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 Lévy 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.