Measurements of volatility typically refer to the standard deviation of returns over a specified period. That obviously includes returns both below and *above* the mean. In practice, however, investors tend to be concerned primarily with downside risk, leading them to regard returns differently: positive and negative logarithmic returns that are equally distant from the mean are not treated as such by investors. Negative surprises have a much greater effect on volatility than do positive ones – witness the explosion of interest in 2008 in all things VIX and volatility-related.

This helps explain the phenomenon of vertical volatility skew. In equity markets, the long-only bias resulting from the structure of mutual funds and other institutional factors means that investors are considerably more nervous, on any given day, about a potential 5% decline than they are fearful (or greedy) about the possibility of a 5% rally. That uneven fear causes investors to overpay for put protection and creates a persistent “volatility smile” in which the implied volatility for deep out-of-the-money options will tend to be significantly higher than at- and in-the-money options in the same expiration cycle – with a volatility “smirk” occurring when skew is more exaggerated on the put side. In commodities, expectations can differ dramatically, such that the smirk is tilted to the call side.

In “How Asymmetric is U.S. Stock Market Volatility?“, Ederington and Guan explore the asymmetry of volatility not by analyzing skewness, but by tracing the effects of equally large positive and negative return shocks on implied volatility, realized volatility, and models that attempt to predict volatility for asymmetric time series:

This paper explores differences in the impact of equally large positive and negative surprise return shocks in the aggregate U.S. stock market on: 1) the volatility predictions of asymmetric time series models, 2) implied volatility, and 3) realized volatility. Both asymmetric time series models and implied volatility predict an increase in volatility following large negative surprise returns and ex post realized volatility normally rises as predicted. However, while asymmetric time series models, such as the EGARCH and GJR models, predict an increase in volatility following a large positive return shocks (albeit a much smaller increase than following a negative shock of the same magnitude), both implied and realized volatility generally fall sharply. While asymmetric time-series models predict a decline in volatility following near-zero returns,

both implied and realized volatility are normally little changed from levels observed prior to the stable market. Reasons for the differences are explored.

So the problem tackled here is that the stochastic models developed precisely to deal with the asymmetry of stock markets don’t seem to make correct predictions when it comes to these discrete events. Which metric does the best job? The implied volatility read straight from option prices – the VIX, actually – appears to correspond to future realized volatility more closely in the types of situations under review. Every approach gets the impact of large negative shocks correct: volatility rises sharply. But following a large positive return, the time series models predict a small increase in volatility even though implied and realized volatility tend to decline afterward. The authors offer an explanation that the GARCH models may tend to overweight extreme return observations. They conclude that the adage that “volatile markets beget volatile markets” may not be exactly right: “Volatile markets do tend to follow bear markets but implied and realized volatility both tend to fall following bull markets. Following stable markets (near-zero returns), implied and realized volatility are little changed from the levels observed prior to the near-zero return.”

Most traders we know aren’t relying heavily on GARCH or related models for volatility predictions anyway, but it is helpful to get some confirmation of the usefulness of options prices for estimating future volatility. Microcosmically, then, options markets appear efficient in the weak sense (even if markets in general seem these days like a maniacally inefficient means for structuring society).