Abstract:
 The raw data set, downloaded from the Wharton Research Data Services, consists of ticfortic actual trade price per share for S&P500 constituents for 2006, 2005, 2001, and 1998. The returns are considered at fiveminute and oneminute intervals for each year. For each stock the empirical distribution of the waiting time to hit the upper (lower) ten percentile of the returns is considered. Most of the empirical distributions are close to each other under two different comparison criteria, ROC area and KolmogorovSmirnov distance. This may be regarded as an empirical invariance. The financial and mathematical implications, other percentiles, and some other markets are discussed. Consider the following mathematical setup. Given a triangular array of row wise finitely exchangeable random variables, for each row consider the empirical distribution of the waiting time to hit a percentile range of a fixed d percent, e.g. upper ten percentile. We prove that the empirical distributions converge a.s. to a geometric distribution. This result is applicable to the case when the return follows a Levy process but not to the fractional Brownian motion with index different from 1 or 1/2. Note that the empirical invariance found in the market is quite far away from the geometric distribution as well as the empirical distributions obtained from fractional Brownian motion
