我和Quant Alex @StochAlex07的讨论：Vega*∂Sigma/∂F是否可以被称为Vanna项？ **Compressed Summary: Jeff & Alex Wu Discussion on “whether Vega × ∂Sigma/∂F is a Vanna term?”** ### Core Question (Jeff) Jeff asks why many traders call **Vega × ∂Sigma/∂F** (Smile Delta / Shadow Delta) a “Vanna term” under implied vol surface context. He notes this term represents **spot-induced IV change due to moneyness shift**, but it differs from the classic cross-derivative ∂²P/(∂F ∂Σ). He wants to know if this broader usage is valid. ### Alex’s Clear Verdict **No — it should not be called Vanna.** The “broad interpretation” is **sloppy and mixes two different concepts**: - **Classical BS Vanna**: Second-order Greek — how Vega changes with Spot, or how Delta changes with IV (inside BS PDE). - **Vega × ∂Sigma/∂F (Smile Delta)**: Describes **realized spot–IV dynamics** (how the implied volatility surface itself moves when spot moves). Alex: “They are not the same thing — their explanatory targets are completely different.” ### Why the Confusion Exists - In pure BS, Vanna and Volga were introduced to handle Greeks sensitivity to spot/IV moves. - Spot-induced IV changes are a **separate phenomenon**, addressed by market conventions: - Sticky Strike / Sticky Delta / Sticky Local Vol - Implied Skew (quick proxy for spot-vol linkage) - Calculating full strike-by-strike spot-IV correlations is impractical, so desks focus on **Spot vs ATM IV dynamics**. ### Modern Practical Approaches (Beyond BS Vanna) 1. **Parametric Vol Dynamics** (SEPP / SVI / Vola Dynamics path) Regress dSpot & dATM Vol → derive each strike’s dIV/dF. Focuses on **smile shape**, not stochastic dynamics. Naturally compatible with sticky rules. 2. **Stochastic Volatility Models** (e.g. SABR) Directly links implied skew to **spot-vol covariance**. A Vanna-like term appears in the PDE, but **Bartlett Delta contains no Vanna**. ### Key Takeaways - Smile Delta’s essence is **spot-vol covariance + smile shape**, unrelated to classical BS Vanna. - The popular “Vanna = Smile Delta” shorthand is convenient market talk but **technically inaccurate**. - Real trading desks have long moved beyond pure BS Greeks precisely because BS Vanna alone cannot capture actual vol-surface behavior. **Conclusion**: The terminology is misleading. Vega*∂Sigma/∂F and Vanna explain different objects and should not be conflated.