Knowledge to work

Using the relationship of VIX futures to the index, we can determine how well the contracts will replicate the index. We can calculate the net exposure of different futures and options contracts in terms of the VIX as an aggregate risk measure, or to cross-hedge different futures and options contracts.

Consider, for example, a trader who has multiple positions in VIX options but desires to hedge directional exposure. He does not need to hedge each maturity with its futures contract, but can hedge all of the positions with the most liquid near-term contracts.

Let’s say the trader is short 1,000 delta in May 2011 options. Instead of buying 1,000 May 2011 futures contracts, he can figure out his VIX exposure in terms of near-term November 2010 contracts. If May11 futures delta is 0.5, and Nov10 futures delta is 0.9, the trader has the equivalent of 1,000 * 0.5 = 500 delta of the VIX underlying. In terms of November 2010 futures, it is 500 / 0.9 = 555 delta. So, instead of buying 1,000 May 2011 futures, the trader can buy 555 November 2010 futures, taking advantage of higher liquidity of near-term futures.

These findings also explain why volatility ETNs (VXX, VXZ, XXV in the United States and VIXS, VSXX in Europe) have somewhat low correlation with the VIX itself. For example, the VXX ETN holds the front and second month futures with average maturity of about 30 calendar days and has a return correlation of about 50%, according to our calculation, which is consistent with "Underlying issues" (below).

Other Greeks for the futures (Γ, Θ, and vega with respect to volatility of the VIX) can be calculated in a similar manner. It may be surprising to think of Γ for futures, but remember that there is no arbitrage relationship between the index and futures. The only connection is a non-linear statistical relationship that is dynamic in time, which gives non-zero values to Greeks other than delta. That being said, delta remains by far the most dominant factor, and the relationship between the index and futures can be thought of as "almost" linear.

However, it would be an oversimplification to treat the relationship between the index and futures as completely so. One of the consequences of non-linearity is non-zero expected costs for rolling a futures contract from one month to the next. Intuitively, there is a cost to rolling a long position because there is no riskless arbitrage possible between the two months, so the relationship is purely statistical and the distribution of VIX is positively skewed; there is a price to pay for convexity.

Empirical observations support this. Given the data from the inception of trading in VIX futures until the end of 2009, we estimated the average difference between the front and second month futures on the expiration day to be 0.8 VIX points, or 6.7% of value, and the second month futures is more expensive than the front in 82% of cases. These costs do not include the bid-ask spread; they are statistically and economically significant. These findings are in line with Standard & Poor’s (2009), which reports that a portfolio of short-term VIX futures is "expected to suffer from roll loss due to term structure decay" and provide a complementary analysis of rolling costs.

Volatility leak

The implied volatility of options will decrease as time to expiration increases. Because the index is mean-reverting, its volatility will have a term structure with a downward slope. That means if you calculate the annualized standard deviation using one-day returns, then using two-day returns, three-day returns, etc., you will get progressively decreasing numbers.

Using the last 19 years of data from the beginning of 1990 through the end of 2009, we can calculate that one-day volatility of the VIX is 5.9%, or 93% per year, one-month annualized volatility is about 60%, half-year annualized volatility is about 45%, and one-year volatility is only about 35% (see "Volatility through time," below).

Included on the chart are recent implied volatility levels from at-the-money VIX options (for available maturities). As with other equity indexes, implied volatility is usually higher than historical; however, the general shape and slope of implied and historical volatilities are comparable.

This increase in volatility (as options get closer to expiration) is completely in-the-model behavior caused by mean-reversion and does not correspond to the increase in implied volatilities as normally observed in equity options right before expiration, where it is an out-of-the-model phenomenon.