What is Volga (Vega Convexity)?
Volga — also called vomma — is the rate of change of vega with respect to implied volatility. It is the second-order vol Greek and the source of P&L on butterflies and variance.
Definition
Volga is ∂²V/∂σ² — how much your vega exposure changes per 1-vol-point move in implied volatility. A position with positive volga gets longer vega as vol rises and shorter vega as vol falls — it is "long convexity in vol." Out-of-the-money options have the highest volga, which is why wing-heavy structures (like butterflies, condors, or risk reversals weighted toward the wings) are essentially volga trades. ATM options have very little volga. A vol book that buys cheap wings and sells fairly priced ATM is harvesting positive volga: when vol explodes, the wings get vega-long fast and the trade pays asymmetrically.
Why it matters & how it's calculated
Under a standard pricing model, volga = vega · d₁ · d₂ / σ. The intuition: vega for ATM options is roughly insensitive to vol level (high vega, low volga), but vega for far OTM options is highly sensitive — they're worth almost nothing at low vol and become very vega-rich at high vol. This is why variance swaps, which by construction have a static replication weighting roughly proportional to 1/K², are essentially long volga: they overweight the wings where volga lives. The relationship volga ↔ smile is also direct: a market with a steep "smile" (high IV for both far OTM puts and far OTM calls relative to ATM) implies the market is willing to pay up for volga, which is exactly what happens when realised vol-of-vol is high. On a desk, hedging volga matters when you sell variance — every dynamic variance replication has residual volga risk that shows up as gamma-of-vol P&L during sharp vol moves.
Formula
Volga = ∂²V/∂σ² = vega · d₁ · d₂ / σ
Worked example
You own 100 SPX 25-delta puts with vega $8 per vol point and volga $2 per vol-point-squared. Vol is at 18%. If vol explodes to 28% overnight (10 points), naive P&L from vega alone = 100×100×8×10 = $800,000. But your vega itself grew by volga × 10 = $20 to $28 per vol point. The actual P&L is closer to ½ × (8+28) × 10 × 100×100 = $1,800,000. The extra $1M is your volga P&L — the convexity payoff.
Related concepts
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