Bernd Rosenow

Universal and Nonuniversal 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.

Order book approach to price impact

Buying and selling stocks causes price changes, which are described by the price impact function. To explain the shape of this function, we study the Island ECN orderbook. In addition to transaction data, the orderbook contains information about potential supply and demand for a stock. The virtual price impact calculated from this information is four times stronger than the actual one and explains it only partially. However, we find a strong anticorrelation between price changes and order flow, which strongly reduces the virtual price impact and provides for an explanation of the empirical price impact function.

Econophysics: financial time series from a statistical physics point of view

In recent years, physicists have started applying concepts and methods of statistical physics to study economic problems. The word “Econophysics” is sometimes used to refer to this work. Much recent work is focused on understanding the statistical properties of financial time series. One reason for this interest is that financial markets are examples of complex interacting systems for which a huge amount of data exist and it is possible that financial time series viewed from a different perspective might yield new results. This article reviews the results of three recent phenomenological studies — (i) The probability distribution of stock price fluctuations: Stock price fluctuations occur in all magnitudes, in analogy to earthquakes — from tiny fluctuations to drastic events, such as market crashes. The distribution of price fluctuations decays with a power-law tail well outside the Levy stable regime and describes fluctuations that differ by as much as eight orders of magnitude. In addition, this distribution preserves its functional form for fluctuations on time scales that differ by three orders of magnitude, from 1 min up to approximately 10 d. (ii) Correlations in financial time series: While price fluctuations themselves have rapidly decaying correlations, the magnitude of fluctuations measured by either the absolute value or the square of the price fluctuations has correlations that decay as a power-law and persist for several months. (iii) Correlations among different companies: The third result bears on the application of random matrix theory to understand the correlations among price fluctuations of any two different stocks. From a study of the eigenvalue statistics of the cross-correlation matrix constructed from price fluctuations of the leading 1000 stocks, we find that the largest ≈ 1% of the eigenvalues and the corresponding eigenvectors show systematic deviations from the predictions for a random matrix, whereas the rest of the eigenvalues conform to random matrix behavior — suggesting that these 1% of the eigenvalues contain system-specific information about correlated time evolution of different companies.

A random matrix theory approach to financial cross-correlations

It is common knowledge that any two firms in the economy are correlated. Even firms belonging to different sectors of an industry may be correlated because of “indirect” correlations. How can we analyze and understand these correlations? This article reviews recent results regarding cross-correlations between stocks. Specifically, we use methods of random matrix theory (RMT), which originated from the need to understand the interactions between the constituent elements of complex interacting systems, to analyze the cross-correlation matrix C of returns. We analyze 30-min returns of the largest 1000 US stocks for the 2-year period 1994–1995. We find that the statistics of approximately 20 of the largest eigenvalues (2%) show deviations from the predictions of RMT. To test that the rest of the eigenvalues are genuinely random, we test for universal properties such as eigenvalue spacings and eigenvalue correlations, and demonstrate that C shares universal properties with the Gaussian orthogonal ensemble of random matrices. The statistics of the eigenvectors of C confirm the deviations of the largest few eigenvalues from the RMT prediction. We also find that these deviating eigenvectors are stable in time. In addition, we quantify the number of firms that participate significantly to an eigenvector using the concept of inverse participation ratio, borrowed from localization theory.

Large stock price changes: volume or liquidity?

We analyse large stock price changes of more than five standard deviations for (i) TAQ data for the year 1997 and (ii) order book data from the Island ECN for the year 2002. We argue that a large trading volume alone is not a sufficient explanation for large price changes. Instead, we find that a low density of limit orders in the order book, i.e. a small liquidity, is a necessary prerequisite for the occurrence of extreme price fluctuations. Taking into account both order flow and liquidity, large stock price fluctuations can be explained quantitatively.

Influence of interactions on flux and back-gate period of quantum Hall interferometers.

In quantum Hall systems with two narrow constrictions, tunneling between opposite edges can give rise to quantum interference and Aharonov-Bohm-like oscillations of the conductance. When there is an integer quantized Hall state within the constrictions, a region between them, with higher electron density, may form a compressible island. Electron tunneling through this island can lead to residual transport, modulated by Coulomb-blockade-type effects. We find that the coupling between the fully occupied lower Landau levels and the higher-partially occupied level gives rise to flux subperiods smaller than one flux quantum. We generalize this scenario to other geometries and to fractional quantum Hall systems, and compare our predictions to experiments.

Dynamics of cross-correlations in the stock market

Co-movements of stock price fluctuations are described by the cross-correlation matrix C. The application of random matrix theory (RMT) allows to distinguish between spurious correlations in C due to measurement noise and true correlations containing economically meaningful information. By calculating cross-correlations for different time windows, we study the time dependence of eigenvectors of C, which are related to economic sectors, and the time evolution of the largest eigenvalue, which describes the average correlation strength. We use these results to forecast cross-correlations, and test the quality of our forecast by constructing investments in the stock market which expose the invested capital to a minimum level of risk only.

Edge-State Velocity and Coherence in a Quantum Hall Fabry-Pérot Interferometer

We investigate nonlinear transport in electronic Fabry-Pérot interferometers in the integer quantum Hall regime. For interferometers sufficiently large that Coulomb blockade effects are absent, a checkerboardlike pattern of conductance oscillations as a function of dc bias and perpendicular magnetic field is observed. Edge-state velocities extracted from the checkerboard data are compared to model calculations and found to be consistent with a crossover from skipping orbits at low fields to E-vector x B-vector drift at high fields. Suppression of visibility as a function of bias and magnetic field is accounted for by including energy- and field-dependent dephasing of edge electrons.

Collective behavior of stock price movements—a random matrix theory approach

We review recent work on quantifying collective behavior among stocks by applying the conceptual framework of random matrix theory (RMT), developed in physics to describe the energy levels of complex systems. RMT makes predictions for “universal” properties that do not depend on the interactions between the elements comprising the system; deviations from RMT provide clues regarding system-specific properties. We compare the statistics of the cross-correlation matrix C—whose elements Cij are the correlation coefficients of price fluctuations of stock i and j—against a random matrix having the same symmetry properties. It is found that RMT methods can distinguish random and non-random parts of C. The non-random part of C which deviates from RMT results, provides information regarding genuine collective behavior among stocks.

FLUCTUATIONS AND MARKET FRICTION IN FINANCIAL TRADING

We study the relation between stock price changes and the difference in the volume of sell and buy orders. Using a soft spin model, we describe the price impact of order imbalances and find an analogy to the fluctuation–dissipation theorem in physical systems. We empirically investigate fluctuations and market friction for a major US stock and find support for our model calculations.