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Second-Order Greeks and Trading Implications

First-order Greeks tell you your current exposure. Second-order Greeks tell you how that exposure will change, which is what you need when managing a book that cannot be rebalanced continuously.

Vanna

Vanna=∂Δ∂σ=∂ν∂S\text{Vanna} = \frac{\partial\Delta}{\partial\sigma} = \frac{\partial\nu}{\partial S}

How delta responds to volatility, equivalently how vega responds to price. The two derivatives are the same quantity.

This matters because price and volatility move together, and in equities they move in opposite directions: markets fall and implied volatility rises. A book that is delta-hedged for a price move alone will find its delta has shifted because vol moved too.

Vanna is the main driver of skew exposure. A position in out-of-the-money puts carries substantial vanna, and any desk trading skew is trading vanna whether or not it uses the word.

Volga (vomma)

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