Non-trivial scaling of fluctuations in the trading activity of NYSE

Abstract

Complex systems comprise a large number of interacting elements, whose dynamics is not always a priori known. In these cases -- in order to uncover their key features -- we have to turn to empirical methods, one of which was recently introduced by Menezes and Barabasi. It is based on the observation that for the activity f_i(t) of the constituents there is a power law relationship between the standard deviation and the mean value: sigma_i ~ ^alpha. For stock market trading activity (traded value), good scaling over 5 orders of magnitude with the exponent alpha = 0.72 was observed. The origin of this non-trivial scaling can be traced back to a proportionality between the rate of trades and their mean sizes . One finds ~ ^0.69 for the ~1000 largest companies of New York Stock Exchange. Model independent calculations show that these two types of scaling can be mapped onto each other, with an agreement between the error bars. Finally, there is a continuous increase in alpha if we look at fluctuations on an increasing time scale up to 20 days.