Index volatility is lower than the average volatility of its members because the members do not move together. The gap is a function of correlation, and dispersion traders take a view on it: short the index straddle, long a basket of single-name straddles.
The trade wins when stocks move a lot individually but the index does not — an earnings season where winners and losers offset. It loses badly in a crash, when correlation goes to one, the index moves more than the components did separately, and the short index leg is the wrong side of everything.
Example: an index implies 18% volatility while its members average 31%. A dispersion book sells the index straddle and buys member straddles sized to be vega neutral. It bleeds in quiet trending markets and loses fast in a correlated selloff.
Related: relative-value-volatility, correlation, variance-risk-premium, embedded-option