Models of asset returns: changes of pattern from high to low event frequency

In this paper we have analysed asset returns of the New York Stock Exchange and the Helsinki Stock Exchange using various time-independent models such as normal, general stable, truncated Lévy, mixed diffusion-jump, compound normal, Student t distribution and power exponential distribution and the time-dependent GARCH(1, 1) model with Gaussian and Student t distributed innovations. In order to study changes of pattern at different event horizons, as well as changes of pattern over time for a given event horizon, we have analysed high-frequency or short-horizon intraday returns up from 20 s intervals to a full trading day, medium-frequency or medium-horizon daily returns and low-frequency or long-horizon returns with holding period up to 30 days. As for changes of pattern over time, we found that for medium-frequency returns there are relatively long periods of business-as-usual when the return-generating process is well-described by a normal distribution. We also found periods of ferment, when the volatility grows and complex time dependences tend to emerge, but the known time dependences cannot explain the variability of the distribution. Such changes of pattern are also observed for high-frequency or short-horizon returns, with the exception that the return-generating process never becomes normal. It also turned out that the time dependence of the distribution shape is far more prominent at high frequencies or short horizons than the time dependence of the variance. For long-horizon or low-frequency returns, the distribution is found to converge towards a normal distribution with the time dependences vanishing after a few days.