Quantifying fluctuations in economic systems by adapting methods of statistical physics

Abstract

The emerging subfield of econophysics explores the degree to which certain concepts and methods from statistical physics can be appropriately modified and adapted to provide new insights into questions that have been the focus of interest in the economics community. Here we give a brief overview of two examples of research topics that are receiving recent attention. A first topic is the characterization of the dynamics of stock price fluctuations. For example, we investigate the relation between trading activity – measured by the number of transactions N Δ t – and the price change G Δ t for a given stock, over a time interval [t, t+ Δt] . We relate the time-dependent standard deviation of price fluctuations – volatility – to two microscopic quantities: the number of transactions N Δ t in Δ t and the variance W Δ t 2 of the price changes for all transactions in Δ t. Our work indicates that while the pronounced tails in the distribution of price fluctuations arise from W Δ t , the long-range correlations found in ∣ G Δ t ∣ are largely due to N Δ t . We also investigate the relation between price fluctuations and the number of shares Q Δ t traded in Δ t. We find that the distribution of Q Δ t is consistent with a stable Lévy distribution, suggesting a Lévy scaling relationship between Q Δ t and N Δ t , which would provide one explanation for volume-volatility co-movement. A second topic concerns cross-correlations between the price fluctuations of different stocks. We adapt a conceptual framework, random matrix theory (RMT), first used in physics to interpret statistical properties of nuclear energy spectra. RMT makes predictions for the statistical properties of matrices that are universal, that is, do not depend on the interactions between the elements comprising the system. In physics systems, deviations from the predictions of RMT provide clues regarding the mechanisms controlling the dynamics of a given system, so this framework can be of potential value if applied to economic systems. We discuss a systematic comparison between the statistics of the cross-correlation matrix C – whose elements C ij are the correlation-coefficients between the returns of stock i and j – and that of a random matrix having the same symmetry properties. Our work suggests that RMT can be used to distinguish random and non-random parts of C ; the non-random part of C , which deviates from RMT results provides information regarding genuine cross-correlations between stocks.