What Is Vomma (Volga)?

Vomma (also called Volga or vega convexity) is the second derivative of option value with respect to volatility (partial² V / partial sigma²). Equivalently, vomma measures how vega changes when implied volatility moves. In the Black-Scholes model, vomma equals vega × (d₁ × d₂) / sigma. Vomma is the structural exposure traded by butterflies and the analytical signal of vol-of-vol pricing.

What Is Vomma in Options?

Vomma captures the convexity of an option's value as a function of implied volatility. Long options have positive vomma (vega itself increases when IV rises further); short options have negative vomma. Vomma matters operationally because it measures the non-linear part of vol P&L: a vega-only estimate of P&L from a 5-point IV move underestimates the actual move when vomma is large.

Two intuitions. First, vomma is the analog of gamma in the volatility direction - both are second-order convexity measures. Second, vomma scales with vega and with the d₁×d₂ product, meaning it is small for ATM options (where d₁×d₂ is small) and grows in the wings. This is the structural reason butterfly trades have explicit vomma exposure: the wings of the butterfly carry the vomma.

Worked Example

SPY at $500, 60-day OTM call (K=540, about 8% OTM), IV 16%, rate 4%. Black-Scholes inputs:

d₁ = -1.053, d₂ = -1.118, phi(d₁) = 0.229
Vega = 500 × 0.229 × sqrt(60/365) = 46.5 (per share, per unit-vol-change)
Vomma = 46.5 × (-1.053 × -1.118) / 0.16 = 46.5 × 1.177 / 0.16 = 342 (per share, per unit-vol-squared)
Per-1%-IV scaling on vega: divide by 100 = 0.465 per share
Per-1%-IV-squared scaling on vomma: divide by 10000 = 0.0342

Operational reading: an IV move from 16% to 21% (5 vol points up) produces a vega P&L of approximately 5 × 0.465 = $2.33 per share. Plus a vomma P&L of approximately 0.5 × 0.0342 × 5² = $0.43 per share. Total: $2.76 per share - the vomma adds about 18% on top of the linear vega estimate. For a 10-vol-point move, the linear vega estimate is $4.65 and vomma adds roughly $1.71 (about 37%), because the vomma contribution scales as the square of the move size.

How Pricing Models Compute Vomma

Black-Scholes: closed-form vomma vega × (d₁ × d₂) / sigma. Same vomma applies to calls and puts at the same strike (by put-call parity vega is the same; vomma inherits this).
Heston (stochastic volatility): Heston has multiple vol parameters, so "vomma" decomposes: sensitivity to v₀ squared, sensitivity to nu, etc. The aggregate vomma in Heston is computed via Fourier inversion of the second derivative.
SABR: SABR has explicit vol-of-vol structure (nu parameter), so vomma is directly captured. SABR vomma in the Hagan formula has both alpha and nu contributions.
Local volatility (Dupire): vomma computed by re-pricing under bumped surfaces; LV vomma is an artifact of the calibration choice rather than a structural model parameter.
Jump diffusion: vomma includes diffusion-component vomma and jump-component vomma. Bates (Heston + jumps) produces vomma profiles closer to observed market behavior than pure Heston.

Vomma in Trading Strategies

Vomma is the structural exposure of vol-curvature trades:

Long butterfly (sell ATM straddle, buy OTM strangle): long vomma. Profits when realized IV move is larger than expected (vol regime expands).
Short butterfly: short vomma. Profits when IV stays compressed.
Long calendar (sell front, buy back): generally long vomma at long-back-leg.
Long diagonal: mixed vomma profile depending on strikes and expirations.
Long out-of-the-money call or put: long vomma. Wing options carry the bulk of vomma exposure.

Vomma and Vol-of-Vol

The pricing of vomma in the market is the pricing of vol-of-vol. The VVIX index measures implied volatility of VIX itself - it is essentially an aggregate measure of priced vomma in the SPX option chain. When VVIX is elevated (above 110), vomma trades are expensive; when VVIX is compressed (below 80), vomma is cheap. Vol-of-vol regime is one of the structural inputs to vomma valuation.

Special Cases

ATM options: vomma near zero. d₁×d₂ approaches zero at the money, killing the vomma magnitude.
OTM options (calls or puts): vomma peaks. Wings carry the convexity exposure.
Deep OTM: vomma falls off again as the option approaches worthlessness.
Long-dated options: vomma scales with vega (which scales with sqrt(T)), so 1-year ATM has larger vomma magnitude than 30-day.

Related Greeks

Vomma is the second derivative in the vol direction. Its sibling cross-derivatives are vanna (cross with spot) and veta (cross with time). The third-order extension is ultima (vomma's sensitivity to vol). Together, vomma, ultima, vanna, and veta describe the second- and third-order vol-direction structure of an option's value.

Related Concepts

Vega · Ultima · Vanna · Veta · Vol of Vol · Volatility Smile · Heston · SABR · All 17 Greeks

References & Further Reading

Hull, J. (2018). Options, Futures, and Other Derivatives, 10th ed. Pearson.
Castagna, A. (2010). FX Options and Smile Risk. Wiley. Practitioner reference on vanna, vomma, and volatility-smile hedging.
Gatheral, J. (2006). The Volatility Surface: A Practitioner's Guide. Wiley.

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This page is part of the 17 Greeks reference covering every options Greek with formula, intuition, worked example, and how each pricing model computes it. Browse the full documentation.