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What is the difference between gamma and vanna?
Gamma and vanna are both second-order Greeks in the same family, and both do the same job at a high level: they update delta when a model input changes. The difference is which input drives the change. Gamma measures how delta changes when the underlying price moves. Vanna measures how delta changes when implied volatility moves. That single distinction, price versus volatility, separates the two Greeks and decides when each one matters.
Gamma is the curvature of an option against price. If a trader knows the current delta and gamma, the estimated new delta after a move is the old delta plus gamma multiplied by the size of the move. A 0.50 delta with 0.06 gamma drifts toward 0.68 after a three-point rally. Nothing about that calculation involves volatility. It is a pure price story.
Vanna is the curvature that links volatility and direction. It reads the change in delta per one-point change in implied volatility, and by the equality of cross-derivatives it equally reads the change in vega per one-point change in the underlying price. When volatility reprices, vanna is what makes a delta move even if the stock has not.
How gamma and vanna differ in practice
For a long vanilla option, gamma is positive. Vanna’s sign follows the pricing inputs and reverses when the position is short. Matched European calls and puts have equal gamma and vanna under the same inputs, because put-call parity ties the pair to stock and cash, which carry no curvature and no volatility exposure.
Gamma concentrates near the money. Vanna crosses zero near the money, where d2 = 0 in Black-Scholes, so volatility, time and rates set the exact point. With zero rates and dividends, S = K = $100, one year to expiration, and 50% volatility, vanna is 0.1933 per unit of volatility, or 0.0019 delta per volatility percentage point, and the zero-vanna strike is $88.25.
The worked figures in each term page show this plainly. Gamma turns a static delta into a moving target as price runs. Vanna, holding spot fixed at 100 and dropping implied volatility from 45 percent to 35 percent, pushes in-the-money call deltas up and out-of-the-money call deltas down while the at-the-money delta barely moves. Same delta, two different reasons to hedge.
When to use gamma vs vanna
The choice comes down to what is moving. When the concern is a fast directional move, especially in short-dated at-the-money contracts, gamma is the lens. It sizes how reactive a position is and how often a delta hedger must rebalance as price travels. Long gamma is stabilizing because it means selling into strength and buying into weakness; short gamma is destabilizing.
When the concern is a volatility event, vanna is the lens. Around earnings, an index rebalance, or a broad volatility shock, implied volatility can move sharply, and a book that is delta neutral at the current volatility can become directional after the surface reprices. Vanna quantifies that residual delta. Volatility traders often reprice spot and volatility together precisely so they can separate the gamma-driven delta change from the vanna-driven delta change.
Most complete risk work uses both. Gamma handles the price axis, vanna handles the volatility axis, and real markets move on both at once, often in correlated ways where declines pair with rising volatility and rallies with falling volatility.
Bottom line
Use gamma to measure how delta changes when the underlying moves. Use vanna to measure how delta changes when volatility reprices. Together they show how the same position responds to two inputs that move throughout the trading day.
Common questions
Gamma vs Vanna: FAQ
- Are gamma and vanna both second-order Greeks?
- Yes. Both are second derivatives of option value and both describe how the first-order Greek delta changes when an input moves. Gamma is the change in delta for a change in the underlying price, so it is the second derivative of value with respect to price. Vanna is the change in delta for a change in implied volatility, which by the equality of cross-derivatives also equals the change in vega for a change in price. Gamma responds to price; vanna responds to volatility.
- Do a call and a put have the same gamma and the same vanna?
- Yes, matched European calls and puts have equal gamma and vanna under the same pricing inputs. Their parity difference is linear in spot and independent of volatility, so its second spot derivative and mixed spot-volatility derivative are zero. Position direction then determines the sign of the exposure held by the trader.
- Why does gamma peak at the money while vanna does not?
- Gamma measures how rapidly delta changes with price and concentrates near the money. Vanna measures how much delta changes for a 1-point move in implied volatility, and it crosses zero near the money, where d2 = 0 in Black-Scholes, so volatility, time and rates set the exact point. A delta sitting at that crossing barely moves when volatility changes. On either side vanna grows with opposite signs, so rising volatility lifts the delta of a far out-of-the-money call and lowers the delta of a deep in-the-money call.
- How do gamma and vanna each affect delta hedging?
- Gamma sets how much a hedger must rebalance as the underlying price moves, because each price move changes delta. Vanna sets how much a hedger must rebalance when implied volatility moves, because a volatility shift can change delta even with spot unchanged. A book can be delta neutral at the current volatility and become directional after the surface reprices, and that residual delta is vanna driven. Managing both requires modeling price and volatility together, not one at a time.