What is Vanna? Definition, Formula, and Example
Vanna is a second-order option greek that measures how an option's delta changes when implied volatility changes, or equivalently how vega changes when the underlying price moves.
Plain-English Definition
Vanna is a second-order option greek that measures the cross-sensitivity of delta to implied volatility. Equivalently — by Schwarz's theorem on mixed partial derivatives — it measures how vega changes when the underlying price moves. Vanna sits at the intersection of two first-order greeks: delta (price sensitivity) and vega (volatility sensitivity). It is critical for understanding dealer hedging flows and explains many of the persistent calendar effects in index options, including the so-called "vanna rally" around monthly options expiration.
How It Is Calculated
Vanna is the second partial derivative of option value V with respect to underlying spot S and implied volatility σ:
- Vanna = ∂Δ/∂σ = ∂ν/∂S = ∂²V/∂S∂σ
For a European call under Black-Scholes:
- Vanna = −e^(−qT) · φ(d₁) · (d₂ / σ)
where φ is the standard normal probability density function, d₁ and d₂ are the standard Black-Scholes terms, q is the dividend yield, T is time to expiry, and σ is implied volatility.
A positive vanna means delta rises as volatility rises. Out-of-the-money calls have positive vanna; OTM puts have negative vanna. Vanna is zero exactly at-the-money under Black-Scholes (because d₂ is approximately zero) and peaks at strikes about one standard deviation out.
Worked Example
Consider an SPY $500 call with 30 days to expiry, spot at $495, implied volatility at 18%, delta 0.45, and vega 0.55. Suppose the option's vanna is +0.012 per 1% IV move per $1 of spot.
Scenario 1 — IV rises 2 points (18% → 20%) with spot unchanged:
- Delta change = +0.012 × 2 = +0.024
- New delta ≈ 0.474
Scenario 2 — Spot rises $1 (495 → 496) with IV unchanged:
- Vega change = +0.012 × 1 = +0.012
- New vega ≈ 0.562
Now scale this to a real dealer book. If a market maker is short 10,000 SPX put contracts at the $5,000 strike with vanna of −0.015 per contract, total book vanna is +150 (short negative = positive). When SPX-implied volatility drops 4 points after a Fed meeting, the dealer's effective delta increases by 150 × 4 = +600 deltas, which must be hedged by buying 600 ES futures contracts. This buying pressure is mechanical — independent of fundamentals — and is the engine behind the post-event drift higher in index futures.
When Traders Use It
Dealer-flow analysts use vanna to predict the direction of mechanical hedging around volatility shifts. The textbook pattern is the vanna rally: into a known event (FOMC, CPI, options expiration), implied volatility is elevated; after the event, IV collapses (IV crush). Dealers short downside puts have positive vanna, so falling IV reduces their effective short delta and forces them to buy the underlying — creating the well-documented post-OPEX rally pattern. Exotic-options traders use vanna to price barrier and cliquet structures where path dependence on the joint distribution of spot and vol matters.
Limitations and Misconceptions
Vanna is model-dependent. Black-Scholes vanna assumes constant volatility, which is empirically false — real markets have stochastic volatility, and vanna under SABR or Heston differs materially from Black-Scholes vanna. For short-dated at-the-money options, vanna is dwarfed by delta and gamma and rarely drives P&L. Vanna only becomes material near the wings of the skew, around large volatility moves, or in books with concentrated OTM exposure. Retail traders should not attempt to vanna-hedge a small portfolio — the bid-ask cost of rebalancing exceeds the second-order P&L it eliminates.