Realistic Processes for Stocks from One Day to One Year

A realistic ARCH process is set so as to duplicate for all practical purposes the properties of stock time series from 1 day to 1 year. The process includes heteroskedasticity with long memory, leverage, fat-tail innovations, relative return, price granularity, and holidays. Its adequacy to describe empirical data is controlled over a broad panel of statistics, including (robust L-statistics) skew, (robust) kurtosis, shape factor for the volatility distribution, and lagged correlations between combinations of return and volatility. These statistics are computed for returns and volatilities with characteristic time intervals ranging from 1 day to 1 year. This wide cross-check between stock time series and simulations ensures that the most important features of the data are correctly captured by the process up to 1 year. The by-products of the statistical analyses and estimations are 1) a positive skew, 2) a cross-sectional relation between kurtosis and heteroskedasticity, 3) the very similar cross-sectional distribution for the statistics evaluated over the empirical data set or for the process with one set of parameters and 4) the heteroskedasticity is very close to an integrated volatility process