## May 9, 2011

### A Useful Risk Index

Recently CBOE introduced SKEW index which many sources have covered and explained extensively. Despite being dubbed Black Swan index and Tail index, values of SKEW index defy simple interpretation. In what follows I hope to construct a simple and intuitive measure of risk that incorporates the same information as the SKEW.

The idea for measure is not mine; credit for inspiration belongs to Mark Sebastian from Optionspit who few weeks ago proposed an index that he called CBOP. CBOP was (originally) defined as CBOP = VIX * SKEW / 100. The index is intended to be a more complete measure of downside risk of the SPX index. By construction CBOP is increasing with both VIX and SKEW. If implied skew from the options is zero – SKEW index equals 100, and CBOP equals to the VIX, signifying no “additional” risk on the downside.

The problem with CBOP that it suffers from arbitrary construction – for example an alternative way to construct an index with the same qualities could have been VIX + SKEW – 100, or infinitely many other ways. The second problem is that it is only comparable to to its own time series, and does not have the same mathematical “meaning” as the VIX.

The solve these two issues I propose to expand on the idea of the VIX with so-called downside VIX, or put VIX, and upside VIX, or call VIX. VIX is defined as standard deviation of returns, or square root of variance, or √(E[R2]). Similarly call VIX and put VIX defined as √(E[R2|R>0]) and √(E[R2|R<0]) or standard deviation condition on return being positive or returns being negative. Just like VIX index corresponds to the value of variance swap, put VIX and call VIX correspond to values of weighted variance swaps, namely down variance swap and up variance swap.

These “partial” risk measures are not original in any way, for example Sortino ratio uses downside volatility in denominator, which is realized analogue to PUT VIX. Again, nothing above is original research, I’m just proposing to use a simple intuitive and existing concept instead of a new one.

Calculating these new indexes is trivial. VIX is calculated as weighted sum of OTM calls and puts, so by definition it contains call VIX (using only calls) and put VIX (using only puts). Please note that VIX contains an adjustment term that converts lowest call (that is ITM) to OTM put. I believe that calculating these indexes in real-time and also providing historical back-calculations should be really easy for CBOE. And as I mentioned before these indexes contain the same information as SKEW index.

But for illustration purposes I created a numerical approximation for these indexes. VIX and SKEW indexes are back-calculated from 1990 and provide information on the second and third moments. Non-zero skew imposes a lower bound on kurtosis, and these can be parametrized with normal inverse gaussian distribution. Then I calculate up-variance and down variance with numerical integration, so its all very straight-forward.

The image above shows the time-series of put VIX (highest line in light blue), VIX - middle, and call VIX (lowest line in dark blue) . The effect of skew is obvious, for example during 2008 VIX topped at 80, which put VIX showed higher number at over 90. Below is the YTD plot of indexes. Raw data is available here.

As I mentioned before, the numbers are intuitive - for example the latest set of numbers is:
call VIX = 14.66, VIX = 18.40, and put VIX 21.50. We can infer that market expects upside volatility of about 15%, but downside volatility of 22%.

#### 1 comment:

1. You mentioned the Sortino Ratio. the Sortino Ratio is losely defined, and several methods of quantifying downside risk exist. The most common appears to be the 2nd order partial lower moment (the method I use this Excel spreadsheet).