Spot is 100 and the 3-month 100-strike implies 20%. The skew slope is -0.10 volatility points per point of strike. Spot falls to 98. What volatility does the 100 strike mark at under sticky strike and under sticky delta, and what is the effective delta if the option has 0.20 vega per point and 0.53 Black-Scholes delta?

Spot is 100 and the 3-month 100-strike implies 20%. The skew slope is -0.10 volatility points per point of strike. Spot falls to 98. What volatility does the 100 strike mark at under sticky strike and under sticky delta, and what is the effective delta if the option has 0.20 vega per point and 0.53 Black-Scholes delta?

Approach: Write the smile as a function of strike minus spot, then differentiate the option value along spot including the volatility that moves with it.

19.8% under sticky delta and 20.0% under sticky strike. Sticky strike holds each strike's volatility fixed as spot moves, so the 100 strike stays at 20.0%. Sticky delta holds the smile fixed in moneyness, so with the skew written as sigma(K) = 20 - 0.10*(K - S) the 100 strike sits 2 points out of the money at spot 98 and marks 19.8%. Sticky delta also changes the hedge: d(sigma)/dS = +0.10 volatility points per point of spot, so multiplying by the 0.20 vega gives an effective delta of 0.53 + 0.02 = 0.55. A desk running sticky strike deltas in a market that actually behaves sticky delta is systematically under-hedged on the way up and over-hedged on the way down, which shows up as a slow bleed rather than a single loss.

Follow-up: Which regime does an equity index usually follow in a calm market, and which does it switch to in a crash?

Key concepts: sticky strike, sticky delta, skew, effective delta, vega.