Second-Order Greeks: Vanna, Charm, Vomma & Veta

Introduction

The first-order Greeks — delta, gamma, theta, vega, rho — each answer a simple question: how much does the option's price move when one input changes? But those sensitivities are not themselves constant. Delta drifts as volatility shifts and as time passes. Vega changes as volatility and time move. The second-order Greeks measure exactly those drifts: they are the derivatives of the first-order Greeks.

If that sounds like splitting hairs, consider that this is the level at which professional options desks actually operate. A market maker running a book of thousands of contracts cannot re-hedge on delta alone, because delta is a moving target — and what moves it is precisely the second-order Greeks. This lesson covers the four that matter most: vanna and charm (how delta moves), and vomma and veta (how vega moves). It assumes fluency with delta and gamma and vega.

A note before we start: these are refinements, not day-one tools. On a single short-dated position their effect is small — that is what "second-order" means. Their power is in understanding how a book behaves over time and across volatility regimes, and in decoding market phenomena like the vanna-charm flows that surround monthly expiration.

The Idea: Greeks Of Greeks

Every second-order Greek is a first-order Greek differentiated again, with respect to one of the inputs. The cleanest way to hold them in your head is a small grid: take the two Greeks whose stability you care about most — delta (direction) and vega (volatility) — and ask how each changes when the underlying, volatility, or time moves.

Vanna: The Bridge Between Direction And Volatility

Vanna measures how delta changes as implied volatility changes. By a symmetry of the pricing model, it also equals how vega changes as the underlying moves — the same number viewed from two directions. It is the hinge that connects an option's directional exposure to its volatility exposure.

Why does delta depend on volatility at all? Recall that delta approximates the probability of finishing in the money. For an out-of-the-money option, raising volatility increases the chance the underlying reaches the strike, so its delta rises. Lowering volatility does the reverse. Vanna captures that coupling: it is largest for out-of-the-money options, and it changes sign either side of the money.

Vanna is the reason a position that looks delta-neutral can quietly become directional when the volatility regime shifts. For a dealer short a large book of out-of-the-money puts (a common structural position, since investors buy puts for protection), a fall in the market that comes with a rise in volatility moves their delta through vanna — forcing them to sell the underlying to re-hedge, which can accelerate the very move that triggered it.

Charm: Delta Decay

Charm measures how delta changes purely because time is passing — often called delta decay. Just as theta bleeds an option's value with time, charm bleeds (or builds) its delta.

The effect is most vivid near expiration and away from the money. An out-of-the-money option that still had a delta of, say, 0.20 a week out sees that delta melt toward zero as expiration approaches and it becomes clear the option will expire worthless. An in-the-money option's delta, conversely, firms toward 1. Charm is the rate of that drift.

For hedgers, charm is a weekend problem. A book hedged to delta-neutral on Friday can open Monday off-neutral, not because anything moved, but because three days of charm have shifted every option's delta. Desks that carry large positions over weekends and holidays pre-adjust for exactly this.

Vomma (Volga): The Convexity Of Vega

Vomma — also called volga — measures how vega changes as implied volatility changes. It is to vega what gamma is to delta: the convexity that makes vega itself a moving target.

Positive vomma means that as volatility rises, your vega grows — a long-volatility position becomes even longer volatility in a spike. This is highly desirable if you are positioned for a volatility explosion: your exposure compounds in your favour. Vomma is largest for options in the wings (well out of the money), whose vega is most sensitive to the level of volatility, and near zero for at-the-money options, whose vega is relatively stable. This is precisely why traders who want to bet on the volatility of volatility — the tendency of vol to move in bursts — reach for out-of-the-money options rather than at-the-money ones.

Veta: The Decay Of Vega

Veta measures how vega changes as time passes. Vega, like everything else in an option, shrinks as expiration nears — a two-year option has far more volatility sensitivity than a two-day one — and veta is the rate of that decline.

Veta explains why long-dated options are the natural home for pure volatility views: their vega decays slowly, so a vega bet placed in a LEAPS survives, while the same bet in a weekly evaporates within days. It is the least-discussed of the four, but it is the quiet reason "trade volatility with time on your side" is sound advice.

(A fifth second-order Greek, vera, measures how rho changes with volatility. It is genuinely obscure and matters only to desks managing long-dated rate-and-vol exposure together; it is named here for completeness and set aside.)

Vanna-Charm Flows: Where This Becomes Visible

The most famous real-world appearance of second-order Greeks is the vanna-charm flow narrative around monthly options expiration.

The setup: options dealers are, in aggregate, often short options to the investing public (who buy calls for upside and puts for protection). To stay directionally neutral, dealers hedge by trading the underlying — and the size of that hedge depends on delta. But delta is being pushed around by vanna (as implied volatility drifts) and charm (as time runs down toward a huge concentration of open interest at the monthly expiration). As those forces move the dealers' delta, they must mechanically buy or sell the underlying to stay hedged.

Because the open interest is large and the expiration date is known, this re-hedging is partly predictable — which is why market commentators watch it. A common pattern: into a calm, drifting-higher tape before monthly expiration, charm and vanna can generate a steady dealer bid that supports the market; after expiration, that support is released. None of this is a trading recommendation — flows are one input among many and frequently overwhelmed by news — but it is a real, mechanical consequence of second-order Greeks operating at market scale.

Why It Matters

• It completes the mental model. Once you know that delta and vega are themselves moving, you stop being surprised when a "neutral" position drifts. The second-order Greeks name the forces doing the drifting.
• It explains dealer-flow phenomena — vanna-charm flows, the market's tendency to firm into monthly expiration and soften after — that are otherwise mysterious.
• It sharpens volatility trading. Choosing wing options for their vomma, or long-dated options for their low veta, is applying these Greeks deliberately rather than by feel.

For a retail trader on a single position, these remain refinements. But they are the vocabulary of professional options risk, and understanding them is the difference between watching the market's machinery and being confused by it.

Common Misconceptions

• "Second-order Greeks are just academic." The vanna-charm flow discussed above moves real money in real markets every monthly expiration. Academic in origin, practical in effect.
• "Vanna and charm are exotic risks I've never had." If you have ever held an out-of-the-money option into expiration and watched its delta collapse, you have felt charm. If your 'neutral' spread turned directional in a volatility spike, that was vanna.
• "Vomma is the same as vega." Vega is your volatility exposure; vomma is how that exposure changes as volatility moves. Confusing them is like confusing delta with gamma.
• "You must track these to trade options." You need not. But the professionals setting the prices you trade against do — and knowing what they are hedging is an edge in understanding, if not in execution.

Real-World Application

A volatility trader wants to bet that a quiet market is about to become turbulent — not a directional view, a volatility view, and specifically a bet that volatility will move in a sharp burst. Rather than buying at-the-money options (high vega, but low vomma, so their vega is stable), they buy out-of-the-money options in the wings, whose high vomma means their vega will expand if volatility spikes — compounding the payoff exactly when it is wanted. They choose a multi-month expiry so veta does not bleed their volatility exposure away before the move arrives, and they note that their position's vanna will turn it mildly directional if the market falls-and-spikes together, which they accept as consistent with their thesis. Every one of those choices is a second-order Greek applied on purpose. The trade may still lose — volatility may stay dead — but it is expressed with a precision that delta and vega alone could not provide.

Key Takeaways

• Second-order Greeks are the derivatives of the first-order Greeks: they measure how delta and vega themselves change as inputs move.
• Vanna — how delta changes with volatility (equivalently, how vega changes with the underlying). It bridges direction and volatility, and drives dealer re-hedging.
• Charm — delta decay: how delta drifts purely with the passage of time. A weekend hazard for hedged books.
• Vomma / volga — the convexity of vega: how vega grows as volatility rises. Largest in the wings; the tool for trading the volatility of volatility.
• Veta — how vega decays with time. Why long-dated options are the home of pure volatility bets.
• At market scale these produce vanna-charm flows, the partly-predictable dealer hedging around monthly expiration. On a single retail position they are refinements — but they are the language of professional options risk.