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Estimating Volatilities

I can observe an asset's return; it is the percentage change in the observed market value of the asset between two well-defined points in time. Strictly speaking, I cannot observe the volatility of the return. I can only estimate it. In Chapter 5, volatility—or standard deviation—was a sample statistic. It was the square root of the return variance. The variance was the mean of the squared deviations of returns from their sample mean. We didn't discuss N—the sample size. The implicit assumption was that N was the size of the historical example. As statisticians, we prefer N to be large, as more information is preferred to less when estimating statistics. The annualized volatility of the return on the S&P 500 since 1950 is 14.7 percent. That is a volatility averaged over 60 years (N = 721). But what if I think that the market today is not average, that is, what if the market is clearly in a high or low volatility regime?

Analysts use several cheap methods to deal with changing volatility. One method is to measure volatility over a short trailing horizon as a moving average, for example, using a trailing three-year or five-year window. The choice of window width is arbitrary but the idea is to provide a long enough window to generate meaningful estimates while at the same time trying to keep the window short enough to capture the present volatility regime. This method is clearly suboptimal for three reasons: first, it throws away data outside the window while ...