Statistical Foundations: Predicting Volatility
Following are the major steps we take to estimate volatility, using the S\&P 500 Index:
Let's look at historical market rates. We start by converting daily prices into log changes. (Daily log changes are conceptually similar to percentage returns, except they are continuously compounded.)
Our goal is to use past returns to predict the volatility of future returns. We can plot changes in market rates onto a histogram and fit a normal distribution. You can see here that the normal distribution is a reasonable but not perfect fit for stock returns. We expect this distribution of returns to stay reasonably stable over time. However, every day we re-estimate the standard deviation of the distribution to predict tomorrow's volatility (hence our predicted distribution of returns changes each day).
By re-estimating volatilities every day, we can get a dynamic estimate of risk. Observe the dynamic nature of risk--several periods of high and low volatility. We expect approximately 5% excessions on both the upside and the downside. Actual daily excessions over this 5-year period were approximately 6% on the upside and 4% on the downside.
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