SVQ · Greeks
Vanna
Also called: DdeltaDvol, DvegaDspot
Vanna is the second-order options Greek that measures how an option's delta changes when implied volatility changes, or equally how vega changes with spot.
Definition
Vanna is the cross second derivative of an option's value with respect to the underlying price and implied volatility. It tells you how much delta moves when implied volatility moves, and, by symmetry of mixed partial derivatives, how much vega moves when the underlying moves.
It is largest in magnitude for out-of-the-money options and is close to zero at the money in the Black–Scholes model.
Intuition
A delta-hedged book is only hedged for the current level of implied volatility. If volatility shifts, deltas shift too, and the hedge drifts. Vanna measures that drift. Traders who hedge large options books care because volatility and spot usually move together; in equities, volatility tends to rise when prices fall.
At the market level, aggregate dealer vanna is one reason volatility changes can trigger hedging flows in the underlying even when price has not moved.
How it's calculated
Vanna = ∂²V / ∂S∂σ = ∂Δ / ∂σ = ∂ν / ∂S
In Black–Scholes with dividend yield q, the vanna of a call and a put is the same:
Vanna = −e−qT · φ(d1) · d2 / σ
where φ is the standard normal density and d2 = d1 − σ√T. This is per unit of volatility; divide by 100 for a one-volatility-point change. Because d2 changes sign near the money, vanna is positive for out-of-the-money calls and negative for out-of-the-money puts.
Example: an out-of-the-money call with delta 0.25 and vanna of 0.004 per volatility point will have a delta of about 0.27 if implied volatility rises 5 points. A hedger short 1,000 such contracts with a 100-share multiplier would need to buy roughly 2,000 more shares to stay neutral.
How to read it
Positive vanna means delta rises when implied volatility rises. For a long out-of-the-money call that is intuitive: higher volatility makes finishing in the money more likely. For a long out-of-the-money put, delta becomes more negative as volatility rises.
Read vanna together with the expected co-movement of spot and volatility. A book with large vanna is effectively exposed to the skew dynamics of the underlying, not just to its level.
Common mistakes
- Assuming a delta hedge stays valid when implied volatility moves.
- Mixing units: per unit of volatility versus per volatility point.
- Ignoring that model vanna depends on the smile; sticky-strike and sticky-delta assumptions give different hedge adjustments.
- Summing vanna across expiries without considering that their volatilities do not move one-for-one.
In Saqrid
SVQ's docked Greek heatmaps include vanna among higher-order sensitivities such as charm, volga, speed, color, zomma and ultima.
Its command terminal accepts queries such as scan butterfly(BTC, 30d) where vanna > 0.5 and draws the result on the surface.
Open SVQ.
Go deeper
Saqrid Institute lesson: Hedging & the Greeks II: Rho, Vanna, Charm, Volga. Lesson text is open to read without an account.