Physical approach to complex systems

The most suitable paradigms and tools for investigating the scaling structure of financial time series are reviewed and discussed in the light of some recent empirical results. Different types of scaling are distinguished and several definitions of scaling exponents, scaling and multi-scaling processes are given. Methods to estimate such exponents from empirical financial data are reviewed. A detailed description of the Generalized Hurst exponent approach is presented and substantiated with an empirical analysis across different markets and assets.

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Multi-scaling in Finance

Using the eigenvalues and eigenvectors of correlations matrices of some of the main financial market indices in the world, we show that high volatility of markets is directly linked with strong correlations between them. This means that markets tend to behave as one during great crashes. In order to do so, we investigate financial market crises that occurred in the years 1987 (Black Monday), 1998 (Russian crisis), 2001 (Burst of the dot-com bubble and September 11), and 2008 (Subprime Mortgage Crisis), which mark some of the largest downturns of financial markets in the last three decades.

Correlation of financial markets in times of crisis

In this paper, the multifractality degree in a collection of developed and emerging stock market indices is evaluated. Empirical results suggest that the multifractality degree can be used as a quantifier to characterize the stage of market development of world stock indices. We develop a model to test the relationship between the stage of market development and the multifractality degree and find robust evidence that the relationship is negative, i.e., higher multifractality is associated with a less developed market. Thus, an inefficiency ranking can be derived from multifractal analysis. Finally, a link with previous volatility time series results is established.

A multifractal approach for stock market inefficiency

Limit order books (LOBs) match buyers and sellers in more than half of the world’s financial markets. This survey highlights the insights that have emerged from the wealth of empirical and theoretical studies of LOBs. We examine the findings reported by statistical analyses of historical LOB data and discuss how several LOB models provide insight into certain aspects of the mechanism. We also illustrate that many such models poorly resemble real LOBs and that several well-established empirical facts have yet to be reproduced satisfactorily. Finally, we identify several key unresolved questions about LOBs.

Limit Order Books

By applying the multifractal detrended fluctuation analysis to the high-frequency tick-by-tick data from Deutsche Borse both in the price and in the time domains, we investigate multifractal properties of the time series of logarithmic price increments and inter-trade intervals of time. We show that both quantities reveal multiscaling and that this result holds across different stocks. The origin of the multifractal character of the corresponding dynamics is, among others, the long-range correlations in price increments and in inter-trade time intervals as well as the non-Gaussian distributions of the fluctuations. Since the transaction-to-transaction price increments do not strongly depend on or are almost independent of the inter-trade waiting times, both can be sources of the observed multifractal behaviour of the fixed-delay returns and volatility. The results presented also allow one to evaluate the applicability of the Multifractal Model of Asset Returns in the case of tick-by-tick data.

Multifractality in the stock market: price increments versus waiting times

We analyzed multifractal properties of 5-min stock returns from a period of over two years for 100 highly capitalized American companies. The two sources: fat-tailed probability distributions and non-linear temporal correlations, vitally contribute to the observed multifractal dynamics of the returns. For majority of the companies the temporal correlations constitute a much more significant related factor, however.

Components of multifractality in high-frequency stock returns

We show that recent stock market fluctuations are characterized by the cumulative distributions whose tails on short, minute time scales exhibit power scaling with the scaling index α > 3 and this index tends to increase quickly with decreasing sampling frequency. Our study is based on high-frequency recordings of the S&P500, DAX and WIG20 indices over the interval May 2004–May 2006. Our findings suggest that dynamics of the contemporary market may differ from the one observed in the past. This effect indicates a constantly increasing efficiency of world markets.

Stock market return distributions: From past to present

We analyse the structure of the distribution of eigenvalues of the stock market correlation matrix with increasing length of the time series representing the price changes. We use 100 highly capitalized stocks from the American market and relate the result to the corresponding ensemble of Wishart random matrices. It turns out that systematically more eigenvalues remain beyond the borders prescribed by this variant of the random matrix theory (RMT). This may indicate that even the bulk of the spectrum of the stock market correlation matrix carries some sort of correlations that are masked by a measurement noise when the time series used to construct the matrix are short. We also study some other characteristics of the “noisy” eigensignals, like their return distributions, temporal correlations or their multifractal spectra, and the results support the above conclusions.

https://arxiv.org/pdf/physics/0505074.pdf

P. Oświe¸cimka

Papers

Multifractality in the stock market: price increments versus waiting times

Components of multifractality in high-frequency stock returns

Stock market return distributions: From past to present

The bulk of the stock market correlation matrix is not pure noise