Second-order Greek · ∂²V/∂S∂σ

Vanna: how delta changes when implied volatility moves

Vanna is the change in delta for a change in implied volatility, and equally the change in vega for a change in spot. When implied vol moves across a whole options book, vanna turns that move into buying or selling.

The formula

d₁ = [ ln(S/K) + (r + σ²/2)·T ] / (σ√T)
d₂ = d₁ − σ√T

Vanna = ∂Δ/∂σ = −φ(d₁)·d₂ / σ

Black-Scholes, European options, no dividends. S spot, K strike, T years to expiry, r the risk-free rate, σ implied volatility, N(·) the standard normal CDF and φ(·) its density.

It's the same for calls and puts. It's positive where d₂ < 0, roughly strikes above spot (out-of-the-money calls, in-the-money puts), and negative where d₂ > 0, strikes below spot. In words: raising implied vol pulls out-of-the-money call deltas up and in-the-money call deltas down, toward the middle.

Vanna flows

If dealers hold a book with net vanna V (in delta per unit of vol), a change Δσ in implied volatility shifts their delta by V·Δσ, and they trade the opposite to stay hedged. The common story: after an event passes, implied vol falls, dealer deltas shift, and their re-hedging buys the index. That's the "vanna rally" you'll hear about around FOMC or OPEX.

What we found when we tested it

On our archive of seven sessions across SPX, SPY, ES, NDX, QQQ, NQ and IWM, a "vanna flow" signal (dealer vanna × the 30-minute change in ATM IV) did correlate with the next 30 minutes' return, but net dealer vanna was positive 88% of the time, so the signal was the IV change in disguise. Holding the IV change fixed, vanna's own contribution was 0.00 on held-out days. Small sample; we'll rerun it as the archive grows.

NDX dealer vanna exposure (VEX) by strike under price, with ATM IV through the session, in Zanshin Flow — VEX by strike beside ATM IV: vanna is how the book's delta moves when IV does.

Questions

What is a vanna rally?

A rise in the index attributed to dealers re-hedging after implied volatility falls: with net vanna on their books, lower IV changes their deltas, and buying back hedges can push the underlying up.

Is vanna the same for calls and puts?

Yes. Under Black-Scholes vanna is identical for a call and a put with the same strike and expiry.

Is vanna ∂Δ/∂σ or ∂vega/∂S?

Both. The two cross-derivatives are equal, so vanna is how delta responds to volatility and how vega responds to spot.

Not financial advice. This article explains how options and dealer hedging work. It is not a recommendation to buy or sell anything.