An index implies 18% volatility while the weighted average of its constituents implies 28%. What average correlation does that pair imply, and what is the trade if you think realised correlation will be lower?

An index implies 18% volatility while the weighted average of its constituents implies 28%. What average correlation does that pair imply, and what is the trade if you think realised correlation will be lower?

Approach: For a large well-diversified index the idiosyncratic terms wash out, so index variance is approximately average constituent variance times average correlation. Invert that.

0.41. For a large index the variance is approximately rho times the square of the average constituent volatility, so rho = (18/28)^2 = 0.4133. If you expect realised correlation below 0.41 the trade is dispersion: sell index volatility and buy the single stock volatility in vega-weighted size, which makes money when the constituents move a lot while offsetting each other and the index stays calm. The position is short correlation, and the exposure it hides is that correlation goes to one in a crash, so the index leg loses far more than the single stock legs gain on a market-wide selloff. Dispersion is therefore a carry trade with a short tail, and its historical profitability comes from index puts being persistently bid by hedgers, which raises index implied volatility relative to the constituents.

Follow-up: How does the answer change if you include the idiosyncratic terms and the index has only 10 names with unequal weights?

Key concepts: implied correlation, dispersion trade, index volatility, index variance.