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.