Mean Reversion Models of Financial Markets

I discuss mean reversion in the first and the second moment of the return distribution. After a discussion of the concepts and a summary of the findings in the literature, I show that investor´s perceptions of mean reversion play a role in stock market crashes. Turning to the second moment of the return distribution, expected volatility, I consider mean reversion using GARCH. The main finding is that in order to properly measure mean reversion, it is crucial to take parameter regime shifts into account. This can but need not necessarily be done by solving the change point detection problem. I present evidence that stock price volatility contains more than a single mean reversion time. After showing that the expectation of the sum of the estimates of the autoregressive coefficients of a GARCH(1,1) model is one when there are unknown parameter changes, I explore the phenomenon in simulations. For parameter changes within realistic ranges for stock price volatility I obtain global estimates indicating high persistence while the average data-generating mean reversion is of the order of a few days. Spectral analysis of the Dow Jones Industrial Average and the S&$P500 index between 1985 and 2001 reveals a short time scale of the magnitude of 5-10 days present in the data. I generalize these results to GARCH(p,q) models with p=1,2 and q=1,2.