By applying the multifractal detrended fluctuation analysis to the high-frequency tick-by-tick data from Deutsche Börse 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.

We analyzed multifractal properties of 5-minute stock returns from a period of over two years for 100 highly capitalized American companies. The two sources: fat-tailed probability distributions and nonlinear 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. The first, simple stock market model by Bachelier [1], based on Gaussian random walk, although it used to be applied for a long time, did not successfully pass practical tests as soon as large data samples were available and it was eventually rejected. The non-Gaussian distributions of price fluctuations, their persistent nonlinear temporal correlations, the intermittent behaviour present on all time scales and even the explicitly shown scaling invariance, together with some other properties of the stock market evolution, required introduction of more advanced and more appropriate models [2,3,4,5,6,7]. What was especially interesting was the apparent similarity of the stock market dynamics to fluid turbulence [8], which led to the development of models based on the multiplicative cascades [9,10,11,12,13]. This sort of processes generate signals which are inherently multifractal with a continuous spectrum of scaling indices f (α) [14,15]. Consistently, the real data from different financial markets (stock, forex and commodity ones) show clear multifractal properties [16,17,18,19,20,21,22,23,24].It has already been pointed out in literature that the two fundamental factors leading to multifractal behaviour of signals are the nonlinear time correlations

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 The so-called financial stylized facts are among the central issues of econophysics research. Much effort has been devoted on both the empirical and the theoretical level to such phenomena like fat-tailed distributions of financial fluctuations, persistent correlations in volatility, multifractal properties of returns etc. Specifically, the interest in the return distributions can be traced back to an early work of Mandelbrot [1] in which he proposed a Lévy process as the one governing the logarithmic price fluctuations. Much later this issue was revisited in [2] based on data with much better statistics and a new model of exponentially-truncated Lévy flights was introduced. Then, in an extensive systematic study of the largest American stock markets [3] the distribution tails for both the prices and the indices were shown to be power-law with the scaling exponent α ≃ 3. The most striking outcome of that study was that despite the fact that the tails were well outside the Lévy-stable regime (α ≤ 2), they were apparently stable under time aggregation up to several days for indices and up to a month for stocks. The existence of return distributions with

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 result to the corresponding ensemble of Wishart random matrices. It turns out that systematically more eigenvalues stay 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.Comment: updated version to appear in Physica