# Speed Greek: How Gamma Changes When the Underlying Moves

Define option Speed as the spot derivative of Gamma, calculate it by finite differences and Black-Scholes, and use its units and limitations in nonlinear hedge analysis.

Canonical: https://wiki.fcontext.com/options/speed-greek/
Fact checked: 2026-07-22

> For educational purposes only; not investment advice.

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## Direct answer

**Speed** is the rate at which an option's Gamma changes when the underlying price changes, holding the model's other inputs fixed:

`Speed = ∂Γ/∂S = ∂³V/∂S³`

Delta is the first spot derivative of option value, Gamma is the second, and Speed is the third. If Gamma is quoted as option-value change per `$1²` move in the underlying, Speed has units of option value per `$1³`. For a standard equity option, multiplying a per-share sensitivity by the contract multiplier converts it to a contract-level approximation.

Speed is useful when a portfolio's Delta and Gamma can change materially across the contemplated spot move. It does not forecast the move, and it is not a separately observable market price. It is a local model sensitivity that depends on spot, strike, time, volatility, rates, dividends, exercise features, and the selected volatility-surface dynamics.

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## From Gamma slope to hedge curvature

For a small spot move `ΔS`, a local expansion of Delta is:

`Δ_new ≈ Δ_old + ΓΔS + ½ Speed(ΔS)²`

The Gamma term assumes a locally linear Delta change. Speed supplies the first correction for Gamma changing along the move. Equivalently:

`Γ_new ≈ Γ_old + Speed × ΔS`

Neither approximation should be stretched across large jumps, barriers, discrete dividends, exercise boundaries, or a changing implied-volatility surface. Repricing is preferable when the move is material.

Under dividend-adjusted Black–Scholes for a European Call or Put:

`Γ = e^(−qT) φ(d₁) / (Sσ√T)`

`Speed = −(Γ/S) × [1 + d₁/(σ√T)]`

where `d₁ = [ln(S/K) + (r − q + ½σ²)T] / (σ√T)`. A Call and Put with the same strike and maturity have the same Gamma and Speed under these assumptions because put-call parity adds only terms linear in spot plus cash.

**Speed is not Color.** Speed is `∂Γ/∂S`; Color is Gamma's sensitivity to time. It is also not Zomma, which measures Gamma's sensitivity to volatility.

### Estimation from a pricing system

When no trusted analytic formula exists, use a symmetric finite difference:

`Speed_h ≈ [Γ(S+h) − Γ(S−h)] / (2h)`

Compute Gamma with identical volatility, clock, curve, dividend, and numerical settings. Repeat with smaller and larger `h`. If the result changes sharply, numerical noise, payoff discontinuity, surface interpolation, or non-smooth exercise logic may dominate the estimate. Higher-order Greeks are generally less stable than price, Delta, or Gamma.

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## Example: analytic value versus a finite difference

Assume a European option with:

- `S = 100`, `K = 100`
- `T = 0.25` year, `σ = 20%`
- `r = 0`, `q = 0`

Black–Scholes gives approximately:

| Spot | Gamma |
| ---: | ---: |
| `$99` | `0.040246` |
| `$100` | `0.039844` |
| `$101` | `0.039060` |

Using `h = $1`:

`Speed_h ≈ (0.039060 − 0.040246) / ($2) = −0.000593 per $³`

At `S = 100`, `d₁ = 0.05`; the closed-form value is:

`Speed = −(0.039844/100) × [1 + 0.05/0.10] = −0.000598 per $³`

The difference is finite-difference and rounding error. For a hypothetical `+$5` move with inputs frozen, the second-order Delta change is:

`ΓΔS + ½ Speed(ΔS)² ≈ 0.039844×5 + 0.5×(−0.000598)×25 = 0.19175`

Gamma alone gives `0.19922`; including Speed lowers the local estimate by about `0.00747` Delta, or `0.747` shares for a `100` multiplier. Direct repricing should decide the actual hedge, especially because real implied volatility can move with spot.

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## How to use Speed without false precision

- State whether sensitivities are per share, per point, per `1%`, or per contract; unit mistakes can be orders of magnitude.
- Freeze and document the inputs used for a partial derivative, then run a separate surface-move scenario.
- Aggregate signed position Speed by underlying, expiry, and scenario; a net total can hide opposing concentrations.
- Compare finite-difference bumps and analytic or automatic-differentiation results where available.
- Reprice after meaningful moves instead of relying on a third-order Taylor approximation.
- Inspect near-expiry and near-strike positions frequently; sensitivity shapes can become narrow and large.
- Include bid-ask, gaps, halts, discrete hedging, transaction costs, assignment, and broker limits.
- Do not hedge Speed in isolation while creating larger Delta, Gamma, Vega, or liquidity exposure.
- Treat American exercise, adjusted deliverables, barriers, and discontinuous payoffs with model-specific methods.
- Record volatility coordinate and sticky rule; holding strike IV fixed and moving an entire surface produce different Speed-like P/L.
- Use executable stress losses and cash requirements for position sizing, not a small local Greek.

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## Common misconceptions

- **“High absolute Speed means the option is expensive.”** Speed describes local curvature, not price level or value.
- **“Positive Gamma means positive Speed.”** Gamma can be positive while its spot slope is negative; sign varies by location.
- **“Speed measures how fast time passes.”** That confuses it with time sensitivities such as Theta or Color.
- **“A Speed-neutral portfolio stays Gamma-neutral.”** Neutrality is local and other inputs, surface dynamics, and large moves change Gamma.
- **“The displayed Greek is market data.”** It is an output of a model and conventions.
- **“Adding Speed makes a Taylor estimate exact.”** Fourth- and higher-order terms plus changing volatility and time remain.

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## Related topics

- [Gamma risk](/options/gamma-risk/)
- [Charm, Vanna, and Vomma](/options/charm-vanna-vomma/)
- [Option model risk](/options/option-model-risk/)
- [Option stress testing](/options/option-stress-testing/)

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## Authoritative sources

- [Haug: Know Your Weapon, Part 1](https://www.espenhaug.com/KnowYourWeapon.pdf)
- [Pagès, Pironneau, and Sall: Vibrato and Automatic Differentiation for High Order Derivatives and Sensitivities of Financial Options](https://arxiv.org/abs/1606.06143)
- [Black and Scholes: The Pricing of Options and Corporate Liabilities](https://doi.org/10.1086/260062)
- [OCC: Characteristics and Risks of Standardized Options](https://www.theocc.com/company-information/documents-and-archives/options-disclosure-document)