We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then described: distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. We then show how these statistical properties invalidate many of the common statistical approaches used to study financial data sets and examine some of the statistical problems encountered in each case.

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Empirical properties of asset returns: stylized facts and statistical issues

We develop a method for the multifractal characterization of nonstationary time series, which is based on a generalization of the detrended fluctuation analysis (DFA). We relate our multifractal DFA method to the standard partition function-based multifractal formalism, and prove that both approaches are equivalent for stationary signals with compact support. By analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series. By comparing the multifractal DFA results for original series with those for shuffled series we can distinguish multifractality due to long-range correlations from multifractality due to a broad probability density function. We also compare our results with the wavelet transform modulus maxima method, and show that the results are equivalent.

Multifractal Detrended Fluctuation Analysis of Nonstationary Time Series

Droplet Barcoding for Single-Cell Transcriptomics Applied to Embryonic Stem Cells

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.

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Introduction to Econophysics: Correlations and Complexity in Finance

This paper develops a simple equilibrium model of CEO pay. CEOs have different talents and are matched to firms in a competitive assignment model. In market equilibrium, a CEO’s pay changes one for one with aggregate firm size, while changing much less with the size of his own firm. The model determines the level of CEO pay across firms and over time, offering a benchmark for calibratable corporate finance. The sixfold increase of CEO pay between 1980 and 2003 can be fully attributed to the six-fold increase in market capitalization of large US companies during that period. We find a very small dispersion in CEO talent, which nonetheless justifies large pay differences. The data broadly support the model. The size of large fi rms explains many of the patterns in CEO pay, across firms, over time, and between countries. (JEL D2, D3, G34, J3)

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Why has CEO Pay Increased So Much

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

We analyze cross correlations between price fluctuations of different stocks using methods of random matrix theory (RMT). Using two large databases, we calculate cross-correlation matrices C of returns constructed from (i) 30-min returns of 1000 US stocks for the 2-yr period 1994-1995, (ii) 30-min returns of 881 US stocks for the 2-yr period 1996-1997, and (iii) 1-day returns of 422 US stocks for the 35-yr period 1962-1996. We test the statistics of the eigenvalues lambda(i) of C against a "null hypothesis" - a random correlation matrix constructed from mutually uncorrelated time series. We find that a majority of the eigenvalues of C fall within the RMT bounds [lambda(-),lambda(+)] for the eigenvalues of random correlation matrices. We test the eigenvalues of C within the RMT bound for universal properties of random matrices and find good agreement with the results for the Gaussian orthogonal ensemble of random matrices-implying a large degree of randomness in the measured cross-correlation coefficients. Further, we find that the distribution of eigenvector components for the eigenvectors corresponding to the eigenvalues outside the RMT bound display systematic deviations from the RMT prediction. In addition, we find that these "deviating eigenvectors" are stable in time. We analyze the components of the deviating eigenvectors and find that the largest eigenvalue corresponds to an influence common to all stocks. Our analysis of the remaining deviating eigenvectors shows distinct groups, whose identities correspond to conventionally identified business sectors. Finally, we discuss applications to the construction of portfolios of stocks that have a stable ratio of risk to return. (Less)

https://arxiv.org/pdf/cond-mat/0108023.pdf

Random matrix approach to cross correlations in financial data.

We study the distribution of fluctuations of the S&P 500 index over a time scale deltat by analyzing three distinct databases. Database (i) contains approximately 1 200 000 records, sampled at 1-min intervals, for the 13-year period 1984-1996, database (ii) contains 8686 daily records for the 35-year period 1962-1996, and database (iii) contains 852 monthly records for the 71-year period 1926-1996. We compute the probability distributions of returns over a time scale deltat, where deltat varies approximately over a factor of 10(4)-from 1 min up to more than one month. We find that the distributions for deltat<or= 4 d (1560 min) are consistent with a power-law asymptotic behavior, characterized by an exponent alpha approximately 3, well outside the stable Levy regime 0<alpha<2. To test the robustness of the S&P result, we perform a parallel analysis on two other financial market indices. Database (iv) contains 3560 daily records of the NIKKEI index for the 14-year period 1984-1997, and database (v) contains 4649 daily records of the Hang-Seng index for the 18-year period 1980-1997. We find estimates of alpha consistent with those describing the distribution of S&P 500 daily returns. One possible reason for the scaling of these distributions is the long persistence of the autocorrelation function of the volatility. For time scales longer than (deltat)x approximately 4 d, our results are consistent with a slow convergence to Gaussian behavior.

Scaling of the distribution of fluctuations of financial market indices.

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.

We present a phenomenological study of stock price fluctuations of individual companies. We systematically analyze two different databases covering securities from the three major U.S. 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 U.S. companies for the 2-yr 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-yr period 1962‐96. We study the probability distribution of returns over varying time scales Dt, where Dt varies by a factor of ’10 5 , from 5 min up to ’4 yr. 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,

https://arxiv.org/pdf/cond-mat/9907161.pdf

Parameswaran Gopikrishnan

Papers

Universal and Nonuniversal Properties of Cross Correlations in Financial Time Series

Random matrix approach to cross correlations in financial data.

Scaling of the distribution of fluctuations of financial market indices.

Statistical properties of the volatility of price fluctuations.

Scaling of the distribution of price fluctuations of individual companies.