Speed Greek: How Gamma Changes When the Underlying Moves
For educational purposes only; not investment advice.
Direct answer
Section titled “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.
From Gamma slope to hedge curvature
Section titled “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
Section titled “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.
Example: analytic value versus a finite difference
Section titled “Example: analytic value versus a finite difference”Assume a European option with:
S = 100,K = 100T = 0.25year,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.
How to use Speed without false precision
Section titled “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.
Common misconceptions
Section titled “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.