Size matters: some stylized facts of the stock market revisited

Abstract

We reanalyze high resolution data from the New York Stock Exchange and find a monotonic (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 grows 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. 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]. PACSStudies 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

Apparent criticality and calibration issues in the Hawkes self-excited point process model: application to high-frequency financial data

“…Finally, we draw attention to the fact that the findings of the present section challenge all studies concerned with inter-event times, such as [55,56,57,58,59,60,61,62] and others, where the observations of heavy tailed distributions, long memory correlation functions and multifractal scaling of the inter-event durations may be strongly biased by the bursty nature of the raw data, subjected to the "bundling" effects.…”

How efficiency shapes market impact

“…If the immediate impact I t = S t − X 0 has a second derivative bounded below zero 21 , in the limit where the number of informed traders K → ∞, any Nash equilibrium must satisfy the fair pricing condition π N = 0 for 1 < N < M . On average market makers profit from orders of length one and take (equal and opposite) losses from orders of length M .…”

“…The system of martingale conditions (Eq. 6) and fair pricing conditions 21 Note that this is sufficient for concavity. 22 The price S M +1 does not exist, so R + M is not needed.…”

Agent-based models of financial markets

“…Fact (iii) has first been discovered by Mandelbrot [138] who proposed the Levy stable model for financial returns. Over the recent years, the majority opinion (see [196,72] for dissent) among researchers in the field has, however, converged to the view that the tails of the cumulative distribution of returns are characterized by a power-law with exponent around 3. The underlying data would, hence, possess finite variance in contradiction to the Levy stable model.…”