Second-Order Greeks - Vanna, Volga, and Charm

By EC Assets Research Team, Derivatives Research · Published 20 May 2026 · Updated 28 May 2026

Second-Order Greeks — Second-order Greeks measure how the first-order Greeks themselves change. Vanna is delta's sensitivity to volatility (and vega's to spot), Volga is vega's sensitivity to volatility, and Charm is delta's decay with time. They are essential risk for large, long-dated, or skew-heavy options books, where a first-order hedge quietly un-hedges itself.

Beyond the First-Order Greeks

The familiar Greeks - delta, vega, theta, rho - measure how an option's price responds to a single variable. But those sensitivities are not constant: delta itself changes as the underlying moves (that is gamma), and it also changes as volatility and time change. The second-order Greeks measure how the first-order Greeks themselves move. For a single vanilla option held briefly they are a refinement; for a large book, a long-dated position, or anything path-dependent, they are essential risk - the difference between a hedge that holds and one that quietly drifts off.

The three that matter most on a volatility desk are Vanna, Volga, and Charm.

Vanna - Delta's Sensitivity to Volatility

Vanna = ∂Delta/∂σ = ∂Vega/∂S

Vanna captures two equivalent things: how an option's delta changes when implied volatility moves, and how its vega changes when the underlying moves. It is the cross-effect between spot and volatility. Vanna matters because spot and volatility move together in the real world - equities fall and volatility jumps - so a position that looks delta-hedged can become un-hedged the moment volatility shifts. Books with significant skew exposure are heavily exposed to vanna, and "vanna hedging" is a known driver of market flows, especially around large option expiries.

Volga - Vega's Sensitivity to Volatility

Volga (Vomma) = ∂Vega/∂σ

Volga is the convexity of an option's value in volatility - how vega itself changes as volatility moves. It is to volatility what gamma is to spot. An option with high volga gains vega as volatility rises, so it benefits convexly from big volatility moves - which is exactly why out-of-the-money options and volatility products like variance swaps carry meaningful volga, and why they are sensitive to the volatility of volatility.

Charm - Delta's Sensitivity to Time

Charm = ∂Delta/∂t

Charm (delta decay) measures how an option's delta drifts purely with the passage of time, holding everything else fixed. It is why a delta hedge set today can be wrong tomorrow even if nothing moved - the delta itself has aged. Charm is largest for near-the-money options approaching expiry, and is a particular headache over weekends and around expiry, when several days of delta decay must be anticipated and hedged in advance.

Worked Example

A volatility desk runs a large, delta-hedged options book and goes home flat on delta on a Friday. Over the weekend, no trading occurs, but:

Charm has aged the book's delta - the hedge that was flat on Friday is no longer flat on Monday, purely from time passing. If Monday opens with a volatility gap, Vanna means the book's delta has also shifted because implied volatility moved, not just spot. And Volga means the book's vega itself is different at the new volatility level, changing how the next volatility move will be felt.

A desk that hedges only the first-order Greeks wakes up exposed on all three counts. This is why institutional volatility books monitor and hedge the second-order Greeks explicitly - the first-order hedge is not self-maintaining.

[!key] First-order Greeks tell you today's exposure; second-order Greeks tell you how that exposure changes as spot, volatility, and time move. Vanna (delta vs vol), Volga (vega vs vol), and Charm (delta vs time) are what make a "hedged" book quietly un-hedge itself - which is why they matter most for large, long-dated, and skew-heavy positions.

[!warning] A book hedged only on delta and vega is not actually hedged through a volatility move or the passage of time. Vanna and charm can leave it materially exposed after a weekend or a volatility gap, and dealer vanna/charm hedging is itself a documented amplifier of market moves around major expiries. Ignoring the second-order Greeks is a common, expensive mistake on any sizeable options book.

Why It Matters for Institutional Investors

• Real hedging on real books. For market-makers and volatility funds, the second-order Greeks are everyday risk - the reason a hedge must be re-examined as volatility and time move, not just as spot moves.
• Skew and surface risk. Vanna and volga are how exposure to the shape of the volatility surface - skew and the volatility of volatility - shows up in the Greeks, central to anyone trading skew or variance.
• Market-structure awareness. Aggregate dealer vanna and charm positioning around large expiries is now widely watched, because the hedging it forces can move the underlying market - useful context even for investors who never trade an option.

References

1. Taleb, N. N. (1997). Dynamic Hedging: Managing Vanilla and Exotic Options. Wiley.
2. Castagna, A., & Mercurio, F. (2007). The Vanna-Volga Method for Implied Volatilities. Risk.
3. Bossu, S. (2014). Advanced Equity Derivatives: Volatility and Correlation. Wiley.
4. Hull, J. C. (2022). Options, Futures, and Other Derivatives (11th ed.). Pearson. Chapter 19.