This book concerns the use of concepts from statistical physics in the description of financial systems. The authors illustrate the scaling concepts used in probability theory, critical phenomena, and fully developed turbulent fluids. These concepts are then applied to financial time series. The authors also present a stochastic model that displays several of the statistical properties observed in empirical data. Statistical physics concepts such as stochastic dynamics, short- and long-range correlations, self-similarity and scaling permit an understanding of the global behaviour of economic systems without first having to work out a detailed microscopic description of the system. Physicists will find the application of statistical physics concepts to economic systems interesting. Economists and workers in the financial world will find useful the presentation of empirical analysis methods and well-formulated theoretical tools that might help describe systems composed of a huge number of interacting subsystems.

/pdf/introduction-to-econophysics-correlations-and-complexity-in-4a1wviggsu.pdf

Introduction to Econophysics: Correlations and Complexity in Finance

Analyses of firm sizes have historically used data that included limited samples of small firms, data typically described by lognormal distributions. Using data on the entire population of tax-paying firms in the United States, I show here that the Zipf distribution characterizes firm sizes: the probability a firm is larger than size s is inversely proportional to s. These results hold for data from multiple years and for various definitions of firm size.

https://science.sciencemag.org/content/sci/293/5536/1818.full.pdf

Zipf distribution of U.S. firm sizes.

We use methods of random matrix theory to analyze the cross-correlation matrix C of stock price changes of the largest 1000 US companies for the 2-year period 1994 ‐ 1995 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 ratios at both edges of the eigenvalue spectrum — a situation reminiscent of localization theory results

/pdf/universal-and-nonuniversal-properties-of-cross-correlations-ojky0l88pe.pdf

Universal and Nonuniversal Properties of Cross Correlations in Financial Time Series

A comprehensive account of economic size distributions around the world and throughout the years In the course of the past 100 years, economists and applied statisticians have developed a remarkably diverse variety of income distribution models, yet no single resource convincingly accounts for all of these models, analyzing their strengths and weaknesses, similarities and differences. Statistical Size Distributions in Economics and Actuarial Sciences is the first collection to systematically investigate a wide variety of parametric models that deal with income, wealth, and related notions. Christian Kleiber and Samuel Kotz survey, compliment, compare, and unify all of the disparate models of income distribution, highlighting at times a lack of coordination between them that can result in unnecessary duplication. Considering models from eight languages and all continents, the authors discuss the social and economic implications of each as well as distributions of size of loss in actuarial applications. Specific models covered include: Pareto distributions Lognormal distributions Gamma-type size distributions Beta-type size distributions Miscellaneous size distributions Three appendices provide brief biographies of some of the leading players along with the basic properties of each of the distributions. Actuaries, economists, market researchers, social scientists, and physicists interested in econophysics will find Statistical Size Distributions in Economics and Actuarial Sciences to be a truly one-of-a-kind addition to the professional literature.

Statistical Size Distributions in Economics and Actuarial Sciences

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 $\ensuremath{\Delta}t.$ We analyze the S 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 $\ensuremath{\mu}\ensuremath{\approx}3,$ 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.

https://arxiv.org/pdf/cond-mat/9903369.pdf

Statistical properties of the volatility of price fluctuations.

By analyzing international company data we find that company size distributions are not universal but clearly depend on country. In each country, the size distributions for different categories of business are quite similar. In order to understand the country dependence we introduce a numerical model of company size which is based on two effects: stochastic competitive growth by capital exchange, and deterministic protection of small companies by equi-partition of taxed wealth. A power law distribution is realized when the protection effect is negligible. The model is also consistent with the empirical laws for company's growth rates.

Country Dependence on Company Size Distributions and a Numerical Model Based on Competition and Cooperation

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Country Dependence on Company Size Distributions and a Numerical Model Based on Competition and Cooperation