Evaluation of common models used in the estimation of the historical volatility applied to trade-by-trade stock returns data from the U.S. and Mexico

The objective of the investigation is to characterize the fundamental statistical properties of various measures that have been used to model the volatility of the trade-by-trade returns process of common stock. Ideally we would like the observations of the returns process or the logarithm of the return process to have been generated as independent and identically distributed Gaussian variates such that its mean is a constant and the variance is a constant or a linear function of time. This is because the stationary Gaussian distribution is the only probability distribution that is completely characterized by two parameters (mean and variance). When the logarithm of the percentage change in price is used to measure volatility it lends itself to the use of the geometric Brownian motion (which is based on the logarithm of the percentage changes having a Gaussian distribution).