The Greeks Have Greeks

The Coordinates, Not Dials post read the five basic Greeks two ways: on your position and on the structure. It closed on a promise, that the higher-order Greeks are not exotic decorations but the velocity of those coordinates: how delta drifts as time passes, how it shifts when vol moves, how vega itself responds to vol. This post makes good on that promise for the second-order Greeks.

The first post opened the series with the five basic coordinates. This is Part 2, the four second-order Greeks that were not already covered as basics: vanna, charm, vomma, and veta. A Part 3 will take the third-order surface (speed, zomma, color, and the rest). The reason for the split is that the second-order set earns its own post: this is where the coordinates start to move on their own, and where the market's mechanical hedging turns your risk numbers into flows that push price with no headline behind them.

How a Coordinate Moves

The five basic Greeks answered two questions: where you sit in the priced distribution, and how sensitive you are to each input. But those sensitivities are not fixed. Your delta is not a constant. It changes as spot moves, which is gamma and which Part 1 already covered, and it also changes as implied volatility moves and as time passes, with spot frozen. Your vega is not a constant either: it moves as vol moves and as time passes. The second-order Greeks are the map of that motion.

There are only three axes anything can move along: spot, volatility, and time. Apply them to the two first-order Greeks that actually carry the money, delta and vega, and the whole second-order set falls out. Delta's motion is gamma with spot, vanna with vol, and charm with time. Vega's motion is vomma with vol, veta with time, and, with spot, vanna again. Vanna is the same second derivative read from either end: how delta moves with vol is identical to how vega moves with spot. That is why it sits in the middle of everything. It is the hinge that couples the delta world to the vega world.

As in Part 1, each of these carries two readings. On your position, a second-order Greek tells you how your hedge decays when you do nothing: how much delta you will silently accumulate overnight (charm), how much your delta will lurch if the surface reprices (vanna), whether your vega grows or shrinks as vol moves (vomma). On the structure, the same numbers summed across the chain are the source of the mechanical dealer flows that professionals track: vanna flows and charm flows into monthly expiration, the price drift that has no news attached to it. We walk the four in the order they couple, delta's movers first.

Vanna: Your Delta Moving With the Surface

On your position. Vanna is how your delta changes when implied volatility changes, and equivalently how your vega changes when spot moves. It is the reason a delta-neutral position does not stay neutral through a vol move. Sell a put spread, hedge to zero delta, then watch the surface drop: the deltas of the options you are short shift, and you are no longer flat, even though spot never moved an inch. Vanna is usually largest around the moderately out-of-the-money wings, and its sign follows the moneyness and skew convention, so any book with a skew position, which is nearly every real options book, carries vanna whether the trader named it or not. It is the first Greek that punishes people who hedge delta but forget that delta is itself a function of vol.

On the structure. Vanna is where the skew becomes flow. Because equity-index skew is steep, with downside puts carrying higher implied vol, dealers who are net short those puts hold a large, one-sided vanna. When vol falls, which is the typical grind-higher tape or the morning after an event when IV crush hits, the deltas of those short puts shrink toward zero and dealers buy back the hedges they no longer need. That mechanical buying is the vanna flow, and it is one of the cleaner explanations for why a market can melt up on no news at all, simply because volatility is bleeding out of the surface. When vol spikes the flow reverses and becomes an accelerant to the downside. Read on the chain, vanna is the coupling term that lets a change in the vol surface produce a change in spot, which is why it sits alongside gamma exposure in any serious dealer-flow model. The vanna reference and the SABR page develop the skew mechanics.

Charm: Your Delta Moving With the Clock

On your position. Charm is delta decay: how much your delta drifts purely because time passed, with spot and vol held still. An in-the-money option's delta creeps toward 1 (or toward −1 for a put) as expiration nears; an out-of-the-money option's delta bleeds toward 0. If you are hedged today and do nothing, you will wake up tomorrow slightly un-hedged, and the closer to expiration you are, the more delta the clock hands you. It is worst over a weekend, when three calendar days of decay accrue against two trading days of opportunity to re-hedge. Charm is the Greek that makes end-of-day and Friday-afternoon rebalancing a real task rather than a formality.

On the structure. Charm is the most flow-heavy of the second-order Greeks, because it is driven by the one variable that always moves: the clock. Every desk hedging a large book has to re-trade delta as charm accrues, and because expiring open interest is concentrated at monthly and quarterly expiration, those charm flows concentrate there too. When dealers are long gamma and positioned the usual way, charm generates a stabilizing, mean-reverting bid into expiration, part of the "OPEX drift" and the pinning the max pain idea gestures at. It is heaviest in the closing hour and across weekends, and it is a large part of why 0DTE books, where charm is enormous, have reshaped intraday flow. Where theta is the clock translated into value decay, charm is the clock translated into hedging pressure. The charm reference has the model-by-model detail.

Vomma: Your Vega Moving With Vol

On your position. Vomma is vega convexity: how your vega itself changes as implied vol changes. If vega is your exposure to the width of the distribution, vomma is whether that exposure grows or shrinks as the width changes. Long vomma, which is what the wings of a butterfly or the legs of a strangle give you, means your vega increases as vol rises: you get longer volatility exactly when volatility is moving your way, and shorter it when it is not. That convexity is a good thing to own, and it is not free, because it is priced. Vomma is near zero at the money and largest in the wings, which makes it fundamentally a tails-and-smile Greek. Ignore it and you will misjudge how a big vol move actually lands on your book, because the vega you measured at today's implied vol is not the vega you will have after a five-point repricing.

On the structure. Vomma is your exposure to vol of vol. A trader who is long vomma is long the volatility of volatility itself, and the price of vomma across strikes is a large part of what makes the volatility smile curved rather than flat: it is the market paying up for convexity in the wings. This is where the second-order Greeks cross from the Flow story into the Density story. Vomma is priced by, and helps price, the fatness of the tails the market is carrying, which is the structural exposure that stochastic-volatility models such as Heston and SABR are built to capture. The vomma reference covers the vega-convexity math.

Veta: Your Vega Moving With Time

On your position. Veta is vega decay: how your vega shrinks as expiration approaches. Just as theta bleeds an option's time value, veta bleeds its sensitivity to vol, and the two are cousins. A near-dated option barely responds to a change in the vol surface; a long-dated one responds a great deal. Veta is the gradient between them, read on a single position. It is the Greek that quietly sets the shape of a calendar spread's payoff: when you sell the near leg and buy the far leg, you are, among other things, harvesting the difference in how fast vega decays across the two expirations.

On the structure. Veta reads the term structure of volatility. The vol surface is not flat across tenor, since near-dated and far-dated options price different levels of vol, and veta is how each position's vega sensitivity travels along that tenor axis as time carries it from long-dated toward short-dated. It is the natural companion to theta from Part 1: theta is the time-collapse of value, veta is the time-collapse of vega, and both are gradients of the same term structure. When an event passes and the near-term surface deflates, veta is part of what a calendar or diagonal is positioned to capture. The veta reference develops the tenor mechanics.

The Velocity of the Coordinates

Put the four together and the picture from Part 1 gains a second layer. There, a position was a bundle of distributional exposures: delta to where the distribution sits, gamma to its curvature, theta to its collapse, vega to its width. The second-order Greeks say those exposures are not parked. They drift, along the only axes that move. Vanna is your delta moving with the surface. Charm is your delta moving with the clock. Vomma is your vega moving with vol. Veta is your vega moving with time. If the first-order Greeks are your coordinates, the second-order Greeks are their velocity, and a position you have not touched in a day is not the position you put on.

And the same four, summed across the chain, are the second layer of the market-structure read. Part 1 gave DEX and GEX: which way dealers lean, and whether their hedging fades moves or chases them. Vanna and charm add the mechanics of when. Vanna flows are the surface repricing into spot pressure; charm flows are the clock repricing into spot pressure; and both concentrate into monthly expiration, where they become some of the most-tracked non-fundamental flows on the calendar. Vomma and veta round out the vol side: how the tails and the term structure of vega move as the regime and the clock change. This is why a Greek is a coordinate and not a dial, all the way up the derivative stack. The same numbers that tell you how your hedge decays tell you how the market's hedge decays, and the market's book is large enough to move the tape.

Part 3 will take the third-order Greeks, speed, zomma, color, ultima, and dcharmdvol, which are the velocity of this velocity: how gamma, vanna, and vomma themselves move. They are where the practical trader and the risk desk part ways, useful mostly for stress work and large-book hedging, but they finish the picture of a position as an object that is not just curved but curving. For now, the second-order set is the working vocabulary. Once you can read how your coordinates move, you are reading the market the way the desks that hedge it do.

Options Analysis Suite computes the second-order Greeks live, vanna, charm, vomma, and veta across 17 pricing models and laid over the implied volatility surface, and shows aggregate dealer vanna and charm alongside gamma exposure on the strike-tenor grid, so the position reading and the structure reading sit on the same screen.