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

This manuscript is a brief summary of a talk designed to address the question of whether two of the pillars of the field of phase transitions and critical phenomena—scale invariance and universality—can be useful in guiding research on interpreting empirical data on economic fluctuations. Using this conceptual framework as a guide, we empirically quantify the relation between trading activity—measured by the number of transactions N—and the price change G(t) for a given stock, over a time interval [t,t+Δt]. We relate the time-dependent standard deviation of price changes—volatility—to two microscopic quantities: the number of transactions N(t) in Δt and the variance W2(t) of the price changes for all transactions in Δt. We find that the long-ranged volatility correlations are largely due to those of N. We then argue that the tail-exponent of the distribution of N is insufficient to account for the tail-exponent of P{G>x}. Since N and W display only weak inter-dependency, our results show that the fat tails of the distribution P{G>x} arises from W. Finally, we review recent work on quantifying collective behavior among stocks by applying the conceptual framework of random matrix theory (RMT). RMT makes predictions for “universal” properties that do not depend on the interactions between the elements comprising the system, and 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. We also discuss results that are reminiscent of phase transitions in spin systems, where the divergent behavior of the response function at the critical point (zero magnetic field) leads to large fluctuations, and we discuss a curious “symmetry breaking”, a feature qualitatively identical to the behavior of the probability density of the magnetization for fixed values of the inverse temperature.

Quantifying economic fluctuations

We survey a theory of the economic underpinnings of the fat-tailed distributions of a number of financial variables, such as returns and trading volumes. Our theory posits that they have a common origin in the strategic trading behavior of very large financial institutions in a relatively illiquid market. We show how the fat-tailed distribution of fund sizes can indeed generate extreme returns and volumes, even in the absence of fundamental news. Moreover, we are able to replicate the individually different empirical values of the power law exponents foreachdistribution.Largeinvestorsmoderatetheirtradestoreducetheirpriceimpact;coupled with a concave price impact function, this leads to volumes being more fat-tailed than returns but less fat-tailed than fund sizes. The trades of large institutions offer a unified explanation for apparently disconnected empirical regularities that are otherwise a challenge for economic theory. (JEL: G12, G14, G23)

A theory of limited liquidity and large investors causing spikes in stock market volatility and trading volume

We address the question of how to precisely identify correlated behavior between different firms in the economy by applying methods of random matrix theory (RMT) Specifically, we use methods of random matrix theory to analyze the cross-correlation matrix of price changes of the largest 1000 US stocks for the 2-year period 1994–1995 We find that the statistics of most of the eigenvalues in the spectrum of agree with the predictions of random matrix theory, but there are deviations for a few of the largest eigenvalues To prove that the rest of the eigenvalues are genuinely random, we test for universal properties such as eigenvalue spacings and eigenvalue correlations We demonstrate that shares universal properties with the Gaussian orthogonal ensemble of random matrices In addition, we quantify the number of significant participants, that is companies, of the eigenvectors using the 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

https://polymer.bu.edu/hes/articles/rpgas00.pdf

Application of random matrix theory to study cross-correlations of stock prices

In recent years, a considerable number of physicists have started applying physics concepts and methods to understand economic phenomena. The term “Econophysics” is sometimes used to describe this work. Economic fluctuations can have many repercussions, and understanding fluctuations is a topic that many physicists have contributed to in recent years. Further, economic systems are examples of complex interacting systems for which a huge amount of data exist and it is possible that the experience gained by physicists in studying fluctuations in physical systems might yield new results in economics. Much recent work in econophysics is focused on understanding the peculiar statistical properties of price fluctuations in financial time series. In this talk, we discuss three recent results. The first result concerns the probability distribution of stock price fluctuations. This distribution decreases with increasing fluctuations with a power-law tail well outside the Levy stable regime and describes fluctuations that differ by as much as 8 orders of magnitude. Further, this non stable distribution preserves its functional form for fluctuations on time scales that differ by 3 orders of magnitude, from 1 min up to approximately 10 days. The second result concerns the accurate quantification of volatility correlations in financial time series. While price fluctuations themselves have rapidly decaying correlations, the volatility estimated by using either the absolute value or the square of the price fluctuations has correlations that decay as a power-law and persist for several months. The third result bears on the application of random matrix theory to understand the correlations among price fluctuations of any two different stocks. We compare the statistics of the cross-correlation matrix constructed from price fluctuations of the leading 1000 stocks and a matrix with independent random elements, i.e., a random matrix. Contrary to first expectations, we find little or no deviation from the universal predictions of random matrix theory for all but a few of the largest eigenvalues of the cross-correlation matrix.

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Econophysics: what can physicists contribute to economics?

We show empirically a series of sharp patterns in stock market fluctuations, trading activity and their contemporaneous relationships. We link together and explain the following facts: (i) the cubic law of returns: returns follow a power law distribution with exponent 3. This “cubic” law seems to hold both across time and internationally. Stock market “crashes” (e.g. the 1929 and 1987 crashes) are not outliers to this law; (ii) the half cubic law of volumes: volumes follow a power law distribution with exponent 3/2; (iii) the square root law of price impact (the price impact of a volume V is proportional to V 1/2 ); (iv) Zipf’s law for mutual funds: mutual funds size follow a power law distribution with exponent 1. The model also makes predictions about the crossconditional relationships between various trading variables. They all appear to be verified empirically. The model makes a series of other, out of sample, testable predictions. Finally, it shows that a Tobin tax, or circuit breakers, do not affect that size of extreme fluctuations. However, a tax that increases with the size of the transactions does reduce the magnitude of those very large fluctuations.

Parameswaran Gopikrishnan

Papers

Quantifying economic fluctuations

A theory of limited liquidity and large investors causing spikes in stock market volatility and trading volume

Application of random matrix theory to study cross-correlations of stock prices

Econophysics: what can physicists contribute to economics?

A simple theory of the ''cubic'' laws of stock market activity ∗