## Apr 17, 2013

### VIX Mean Reversion Or VIX Median Reversion?

In January 2013 issue of Expiring Monthly Bill Luby wrote an article “Drilling Down on VIX Mean Reversion” where he states: “Before diving headlong into the data, I will go one step further and propose that it is worth challenging the idea of mean reversion in general. It is not that I am challenging the idea of the VIX tending toward a middling value over time, but rather that median reversion might be a better concept to consider than mean reversion. The distinction is not a trivial one, with the lifetime arithmetic mean of the VIX at 20.42 and the median (middle value of the series) at 18.83. If I am trading distant month VIX options at a strike of 20, it certainly would be helpful to know whether that value is above or below the historical record … “

I absolutely agree with the statement, but wanted to elaborate on explanation. The difference between mean and median has to do with skewness of distribution, and VIX prices distribution is very skewed. If we consider a level-reversion model of VIX the result will be as follows – VIX will tend toward some level, but its average will be higher than that level.

Consider a very simple model – first proposed for volatility futures by Detemple (2000). I consider it to be the simplest practical model that explains most (not all) features of VIX futures. In the model logarithm of VIX prices are mean-revering. As one can plainly see while VIX prices have huge right tail, log of VIX is much more symmetric.

The formula for the process is VIX = exp(S), where dS = λ(µ-S)dt + σdW. If we take historical prices of VIX we can recover parameters λ=3.88, µ=2.95, and σ=0.97. This means that according to the model VIX will tend to rise when below exp(µ) = 19.05, and fall when above 19.05, but its average price is going to be exp(µ + σ2 / 4 λ) = 20.25. In fact, looking at the formulas it is clear that exp(µ) <= exp(µ + σ2 / 4 λ) , that is VIX average price cannot be smaller than it’s reversion level. In summary, mean-median difference that Bill Luby pointed out does exist, and the explanation for the difference is skew in the prices; mean-reversion takes place not in price levels but in some (more symmetric) function of price levels.

Caveat: the model above it really too simplistic to describe all the fine details of VIX behaviour. One critical issue that is clear in the charts provided by Bill Luby is non-linearity in the 100-day reversion of the VIX.

Here is the chart of 100-day VIX returns as a function of starting VIX level. I removed values that have less than 100 observations. If VIX were to follow a simple reversion process as above we would expect to see a monotonically declining function that intersects vertical axis at around 19.05. What we see in the data is that there are two critical levels - lower around 14, and higher one around 24, suggesting bimodal distribution.

Intuitively, if expected (average) return at some level is zero, it means that VIX is “indifferent” whether to go up or down, and is at some sort of equilibrium level. Data suggests that instead of some “population average” level of 19, there may be two “local” levels of mean-reversion that the model does not show. Over the weekend I tried statistical clustering techniques to identify these levels but unfortunately without results. If one looks at the histograms above it is hard to see two distinct peaks – there is simply more “noise” in the data. I believe this approach is unlikely to yield a clear picture for VIX dynamics.