INDEX IMPLIED LESS THAN SINGLE STOCKS DUE TO DIVERSIFICATION

The volatility of an index is capped by the weighted average volatility of its members. In order to show this we shall construct a simple index of two equal weighted members who have the same volatility. If the two members are 100% correlated with each other, then the volatility of the index is equal to the volatility of the members (as they have the same volatility and weight, this is the same as the weighted average volatility of the constituents).

Volatility of index has floor at zero when there is very low correlation
If we take a second example of two equal weighted index members with the same volatility, but with a negative 100% correlation (ie, as low as possible), then the index is a straight line with zero volatility.

Index volatility is bounded by zero and weighted average single stock volatility

While the simple examples above have an index with only two members, results for a bigger index are identical. Therefore, the equation below is true. While we are currently examining historical volatility, the same analysis can be applied to implied volatility. In this way, we can get an implied correlation surface from the implied volatility surfaces of an index and its single-stock members. However, it is usually easiest to look at variance swap levels rather than implied volatility to remove any strike dependency.

where

σI = index volatility
σi = single stock volatility (of ith member of index)
wi = single stock weight in index (of ith member of index)
n = number of members of index

CORRELATION OF INDEX CAN BE ESTIMATED FROM VARIANCE

If the correlation of all the different members of an index is assumed to be identical (a heroic assumption, but a necessary one if we want to have a single measure of correlation), the correlation implied by index and single-stock implied volatility can be estimated as the variance of the index divided by the weighted average single-stock variance. This measure is a point or two higher than the actual implied correlation but is still a reasonable approximation.

where
ρimp = implied correlation (assumed to be identical between all index members)

Proof implied correlation can be estimated by index variance divided by single stock variance. The formula for calculating the index volatility from the members of the index is given below.

where
ρij = correlation between single stock i and single stock j

If we assume the correlations between each stock are identical, then this correlation can be implied from the index and single stock volatilities.

Assuming reasonable conditions (correlation above 15%, approx. 20 members or more, reasonable weights and implied volatilities), this can be rewritten as the formula below.

This can be approximated by the index variance divided by the weighted average single-stock variance.

This approximation is slightly too high (approx. 2pts) due to Jensen’s inequality (shown below).

• Dispersion trading. Going short index implied volatility and going long single-stock implied volatility is known as a dispersion trade. As a dispersion trade is short Volga, or vol of vol, the implied correlation sold should be approx. 10pts higher value than for a correlation swap. A dispersion trade was historically put on using variance swaps, but the large losses from being short single stock variance led to the single stock market becoming extinct. Now dispersion is either put on using straddles, or volatility swaps. Straddles benefit from the tighter bid-offer spreads of ATM options (variance swaps need to trade a strip of options of every strike). Using straddles does imply greater maintenance of positions, but some firms offer delta hedging for 5-10bp. A disadvantage of using straddles is that returns are path dependent. For example, if half the stocks move up and half move down, then the long single stocks are away from their strike and the short index straddle is ATM.
• Correlation swaps. A correlation swap is simply a swap between the (normally equal weighted) average pairwise correlation of all members of an index and a fixed amount determined at inception. Market value-weighted correlation swaps are approx. 5 correlation points above equal weighted correlation, as larger companies are typically more correlated than smaller companies. While using correlation swaps to trade dispersion is very simple, the relative lack of liquidity of the product is a disadvantage. We note the levels of correlation sold are typically approx. 5pts above realized correlation.
• Covariance swaps. While correlation swaps are relatively intuitive and are very similar to trading correlation via dispersion, the risk is not identical to the covariance risk of structured product sellers (from selling options on a basket). Covariance swaps were invented to better hedge the risk on structure books, and they pay out the correlation multiplied by the volatility of the two assets.
• Basket options. Basket options (or options on a basket) are similar to an option on an index, except the membership and weighting of the members does not change over time. The most popular basket option is a basket of two equal weighted members, usually indices.
• Worst-of / Best-of option. The pricing of worst-of and best-of options has a correlation component.
  
• Outperformance options. Outperformance options pricing has as an input the correlation between the two assets.