Size matters : some stylized facts of the market revisited

We reanalyze high resolution data from the New York Stock Exchange and find a monotonous (but not power law) variation of the mean value per trade, the mean number of trades per minute and the mean trading activity with company capitalization. We show that the second moment of the traded value distribution is finite. Consequently, the Hurst exponents for the corresponding time series can be calculated. These are, however, non-universal: The persistence increases with larger capitalization and this results in a logarithmically increasing Hurst exponent. A similar trend is displayed by intertrade time intervals. Finally, we demonstrate that the distribution of the intertrade times is better described by a multiscaling ansatz than by simple gap scaling. PACS. 89.75. -k Complex systems – 89.75. Da Systems obeying scaling laws – 05.40. -a Fluctuation phenomena, random processes, noise, and Brownian motion – 89.65. Gh Economics; econophysics, financial markets, business and management Understanding the financial market as a self-adaptive, strongly interacting system is a real interdisciplinary challenge, where physicists strongly hope to make essential contributions [1, 2, 3]. The enthusiasm is understandable as the breakthrough of the early 70’s in statistical physics taught us how to handle strongly interacting systems with a large number of degrees of freedom. The unbroken development of this and related disciplines brought up several concepts and models like (fractal and multifractal) scaling, frustrated disordered systems, or far from equilibrium phenomena and we have obtained very efficient tools to treat them. Many of us are convinced, that these and similar ideas and techniques will be helpful to understand the mechanisms of the economy. In fact, there have been quite successful attempts along this line [4, 5, 6]. An ubiquitous aspect of strongly interacting systems is the lack of finite scales. The best understood examples are second order equilibrium phase transitions where renormalization group theory provides a general explanation of scaling and universality [7]. It seems that some features of the stock market can indeed be captured by these concepts: For example, the so called inverse cube law of the distribution of logarithmic returns shows a quite convincing data collapse for different companies with a good fit to an algebraically decaying tail [8, 9]. Studies in econophysics concentrate on the possible analogies, although there are important differences between physical and financial systems. This is, of course, a trivial statement – it is enough to refer to the abovementioned self-adaptivity, to the possibility of influencing a e-mail: eisler@maxwell.phy. bme.hu the system by its characterization or to the intrinsic nonstationarity of economic processes. Here we would like to emphasize the discrepancy in the levels of description. In the case of a physical system undergoing a second order phase transition, it is natural to assume scaling on profound theoretical grounds and the (experimental or theoretical) determination of, e.g., the critical exponents is a fully justified undertaking. There is no similar theoretical basis for the financial market whatsoever, therefore in this case the assumption of power laws should be considered only as one possible way of fitting fat tailed distributions. Also, the reference to universality should not be plausible as the robustness of qualitative features – like the fat tail of the distributions – is a much weaker property. Therefore, e.g., averaging distributions over companies with very different capitalization is questionable. While we fully acknowledge the process of understanding based on analogies as an important method of scientific progress, we emphasize that special care has to be taken in cases where the theoretical support is sparse. Motivated by this, the aim of the present paper is to carry out a careful analysis of the high resolution data of the New York Stock Exchange with special emphasis on the effects caused by the size of the companies. The paper is organized as follows. After the introduction of notations in Section 1, Section 2 presents the results on the capitalization dependence of various measures of trading activity. In Section 3 we show that the distribution of the traded values is not Lévy stable as suggested previously [10]. Consequently, the Hurst exponents of the related time series exist, these are analyzed in Section 4. We point out, that correlations in trading activity are 2 Zoltán Eisler, János Kertész: Size matters: some stylized facts of the market revisited strongly non-universal with respect to company size, and that the Hurst exponent of the traded value depends logarithmically on the mean traded value per minute. Section 5 deals with the time intervals between trades and we give indications, that their distribution is better described by a multiscaling ansatz than by gap scaling proposed earlier [11]. Finally, Section 6 concludes. 1 Notations and data For a given time window size ∆t, let the total traded value (activity, flow) of the ith stock at time t be