greeks
Vanna
Cross-Greek measuring how Delta changes when IV changes (or equivalently, how Vega changes when the stock moves). Key driver of "vol smile dynamics."
Formula
Vanna = -φ(d₁) · d₂ / (S · σ · √T)
Variables
- Vanna
- dΔ/dσ or dν/dS — sensitivity of Delta to volatility
- φ(d₁)
- Standard normal PDF at d₁
- d₂
- Standard Black-Scholes d₂
- S
- Current stock price
- σ
- Implied volatility
- T
- Time to expiration (years)
Worked Example
Inputs
- Scenario
- OTM call, 45 DTE
- Δ
- 0.30
- IV
- 25%
- Vanna
- 0.04
Calculation Steps
- 1
If IV rises 5%: Δ_new ≈ 0.30 + 0.04 × 5 = 0.50 - 2
The OTM call behaves more like an ATM call when vol rises - 3
This is why portfolio delta shifts during vol spikes
Result: Delta shifts by +0.04 per 1% IV increase
Intuition
Vanna explains vol-driven rallies: when the market drops and vol spikes, dealer short gamma positions get more negative delta via vanna, forcing them to sell more stock, amplifying the move.