Why does the volatility of the stock market as implied by the prices of equity index options generally exceed actual (realized) volatility, thereby indicating large returns for sellers of index options? Is the reward of selling such options commensurate with the risk? In the June 2008 version of his paper entitled “The Volatility Premium”, Bjorn Eraker models the volatility risk premium based on the long-run effects of small (normal diffusion) and large (non-normal jumps) shocks to volatility. Using daily returns for the S&P 500 index and daily levels of the CBOE Volatility Index (VIX) over the period 1990-2007, *he concludes that:*

- On average, at-the-money S&P 500 index options with one-month maturities imply an index volatility of 19% (based on standard deviation of annualized returns), about 18% more expensive than options priced for a volatility of 16% (close to the observed volatility of 15.7%). This premium varies over time, generally rising and falling with the level of volatility (see chart below).
- Empirical evidence confirms that the average returns generated by index option sellers are substantial.
- The model presented predicts a volatility risk premium roughly in line with empirical observation, driven more by the risk of violent jumps in volatility than the risk of slow drift in volatility. The model predicts high mean returns for shorting out-of-the-money index put options, regardless of volatility level and whether hedged for S&P 500 movement or not. However, variability of returns is also high.

The following chart, extracted from the paper, shows the volatility risk premium (option-implied volatility minus forecasted volatility, in standard deviations) for the S&P 500 index over the period 1990-2007. It shows that the premium is generally positive but variable.

In summary, *modeling suggests that sharp jumps in stock market volatility drive investors to overprice some equity index options, most consistently out-of-the-money put options.*