Volga - The Convexity of Vega

By EC Assets Research Team, Derivatives Strategy · Published 12 August 2026 · Updated 12 August 2026

Volga: Volga, also called vomma, measures how an option's vega changes as implied volatility moves. It is to volatility what gamma is to spot: the convexity term. Wing options carry large positive volga and at-the-money options carry almost none, which is why the volatility smile exists and why out-of-the-money options are the natural instrument for trading the volatility of volatility.

What Volga Measures

Volga, also written vomma, is the second derivative of option value with respect to implied volatility:

Volga = ∂Vega/∂σ = ∂²V/∂σ²

It answers a question that vega alone cannot: as volatility moves, does the position's sensitivity to volatility grow or shrink? Vega tells you the slope; volga tells you the curvature. The relationship is exactly parallel to the one between delta and gamma, with volatility in place of spot.

In Black-Scholes terms it can be written compactly:

Volga = Vega × (d₁ × d₂) / σ

Where the Convexity Sits

The product d₁ × d₂ determines the sign, and it produces a distinctive shape across strikes.

At the money, volga is close to zero. For an at-the-money-forward option, d₁ and d₂ are equal in size and opposite in sign, so their product is slightly negative and the option's value is very nearly linear in volatility. Doubling volatility roughly doubles the premium of an at-the-money option, which is the intuition behind quoting such options in volatility terms at all.

In the wings, volga is large and positive. For strikes well away from the money, d₁ and d₂ share a sign, and the option's value is strongly convex in volatility. A far out-of-the-money option is worth almost nothing at low volatility and a great deal at high volatility, and it gains vega as volatility rises, so each further increase is felt more than the last.

Why This Creates the Smile

Volga is one of the cleanest explanations for why a volatility smile exists at all. If volatility itself is uncertain, then an instrument whose value is convex in volatility is worth more than one that is linear in it, by exactly the logic that makes any convex payoff valuable under uncertainty.

Wing options carry that convexity; at-the-money options do not. The market therefore charges a higher implied volatility for the wings, and the smile is the visible result. This also explains why the wings are the natural instrument for expressing a view on the volatility of volatility, and why an index that measures the volatility of the VIX exists as a distinct traded concept.

Volga in Volatility Products

A variance swap pays on realised variance, which is volatility squared. That payoff is linear in variance and therefore convex in volatility, giving the instrument structurally positive volga. It is the reason a variance swap and a straddle behave differently in a large volatility move even when they start with similar vega, and the reason variance swaps are replicated with a strip of options weighted toward the wings rather than with a single at-the-money contract.

The same property underpins the vanna-volga method, a practitioner technique originating in foreign exchange that builds a full smile from three quoted points by correcting a flat Black-Scholes price for its vanna and volga exposure. That the method works at all is evidence of how much of the smile these two Greeks account for.

Worked Example

Two positions each carry the same vega at a starting implied volatility of 20 percent: a single at-the-money straddle and a larger quantity of far out-of-the-money options. Volatility then jumps to 35 percent.

The at-the-money position gains roughly what its vega predicted, because its value is close to linear in volatility. The wing position gains considerably more: its vega itself increased as volatility rose, so the second half of the move was worth more per point than the first. The convexity that looked like an accounting detail at 20 percent volatility became the dominant source of profit at 35.

The trade is symmetric in the unpleasant direction too. In a market that grinds quietly lower in volatility, the wing position loses vega as it goes, and its value decays toward nothing while the at-the-money position merely declines in proportion.

[!key] Volga is the curvature of option value in volatility, exactly as gamma is the curvature in spot. At-the-money options are nearly linear in volatility and carry almost no volga; wing options are convex and carry a great deal.

[!warning] Vega alone is an unreliable summary of a wing-heavy book. Two positions can show identical vega and behave completely differently in a large volatility move, because one gains vega as volatility rises and the other does not. Any position sized on vega without regard to volga is sized for a small move only.

Why It Matters for Institutional Investors

• Tail hedges are volga trades. A protective position built from far out-of-the-money options is bought precisely for its convexity in volatility. Judging it on vega, or on its cost in calm markets, misses the property it was purchased for.
• Instrument selection. Variance swaps, straddles and wing strips deliver different payoffs in a large volatility move despite similar headline vega. The difference is volga, and it determines which instrument actually expresses the intended view.
• Understanding smile pricing. The premium charged for wing options is not a market inefficiency to be harvested casually. It is compensation for selling convexity in volatility, and a seller of the wings is short exactly the property that pays off in a crisis.

References

1. Castagna, A., & Mercurio, F. (2007). The Vanna-Volga Method for Implied Volatilities. Risk.
2. Gatheral, J. (2006). The Volatility Surface: A Practitioner's Guide. Wiley.
3. Taleb, N. N. (1997). Dynamic Hedging: Managing Vanilla and Exotic Options. Wiley.
4. Bossu, S. (2014). Advanced Equity Derivatives: Volatility and Correlation. Wiley.
5. Hull, J. C. (2022). Options, Futures, and Other Derivatives (11th ed.). Pearson.