The five Greeks describe an option at one moment, and they do not stay put. Delta changes when the stock moves; that is gamma. But delta also changes when nothing happens except a day passing, or when implied volatility moves. Vega changes when volatility moves. These second-order effects have names, and traders who run hedged books watch them because they explain hedges that drift for no obvious reason.
Charm: delta drifts with time
Charm is the change in delta for one day passing, with the stock unchanged. Picture a call 5% out of the money with a month left and a delta of 0.30. If the stock does nothing for two weeks, that call is less likely to finish in the money than it was, so its delta falls, perhaps to 0.20. An in-the-money call drifts the other way, toward 1. Every option's delta slides toward 0 or toward 1 as expiry approaches, and charm is the speed of that slide.
For a call, delta falls each day below the strike and rises above it, drifting toward 0 or 1. The drift grows sharply in the last days. Hover to read the daily change.
For a hedger it means the hedge set on Friday is wrong on Monday without the stock moving. A market maker short out-of-the-money calls and long stock against them finds, as the days pass, that the calls need less stock, and sells some. Summed across the market into a large monthly expiration, that steady flow is one reason index prices can drift in the days before expiry.
Vanna: delta moves with volatility
Vanna is the change in delta for a one-point rise in implied volatility. Higher volatility flattens the delta curve you saw in the delta lesson: out-of-the-money options become more likely to finish in the money, so their deltas rise toward 0.50, and in-the-money options become less certain, so theirs fall toward it. An out-of-the-money call has positive vanna and an in-the-money call negative vanna.
Positive for out-of-the-money calls, negative for in-the-money ones, near zero at the strike. A put has the same vanna as the call at its strike, so switching to puts leaves the chart unchanged.
Vanna is why a hedged book can suddenly carry direction after a volatility shock. In a sell-off, implied volatility usually jumps. Every out-of-the-money put's delta grows more negative before the stock moves much further, and dealers short those puts have to sell more stock to stay hedged. When volatility falls back, the same flow runs in reverse and buying appears.
Vomma: vega moves with volatility
Vomma, also called volga, is the change in vega for a one-point rise in implied volatility. An at-the-money option's price is close to a straight line in volatility, so its vega hardly changes and its vomma is near zero. An out-of-the-money option is different. At low volatility it is almost worthless and barely reacts; as volatility rises it comes alive and its vega grows. Its price curves upward in volatility, and that curvature is vomma.
Vega against implied volatility. The at-the-money line is nearly flat, so its vomma is close to zero; the wings' vega climbs as volatility rises, which is positive vomma.
The practical point: the wings are convex in volatility. Owning out-of-the-money options means a volatility spike helps you more the bigger it gets, and selling them means the reverse. It is part of why the market charges extra for far out-of-the-money puts, which you will meet as skew in Module 11.
The maths: second derivatives, and how the charts compute them
Each second-order Greek is a second derivative of the option price: charm is ∂Δ/∂t, vanna is ∂Δ/∂v (equally ∂vega/∂S), and vomma is ∂vega/∂v. For a call with no dividends, vanna is -n(d1)·d2 / v per unit of volatility and vomma is vega·d1·d2 / v. The signs follow from d1 and d2. For a call well out of the money both are negative, so their product is positive and so is vomma; at the money they sit either side of zero and vomma is close to zero.
The charts compute them the way many risk systems do, by bumping: charm is delta one day later minus delta today, vanna is delta at one vol point higher minus delta now, and vomma is the same bump applied to vega. That is why their units are per day and per vol point.
Your turn
Open the NVDA chain. Find the call about 5% out of the money on the nearest expiration and read its delta, then do the same on the monthly expiration. The shorter-dated call has the smaller delta, because it has less time to reach its strike. The gap between the two is charm, added up over the days between them.