We analyze cross correlations between price fluctuations of different stocks using methods of random matrix theory (RMT). Using two large databases, we calculate cross-correlation matrices C of returns constructed from (i) 30-min returns of 1000 US stocks for the 2-yr period 1994-1995, (ii) 30-min returns of 881 US stocks for the 2-yr period 1996-1997, and (iii) 1-day returns of 422 US stocks for the 35-yr period 1962-1996. We test the statistics of the eigenvalues lambda(i) of C against a "null hypothesis"--a random correlation matrix constructed from mutually uncorrelated time series. We find that a majority of the eigenvalues of C fall within the RMT bounds [lambda(-),lambda(+)] for the eigenvalues of random correlation matrices. We test the eigenvalues of C within the RMT bound for universal properties of random matrices and find good agreement with the results for the Gaussian orthogonal ensemble of random matrices-implying a large degree of randomness in the measured cross-correlation coefficients. Further, we find that the distribution of eigenvector components for the eigenvectors corresponding to the eigenvalues outside the RMT bound display systematic deviations from the RMT prediction. In addition, we find that these "deviating eigenvectors" are stable in time. We analyze the components of the deviating eigenvectors and find that the largest eigenvalue corresponds to an influence common to all stocks. Our analysis of the remaining deviating eigenvectors shows distinct groups, whose identities correspond to conventionally identified business sectors. Finally, we discuss applications to the construction of portfolios of stocks that have a stable ratio of risk to return.

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.

Insights into the dynamics of a complex system are often gained by focusing on large fluctuations. For the financial system, huge databases now exist that facilitate the analysis of large fluctuations and the characterization of their statistical behaviour. Power laws appear to describe histograms of relevant financial fluctuations, such as fluctuations in stock price, trading volume and the number of trades. Surprisingly, the exponents that characterize these power laws are similar for different types and sizes of markets, for different market trends and even for different countries--suggesting that a generic theoretical basis may underlie these phenomena. Here we propose a model, based on a plausible set of assumptions, which provides an explanation for these empirical power laws. Our model is based on the hypothesis that large movements in stock market activity arise from the trades of large participants. Starting from an empirical characterization of the size distribution of those large market participants (mutual funds), we show that the power laws observed in financial data arise when the trading behaviour is performed in an optimal way. Our model additionally explains certain striking empirical regularities that describe the relationship between large fluctuations in prices, trading volume and the number of trades.

We study the distribution of fluctuations over a time scale ∆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 ∆t, where ∆t varies approximately over a factor of 10 4 -from 1 min up to more than 1 month. We find that the distributions for ∆t ≤ 4 days (1560 mins) are consistent with a power-law asymptotic behavior, characterized by an exponent α ≈ 3, well outside the stable Lévy regime 0 < α < 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 α 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 (∆t)× ≈ 4 days, our results are consistent with slow convergence to Gaussian behavior.

We study the statistical properties of volatility, measured by locally averaging over a time window T, the absolute value of price changes over a short time interval deltat. We analyze the S&P 500 stock index for the 13-year period Jan. 1984 to Dec. 1996. We find that the cumulative distribution of the volatility is consistent with a power-law asymptotic behavior, characterized by an exponent mu approximately 3, similar to what is found for the distribution of price changes. The volatility distribution retains the same functional form for a range of values of T. Further, we study the volatility correlations by using the power spectrum analysis. Both methods support a power law decay of the correlation function and give consistent estimates of the relevant scaling exponents. Also, both methods show the presence of a crossover at approximately 1.5 days. In addition, we extend these results to the volatility of individual companies by analyzing a data base comprising all trades for the largest 500 U.S. companies over the two-year period Jan. 1994 to Dec. 1995.

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 ∆t, where ∆t varies by a factor of ≈ 10 5 -from 5 min up to ≈ 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 α ≈ 3 -well outside the stable Lévy regime 0 < α < 2. For time scales ∆t ≫ (∆t)× ≈ 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.

The probability distribution of stock price changes is studied by analyzing a database (the Trades and Quotes Database) documenting every trade for all stocks in three major US stock markets, for the two year period Jan 1994 -Dec 1995. A sample of 40 million data points is extracted, which is substantially larger than studied hitherto. We find an asymptotic power-law behavior for the cumulative distribution with an exponent α ≈ 3, well outside the Levy regime (0 < α < 2).PACS numbers: 89.90.+n Typeset using REVT E X 1The asymptotic behavior of the increment distribution of economic indices has long been a topic of avid interest [1][2][3][4][5][6]. Conclusive empirical results are, however, difficult to obtain, since the asymptotic behavior can be obtained only by a proper sampling of the tails, which requires a huge quantity of data. Here, we analyze 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 2-year period Jan. 1994 to Dec. 1995 [7].We thereby extract a sample of approximately 4 × 10 7 data points, which is much larger than studied hitherto.We form 1000 time series S i (t), where S i is the market price of company i (i.e. the share price multiplied with the number of outstanding shares), i = 1 . . . 1000 is the rank of the company according to its market price on Jan. 1, 1994. The basic quantity of our study is the relative price change,where the time lag is ∆t = 5 min. We normalize the increments,where the volatility v i ≡ (G i (t) − G i (t) ) 2 of company i is measured by the standard deviation, and . . . is a time average [8].We obtain about 20,000 normalized increments g i (t) per company per year, which gives about 4 × 10 7 increments for the 1000 largest companies in the time period studied. Figure 1a shows the cumulative probability distribution, i.e. the probability for an increment larger or equal to a threshold g, P (g) ≡ P {g i (t) ≥ g}, as a function of g. The data are well fit by a power lawwith exponents α = 3.1 ± 0.03 (positive tail) and α = 2.84 ± 0.12 (negative tail) from two to hundred standard deviations. 2In order to test this result, we calculate the inverse of the local logarithmic slope of P (g),. We estimate the asymptotic slope α by extrapolating γ as a function of 1/g to 0. Figure 1b shows the results for the negative and positive tail respectively, each using all increments larger than 5 standard deviations. Extrapolation of the linear regression lines yield α = 2.84 ± 0.12 for the positive and α = 2.73 ± 0.13 for the negative tail. We test this method by analyzing two surrogate data sets with known asymptotic behavior, (a) an independent random variable with P (x) = (1 + x) −3 and (b) an independent random variable with P (x) = exp(−x). The method yields the correct results 3 and ∞ respectively.To test the robustness of the inverse cubic law α ≈ 3, we perform several additi...

We present a theory of excess stock market volatility, in which market movements are due to trades by very large institutional investors in relatively illiquid markets. Such trades generate significant spikes in returns and volume, even in the absence of important news about fundamentals. We derive the optimal trading behavior of these investors, which allows us to provide a unified explanation for apparently disconnected empirical regularities in returns, trading volume and investor size.