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

Portfolio Optimization and the Random Magnet Problem

Diversification of an investment into independently fluctuating assets reduces its risk. In reality, movement of assets are are mutually correlated and therefore knowledge of cross--correlations among asset price movements are of great importance. Our results support the possibility that the problem of finding an investment in stocks which exposes invested funds to a minimum level of risk is analogous to the problem of finding the magnetization of a random magnet. The interactions for this ``random magnet problem'' are given by the cross-correlation matrix {\bf \sf C} of stock returns. We find that random matrix theory allows us to make an estimate for {\bf \sf C} which outperforms the standard estimate in terms of constructing an investment which carries a minimum level of risk.Comment: 12 pages, 4 figures, revte

The Grounds For Time Dependent Market Potentials From Dealers' Dynamics

We apply the potential force estimation method to artificial time series of market price produced by a deterministic dealer model. We find that dealers' feedback of linear prediction of market price based on the latest mean price changes plays the central role in the market's potential force. When markets are dominated by dealers with positive feedback the resulting potential force is repulsive, while the effect of negative feedback enhances the attractive potential force.Comment: 9 pages, 3 figures, proceedings of APFA

Long-Time Fluctuations in a Dynamical Model of Stock Market Indices

Financial time series typically exhibit strong fluctuations that cannot be described by a Gaussian distribution. In recent empirical studies of stock market indices it was examined whether the distribution P(r) of returns r(tau) after some time tau can be described by a (truncated) Levy-stable distribution L_{alpha}(r) with some index 0 < alpha <= 2. While the Levy distribution cannot be expressed in a closed form, one can identify its parameters by testing the dependence of the central peak height on tau as well as the power-law decay of the tails. In an earlier study [Mantegna and Stanley, Nature 376, 46 (1995)] it was found that the behavior of the central peak of P(r) for the Standard & Poor 500 index is consistent with the Levy distribution with alpha=1.4. In a more recent study [Gopikrishnan et al., Phys. Rev. E 60, 5305 (1999)] it was found that the tails of P(r) exhibit a power-law decay with an exponent alpha ~= 3, thus deviating from the Levy distribution. In this paper we study the distribution of returns in a generic model that describes the dynamics of stock market indices. For the distributions P(r) generated by this model, we observe that the scaling of the central peak is consistent with a Levy distribution while the tails exhibit a power-law distribution with an exponent alpha > 2, namely beyond the range of Levy-stable distributions. Our results are in agreement with both empirical studies and reconcile the apparent disagreement between their results

Exponential distribution of financial returns at mesoscopic time lags: a new stylized fact

We study the probability distribution of stock returns at mesoscopic time lags (return horizons) ranging from about an hour to about a month. While at shorter microscopic time lags the distribution has power-law tails, for mesoscopic times the bulk of the distribution (more than 99% of the probability) follows an exponential law. The slope of the exponential function is determined by the variance of returns, which increases proportionally to the time lag. At longer times, the exponential law continuously evolves into Gaussian distribution. The exponential-to-Gaussian crossover is well described by the analytical solution of the Heston model with stochastic volatility.Comment: 7 pages, 12 plots, elsart.cls, submitted to the Proceedings of APFA-4. V.2: updated reference

Random Matrix Theory and Fund of Funds Portfolio Optimisation

The proprietary nature of Hedge Fund investing means that it is common practise for managers to release minimal information about their returns. The construction of a Fund of Hedge Funds portfolio requires a correlation matrix which often has to be estimated using a relatively small sample of monthly returns data which induces noise. In this paper random matrix theory (RMT) is applied to a cross-correlation matrix C, constructed using hedge fund returns data. The analysis reveals a number of eigenvalues that deviate from the spectrum suggested by RMT. The components of the deviating eigenvectors are found to correspond to distinct groups of strategies that are applied by hedge fund managers. The Inverse Participation ratio is used to quantify the number of components that participate in each eigenvector. Finally, the correlation matrix is cleaned by separating the noisy part from the non-noisy part of C. This technique is found to greatly reduce the difference between the predicted and realised risk of a portfolio, leading to an improved risk profile for a fund of hedge funds.Comment: 17 Page

Variety and Volatility in Financial Markets

We study the price dynamics of stocks traded in a financial market by considering the statistical properties both of a single time series and of an ensemble of stocks traded simultaneously. We use the $n$ stocks traded in the New York Stock Exchange to form a statistical ensemble of daily stock returns. For each trading day of our database, we study the ensemble return distribution. We find that a typical ensemble return distribution exists in most of the trading days with the exception of crash and rally days and of the days subsequent to these extreme events. We analyze each ensemble return distribution by extracting its first two central moments. We observe that these moments are fluctuating in time and are stochastic processes themselves. We characterize the statistical properties of ensemble return distribution central moments by investigating their probability density functions and temporal correlation properties. In general, time-averaged and portfolio-averaged price returns have different statistical properties. We infer from these differences information about the relative strength of correlation between stocks and between different trading days. Lastly, we compare our empirical results with those predicted by the single-index model and we conclude that this simple model is unable to explain the statistical properties of the second moment of the ensemble return distribution.Comment: 10 pages, 11 figure

Stochastic volatility of financial markets as the fluctuating rate of trading: an empirical study

We present an empirical study of the subordination hypothesis for a stochastic time series of a stock price. The fluctuating rate of trading is identified with the stochastic variance of the stock price, as in the continuous-time random walk (CTRW) framework. The probability distribution of the stock price changes (log-returns) for a given number of trades N is found to be approximately Gaussian. The probability distribution of N for a given time interval Dt is non-Poissonian and has an exponential tail for large N and a sharp cutoff for small N. Combining these two distributions produces a nontrivial distribution of log-returns for a given time interval Dt, which has exponential tails and a Gaussian central part, in agreement with empirical observations.Comment: 5 pages, 7 figures, RevTeX, proceedings of APFA-5. V.2: minor typos corrected, 2 references adde

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

A New Method to Estimate the Noise in Financial Correlation Matrices

Financial correlation matrices measure the unsystematic correlations between stocks. Such information is important for risk management. The correlation matrices are known to be ``noise dressed''. We develop a new and alternative method to estimate this noise. To this end, we simulate certain time series and random matrices which can model financial correlations. With our approach, different correlation structures buried under this noise can be detected. Moreover, we introduce a measure for the relation between noise and correlations. Our method is based on a power mapping which efficiently suppresses the noise. Neither further data processing nor additional input is needed.Comment: 25 pages, 8 figure