What Is Vanna?

Vanna is the second-order cross derivative of option value with respect to spot and volatility (partial2 V / partial S partial sigma). Equivalently, vanna measures how delta changes when implied volatility moves, or how vega changes when spot moves. In the Black-Scholes model, vanna equals -exp(-qT) phi(d1) d2 / sigma. Vanna is the structural Greek for skew-driven delta-hedge wobble and dealer-flow analytics.

What Is Vanna in Options?

Vanna captures the cross-effect between two state variables that both individually drive option price: spot price and implied volatility. It has two equivalent interpretations that are equally useful: how the delta of an option changes when IV changes (so it becomes a hedge-ratio adjustment when vol regime shifts), or how the vega of an option changes when spot changes (so it becomes a vol-position drift indicator when the underlying moves).

Three intuitions for vanna. First, vanna is the curvature signal between two first-order Greeks - it tells you their interaction is non-linear. Second, vanna is operationally large when an option is OTM and the wing of the smile matters - skew trading is essentially a vanna play. Third, vanna's sign tracks d2's sign in BS: high strikes (above the forward, including OTM calls) have d2 < 0 and produce positive vanna; low strikes (below the forward, including OTM puts) have d2 > 0 and produce negative vanna. ATM (K near forward) sits near zero vanna.

Worked Example

SPY at $500, 30-day option, IV 14%, rate 4%. We compare vanna for an ATM call (K=500) and an OTM call (K=530, about 6% OTM).

For the ATM call:

For the OTM call (K=530):

Operational reading: when IV rises by 1 vol point, the ATM call's delta drifts by approximately -0.0018 (small move). The OTM call's delta drifts by approximately +0.016 - meaning a 5-vol-point IV move shifts its delta by roughly +0.08, materially repricing the wing's hedge ratio. This is why skew traders care about vanna: skew movements rebalance the wings' deltas more than the center's.

For an OTM put on the downside wing (e.g., K=470), vanna is negative with magnitude similar to the OTM call's positive vanna in the BS-flat-vol case. Empirical equity-index skew is asymmetric, so the absolute magnitudes of OTM put vanna versus OTM call vanna are not equal; the put-side wing typically carries larger |vanna| because the smile is steeper on the downside.

How Pricing Models Compute Vanna

Vanna in Skew Trading

Skew is essentially "vanna of the option chain." A long-skew position (long OTM puts, short OTM calls) is structurally long vanna in the sense that its P&L is positive when skew steepens (OTM put IV rises faster than OTM call IV) and negative when skew flattens. Risk-reversal positions are vanna-isolating: they are constructed to be vega-neutral or near-vega-neutral, leaving primarily vanna and skew exposure.

For dealer desks, aggregate book vanna is a closely-watched metric. A negative-vanna book (typical for option-market makers selling OTM puts to retail) loses on skew-steepening events and gains on flattening. Intra-day skew rotation is tracked because aggregate dealer vanna creates implicit hedging flows in the underlying and in skew-adjacent products.

Vanna and Dealer Flow Analytics

Beyond dealer gamma, dealer vanna is the second-most-tracked aggregate Greek for understanding option-market microstructure. The gamma exposure (GEX) dashboard typically shows vanna profile alongside gamma profile because the two together describe how dealer hedging flows respond to spot moves and IV moves jointly.

Operational rule: when vanna and gamma point the same direction (both positive or both negative), dealer hedging is amplified - small spot moves produce outsized hedging flows. When vanna and gamma point opposite directions, the flows partially cancel. Pre-FOMC, pre-CPI, and pre-earnings windows are notable for vanna structure being a leading indicator of post-event price stability or instability.

Vanna Across Moneyness

Vanna is approximately zero ATM (because the skew effect there is symmetric in IV). It peaks in absolute magnitude at moderate moneyness (10-25 delta) where the smile is steepest. Deep OTM, vanna falls off again because the option's sensitivity to anything diminishes. The structure mirrors what skew curvature reveals: vanna is the analytical Greek for what the skew curve says geometrically.

Special Cases

Related Greeks

Vanna is the cross of delta and vega. Charm is the cross of delta and time. Veta is the cross of vega and time. Together, vanna, charm, and veta form the "cross-Greek triplet" that describes how each first-order Greek interacts with each other state variable.

Related Concepts

Delta · Vega · Charm · Volatility Skew · Dealer Gamma · SABR · Heston · All 17 Greeks

References & Further Reading

View live SPY gamma and vanna profiles →

This page is part of the 17 Greeks reference covering every options Greek with formula, intuition, worked example, and how each pricing model computes it. Browse the full documentation.