Distributions of high-frequency stock market observables

Power laws and scaling are two features that have been known for some time in the distribution of returns (i.e., price fluctuations), and, more recently, in the distribution of volumes (i.e., numbers of shares traded) of financial assets. As in numerous examples in physics, these power laws can be understood as the asymptotic behavior of distributions that derive from nonextensive thermostatistics. Recent applications of the q-Gaussian distribution to returns of exchange rates and stock indices are extended here for individual U.S. stocks over very small time intervals and explained in terms of a feedback mechanism in the dynamics of price formation. In addition, we discuss some new empirical findings for the probability density of low volumes and show how the overall volume distribution is described by a function derived from q-exponentials.