Speed (Option Greek)

What is speed in options?

Speed is the third-order option Greek that measures how fast gamma changes as the underlying price moves. Delta is the first derivative of option value with respect to spot, gamma is the second, and speed is the third. Speed is therefore the partial derivative of gamma with respect to spot and the second derivative of delta. Cboe research on zero-days-to-expiration index options states the chain in exactly those three equivalent forms.

The Greek travels under two other names. DgammaDspot is the literal description used in risk systems, and traders sometimes call it the gamma of the gamma. All three labels refer to the same quantity.

The name collides with a common teaching analogy that calls delta speed and gamma acceleration. The Greek called speed sits one rung above gamma, so in that analogy it corresponds to jerk, the rate at which acceleration changes. In an options risk report, read speed as a third derivative.

Units follow the same convention as the Greeks below it. Delta is quoted per share, gamma is quoted as delta change per one-point move in the underlying, and speed is quoted as gamma change per one-point move. A speed of −0.0106 says the model expects gamma to fall 0.0106 for a one-point rise in the stock. The 100-share deliverable of a standard equity contract converts any of these into full-contract terms.

How does speed work?

The Black-Scholes speed formula

Under Black-Scholes, speed equals the negative of gamma divided by spot, multiplied by the quantity d1 divided by sigma times the square root of time, plus one. Two properties fall straight out of that expression.

The first is the sign. Speed is zero where d1 = −σ√T. Solving for spot gives S = K × exp[−(r − q + 1.5σ²)T], the crest of the gamma profile. A long option has positive speed below that crest and negative speed above it. The crest sits below the strike in the zero-rate, zero-dividend example; the complete expression places it for other carry inputs.

The second is the scale. Gamma sits in the numerator and spot in the denominator, so speed inherits gamma’s concentration near the strike and then divides it by the price level. Because gamma itself compresses into a narrower band around the strike as expiration nears, the slope of that hump steepens rapidly at the same time. A worked Black-Scholes calculation on a 100-strike call at 30 percent volatility gives a peak speed of 0.0036 with thirty days remaining and 0.0147 with seven, four times higher as the clock runs down. Both peaks sit out on the flanks rather than at the strike, near $90 in the thirty-day case and near $96 in the seven-day case, because the crest of the gamma hump is exactly where the slope of that hump is zero.

Where speed shows up in a hedge

The practical home for speed is the expansion of a delta change. Applying Ito’s lemma to delta gives three terms: gamma multiplied by the price move, one half of speed multiplied by the squared price move, and charm multiplied by elapsed time. Cboe research uses precisely this decomposition and notes that the first two terms estimate how a required delta hedge changes when the stock price moves, with the speed term capturing the impact of large price moves on delta, while the third estimates how the hedge changes as time passes if the stock does not move.

Because the squared price move scales with variance multiplied by elapsed time, the speed term enters the rebalancing equation as a drift alongside charm rather than as a separate price-driven jump. That is why traders study the two together.

How traders use speed

Delta hedgers use speed to refine the next hedge estimate as gamma changes across a price move. In the $102-to-$104 example, negative speed shrinks the delta increase: the full reprice adds 7,100 delta-equivalent shares on 500 calls, the speed-aware estimate 7,250, and starting gamma alone projects 8,300.

The measure matters most in zero-days-to-expiration and other very short-dated books, where gamma is enormous but also extremely local. Cboe research reports the useful nuance directly: for 0DTE options, the speed and charm terms are each larger than the gamma term at reasonable parameter values, yet they very nearly offset each other, so their sum stays small and remains proportional to gamma. Reading the speed contribution alone at that horizon exaggerates it.

Structure traders use speed to locate changes in the shape of a position’s gamma profile. Butterflies, condors, and ratio structures combine long and short legs, creating price regions with different net gamma and potential zero crossings. A short strangle has negative gamma throughout the Black-Scholes price range. Calculate signed gamma and speed across the full structure to identify its peaks, slopes, and crossings.

At the desk and vendor level, speed belongs to the same family as the other third-order gamma derivatives. Speed differentiates gamma with respect to spot, color differentiates it with respect to time, and zomma differentiates it with respect to implied volatility. Together the three describe how the curvature of a book decays and shifts along all three axes, which is why market-making risk systems compute them as a group rather than one at a time.

How to use speed in a hedge scenario

Calculate speed from the pricing model used for the contract. State its price unit and sign convention, then combine it with current gamma to measure the local change in curvature.

For a larger price move, reprice gamma at the target and across intermediate prices. Near expiration, the gamma peak changes rapidly across strikes; a scenario grid captures the profile across the whole planned move.

Keep the volatility-surface convention consistent. Sticky-strike and sticky-delta scenarios imply different volatility inputs as spot changes. Combine speed with charm to compare hedge changes across both price and time.

Limitations of speed

• Reprice the full move when it is large. Speed itself changes along the way, which is why the speed-aware estimate still misses the full reprice by 150 shares.
• Add color and zomma when time or volatility moves too. Speed holds implied volatility, time, and rates fixed, while gamma also shifts with time through color and with implied volatility through zomma, so a move that changes several inputs needs a full reprice.
• Fix the volatility convention before you compare. Sticky-strike and sticky-delta scenarios feed different volatilities into the model as spot moves, so the same book shows a different speed under each.
• Read speed with charm near expiration. Speed becomes very large and very local there, and for 0DTE options its term nearly offsets charm, so the speed contribution alone overstates the hedge change.

Bottom line

Speed estimates how gamma will change as the stock moves, which turns a constant-gamma hedge projection into a more accurate one. Use it with gamma and charm in short-dated books, state its unit and sign convention, and reprice gamma across the whole planned move when the move is large or expiration is close.