Cross-Gamma And Correlation Risk

Ordinary gamma measures how an option's delta changes as its own underlying moves. Cross-gamma measures something extra: how the delta of a multi-asset option — like a basket or worst-of option written on two or more stocks — changes when a different underlying in that basket moves. A worst-of option's sensitivity to stock A doesn't stay fixed as stock B rises and falls; the two underlyings' moves interact, and that interaction is cross-gamma.

This matters because cross-gamma can't be hedged by delta-hedging each underlying separately — doing so leaves the cross term unaddressed, since it's a property of how the two assets move together, not of either one alone. Concretely, if two stocks in a worst-of option's basket are highly correlated, a move that pushes both up or both down together changes the option's value differently than the same-sized moves happening in opposite directions would, even though each single-asset delta hedge would look adequately hedged in isolation.

This is why correlation itself becomes a risk to manage on a multi-asset options book, not just each underlying's volatility: a desk can be flat delta and flat single-asset gamma in every name and still lose money if realized correlation between the names shifts.

A simple illustration: a worst-of put on two stocks currently both at $100 is worth more when the stocks are weakly correlated (more chances one of them falls hard) than when they move in lockstep. If stock A drops to $95, the option's sensitivity to stock B changes depending on whether B tends to fall alongside A or not — that dependence of one delta on the other asset's level is exactly the cross-gamma term, and no amount of hedging A and B individually captures it.