What does zomma measure?

Zomma measures how an option gamma changes when implied volatility changes. Gamma is already the second derivative of option value with respect to the underlying price, so differentiating once more with respect to volatility makes zomma a third-order Greek. Risk systems usually label it dgamma/dvol. It answers a practical question: after the volatility surface reprices, is the position more convex or less convex than the gamma number currently on the screen suggests.

Why is zomma negative for at-the-money options?

Zomma carries the sign of d1 times d2 minus 1. Near the money that product sits close to zero, so the bracket is close to negative one and zomma is negative. Economically, a higher volatility input widens the modeled distribution of expiration prices and spreads probability away from the strike sitting under spot, so the gamma peak drops and broadens. The OIC makes the same point from the other direction: at-the-money gamma is largest when implied volatility is low.

Where does zomma turn positive?

Zomma turns positive once d1 times d2 exceeds 1, which happens roughly one standard deviation of the log move away from the forward in either direction. For a $100 stock with 30 days left and 30 percent implied volatility, that band runs from about $92 to $109. Inside it zomma is negative and rising volatility flattens gamma. Outside it zomma is positive, so a volatility spike builds gamma in wing strikes that previously carried almost none.

Do traders hedge zomma directly?

Usually not. Zomma quantifies how an IV change alters gamma, and that volatility-driven change in convexity feeds the gamma hedge calculation rather than being hedged as a separate exposure. The complete book is repriced across spot and volatility scenarios, and the hedge plan is updated using the resulting gamma, transaction costs, and rebalance rules.

Why do zomma values differ between platforms?

Units and models. One vendor quotes the gamma change per 1.00 of volatility while another quotes it per one percentage point, and those differ by a factor of 100. Gamma itself may be reported per share or per contract, which introduces the 100 multiplier again. Systems also differ in day count, dividend treatment, and whether they feed one implied volatility per option or a full surface. State the scaling before you compare two zomma numbers.

Zomma in Options: Gamma Sensitivity to Volatility

On This Page

What is zomma in options?
Zomma vs vanna and vomma
Worked example
How does zomma work?
How traders use zomma
How to reprice gamma after a volatility move
Limitations of zomma
Bottom line
FAQ

Key Takeaways

Zomma is the third-order Greek that measures how an option gamma changes when implied volatility changes, so it is the derivative of gamma with respect to sigma.
Under Black-Scholes, zomma equals gamma multiplied by the quantity d1 × d2 minus 1, divided by sigma, so the entire sign lives in whether d1 × d2 sits above or below 1.
Zomma is negative for strikes within roughly one standard deviation of the forward, which means rising implied volatility flattens the gamma peak even though spot never moves.
Zomma is positive for wing strikes, so a volatility spike builds gamma in contracts that previously carried almost none and widens the range where hedging can bite.
Vanna is the volatility sensitivity of delta, vomma is the volatility sensitivity of vega, and zomma is the volatility sensitivity of gamma; the three are separate exposures.
A gamma figure is only valid at the volatility level it was computed from, and zomma is the correction that says how far it has drifted after the surface reprices.

What is zomma in options?

Zomma is the third-order option Greek that measures how gamma changes when implied volatility changes. Gamma is already the second derivative of option value with respect to the underlying price. Differentiate it once more, this time against volatility, and the result is zomma. Risk systems and pricing libraries usually label it dgamma/dvol.

Placing it in the hierarchy keeps the vocabulary straight. Delta, vega, theta, and rho are first-order sensitivities that map one input to option value. Gamma, vanna, vomma, and charm are second-order: each measures how a first-order Greek moves. The OIC describes that second tier explicitly, naming vanna as the change in delta as implied volatility changes and vomma as the change in vega as implied volatility changes. Zomma sits one level above, in the third tier with speed and color.

The practical content of the Greek is simple. Gamma is quoted at one volatility level. If that level changes, the gamma number changes with it, and zomma says by how much and in which direction.

Zomma vs vanna and vomma

Vanna, vomma, and zomma all begin with a volatility shock, which is why they get confused. The difference is only in what the shock acts on:

Vanna: what a volatility move does to delta.
Vomma: what a volatility move does to vega.
Zomma: what a volatility move does to gamma.

Zomma is also the derivative of raw vanna with respect to spot. Under Black-Scholes with the other inputs fixed, vanna depends on S/K, so its derivative with respect to strike equals −(S/K) × zomma. At S = K = $100, this reduces to −zomma. At S = $100 and K = $120 in the same 30-day, 30% IV example, zomma is +0.0625 and the strike derivative of vanna is −0.0521. Include the S/K factor when moving between the two axes.

Formula

Zomma = change in gamma / change in implied volatility = Gamma × (d₁ × d₂ − 1) / σ

Gamma is the Black-Scholes gamma, e^(−qT) × φ(d₁) divided by S × σ × √(T), where phi is the standard normal density. The dividend factor drops out only when q is zero.
d₁ = [ln(S/K) + (r − q + σ squared / 2) × T] / (σ × √(T)), and d₂ = d₁ − σ × √(T).
S is spot, K is strike, T is time to expiration in years, σ is implied volatility as a decimal, r is the continuously compounded risk-free rate, and q is the continuous dividend yield.
The raw expression gives the gamma change per 1.00 of volatility; divide by 100 for the change per one volatility point.
Zomma is zero where d₁ × d₂ equals 1, negative inside that band and positive outside it.

Worked Example

A book holds 100 long at-the-money calls on a $100 stock with 30 days to expiration, 30 percent implied volatility, no dividend and a zero rate. Black-Scholes gamma is 0.0463 per share and zomma is −0.155 per 1.00 of volatility, which is −0.00155 of gamma per volatility point.

Position gamma at 30 percent implied volatility = 100 contracts × 0.0463 × 100 shares = 463 deltas per $1 move.
Implied volatility rises 5 points to 35 percent with spot unchanged, so the estimated gamma change = −0.00155 × 5 = −0.0078 per share.
Estimated new gamma = 0.0463 − 0.0078 = 0.0385 per share, while a full reprice at 35 percent returns 0.0397 per share.
Position gamma at 35 percent = 100 × 0.0397 × 100 = 397 deltas per $1 move.

The book shed about 66 deltas of gamma, close to 14 percent of its convexity, without the stock moving at all. A rebalancing schedule sized on the 30 percent gamma would now trigger more often than the remaining convexity justifies. The gap between the linear estimate of 0.0385 and the repriced 0.0397 also shows that zomma itself changes across the move, so reprice gamma in full when the volatility move is large.

How does zomma work?

The Black-Scholes zomma formula

Under Black-Scholes, zomma equals gamma multiplied by the quantity d1 times d2 minus one, all divided by the volatility input. Gamma is positive for a long option and sigma is positive, so the entire sign of zomma is carried by the bracket. The question is whether the product d1 times d2 sits above or below 1.

The sign map across the chain

Near the money, d1 and d2 are both close to zero. Their product is close to zero, the bracket is close to negative one, and zomma is negative. Raising implied volatility widens the modeled distribution of expiration prices and pulls probability away from the single strike sitting under spot, so the gamma peak drops. This is the same relationship the OIC states from the other side: at-the-money gamma is at its largest when implied volatility is low, and it falls as volatility rises.

Out in the wings, d1 and d2 are both large and share a sign, so their product climbs past 1 and zomma turns positive. A far out-of-the-money strike that carries almost no gamma at a quiet volatility level starts to accumulate gamma when volatility spikes, because the model now assigns real probability to spot reaching it.

The crossover happens where d1 times d2 equals 1, which is close to where the absolute value of d1 equals 1. That is roughly one standard deviation of the log move away from the forward. For a $100 stock with 30 days left and 30 percent implied volatility, one standard deviation is about 8.6 percent, so the sign flips near $92 and near $109. Inside that band zomma is negative, outside it zomma is positive.

Zomma example: gamma at 30% vs 35% implied volatility

Take the same $100 stock, 30 days out, and lift implied volatility from 30 percent to 35 percent with spot pinned. The $100 strike sees gamma fall from about 0.0463 to 0.0397 per share. The $120 strike sees gamma rise from about 0.0054 to 0.0083 per share. The peak sags while the shoulders lift, and the whole gamma profile widens without spot contributing anything.

Two structural facts follow from the formula. A call and a put on the same strike and expiration share gamma, so they share zomma as well. And because gamma concentrates as expiration approaches, near-dated at-the-money contracts carry the largest negative zomma on a chain, which is why short-dated books see their convexity reprice fastest when volatility moves.

Zomma is also one of three ways to differentiate gamma, which is why risk engines compute it alongside its siblings rather than on its own. Speed differentiates gamma against spot, color differentiates it against time, and zomma differentiates it against volatility. Between them the three describe every axis along which the curvature of a book can shift while the position itself stays untouched.

How traders use zomma

The first use is keeping the gamma input current in the hedge plan. When IV changes, recompute the position’s gamma and apply the book’s hedge-band or rebalance rules to that new exposure. Zomma identifies the direction and local size of the gamma change; the strategy’s costs and rules determine the next hedge.

The second use is scheduling. Gamma scalping monetizes the difference between realized moves and the volatility paid for. The expected harvest per rebalance scales with gamma. If implied volatility has run up 10 points, an at-the-money position is holding materially less gamma than the entry sheet showed, and the scalping cadence built on the old number will churn commissions against thinner convexity. The reverse also holds: a volatility collapse into an at-the-money strike sharpens gamma and raises both the harvest and the hedging burden.

The third use is reading dealer maps. Aggregate gamma exposure profiles are computed strike by strike from a volatility surface. A parallel lift in that surface does not scale every strike equally; it drains the peak and feeds the wings. Two gamma maps built on the same open interest at different volatility levels can therefore imply different pin levels and different hedge intensity. Analysts who rebuild the map after a volatility event rather than reusing yesterday’s are, in effect, respecting zomma.

The fourth use is structure selection. A wide strangle and an at-the-money straddle can look similar on a delta screen and behave very differently through a volatility spike, because one sits on positive zomma and the other on negative zomma. Traders positioning for a volatility regime change, rather than a single directional move, use that split deliberately.

A retail-scale version of the same discipline is to reprice the option Greeks for a position at the volatility level expected after an event, not the level quoted before it. Earnings, a Federal Reserve decision, or an index volatility shock all move the surface far enough for the difference to show up in the hedge count.

How to reprice gamma after a volatility move

Keep the volatility unit and gamma unit explicit. Per-percentage-point and per-decimal zomma differ by a factor of 100; per-share and per-contract gamma introduce the deliverable multiplier.

Read the sign across moneyness. Near-the-money and wing contracts respond differently to the same volatility change, so apply the strike-specific reading rather than one sign to the entire chain.

For a larger volatility move, reprice gamma across a spot-volatility grid. In the worked example, the local estimate is 0.0385 and the full reprice is 0.0397. Use the repriced gamma to update the hedge band, then compare the resulting exposure with delta, theta, and vega.

Limitations of zomma

Reprice gamma on a large IV move. Zomma is a local reading: on the worked 5-point IV move, the linear estimate of 0.0385 misses the repriced 0.0397 because zomma itself changes across the move.
Run the full spot, volatility and time grid. Zomma moves one implied volatility with spot and time fixed, while real surfaces shift skew and term structure unevenly, a spot move adds speed effects, and passing time adds color.
State the units. Per 1.00 of volatility and per volatility point differ by a factor of 100, and per-share versus per-contract gamma adds the multiplier again.
Recheck each strike’s sign as inputs move. The sign boundaries sit roughly one standard deviation from the forward, so they shift with volatility, time, and carry.

Bottom line

Zomma shows how a gamma reading drifts when implied volatility moves: on a rise, down near the money and up in the wings, with the crossover about one standard deviation from the forward. Use it to keep hedge bands, scalping schedules, and dealer maps current after the surface reprices, state its units, and reprice gamma on a spot-volatility grid when the move is large.

Clear answers

Frequently Asked Questions

Sources and Further Reading

How entries are sourced, calculated and updated: editorial methodology.