential distribution of absolute log-returns is generated.Comment: 12 pages, 5 figure

Scaling of the distribution of price fluctuations of individual companies

We present a phenomenological study of stock price fluctuations of individual companies. We systematically analyze two different databases covering securities from the three major US stock markets: (a) the New York Stock Exchange, (b) the American Stock Exchange, and (c) the National Association of Securities Dealers Automated Quotation stock market. Specifically, we consider (i) the trades and quotes database, for which we analyze 40 million records for 1000 US companies for the 2-year period 1994--95, and (ii) the Center for Research and Security Prices database, for which we analyze 35 million daily records for approximately 16,000 companies in the 35-year period 1962--96. We study the probability distribution of returns over varying time scales $\Delta t$, where $\Delta t$ varies by a factor of $\approx 10^5$---from 5 min up to $\approx$ 4 years. For time scales from 5~min up to approximately 16~days, we find that the tails of the distributions can be well described by a power-law decay, characterized by an exponent $\alpha \approx 3$ ---well outside the stable L\'evy regime $0 < \alpha < 2$. For time scales $\Delta t \gg (\Delta t)_{\times} \approx 16$days, we observe results consistent with a slow convergence to Gaussian behavior. We also analyze the role of cross correlations between the returns of different companies and relate these correlations to the distribution of returns for market indices.Comment: 10pages 2 column format with 11 eps figures. LaTeX file requiring epsf, multicol,revtex. Submitted to PR

Statistical Properties of Share Volume Traded in Financial Markets

We quantitatively investigate the ideas behind the often-expressed adage `it takes volume to move stock prices', and study the statistical properties of the number of shares traded $Q_{\Delta t}$ for a given stock in a fixed time interval $\Delta t$. We analyze transaction data for the largest 1000 stocks for the two-year period 1994-95, using a database that records every transaction for all securities in three major US stock markets. We find that the distribution $P(Q_{\Delta t})$ displays a power-law decay, and that the time correlations in $Q_{\Delta t}$ display long-range persistence. Further, we investigate the relation between $Q_{\Delta t}$ and the number of transactions $N_{\Delta t}$ in a time interval $\Delta t$, and find that the long-range correlations in $Q_{\Delta t}$ are largely due to those of $N_{\Delta t}$. Our results are consistent with the interpretation that the large equal-time correlation previously found between $Q_{\Delta t}$ and the absolute value of price change $| G_{\Delta t} |$ (related to volatility) are largely due to $N_{\Delta t}$.Comment: 4 pages, two-column format, four figure

Statistical mechanics of the mixed majority-minority game with random external information

We study the asymptotic macroscopic properties of the mixed majority-minority game, modeling a population in which two types of heterogeneous adaptive agents, namely ``fundamentalists'' driven by differentiation and ``trend-followers'' driven by imitation, interact. The presence of a fraction f of trend-followers is shown to induce (a) a significant loss of informational efficiency with respect to a pure minority game (in particular, an efficient, unpredictable phase exists only for f<1/2), and (b) a catastrophic increase of global fluctuations for f>1/2. We solve the model by means of an approximate static (replica) theory and by a direct dynamical (generating functional) technique. The two approaches coincide and match numerical results convincingly.Comment: 19 pages, 3 figure