Speed
Speed is a third-order derivative of an option's price with respect to the price of the underlying asset — specifically, the rate of change of an option's gamma with respect to the underlying price. It measures how rapidly gamma itself changes as the underlying moves, providing options traders and risk managers with insight into the convexity of their gamma exposure and the stability of delta hedging programs near specific price levels.
Key takeaways
- Speed = ∂Gamma / ∂S = ∂³V / ∂S³, making it a third-order sensitivity; it is one of the 'minor Greeks' or 'Greeks of Greeks' (also called 'color,' 'vomma,' and 'ultima' for other higher-order sensitivities).
- Positive speed means gamma is increasing as the underlying rises; negative speed means gamma is decreasing. For a long at-the-money call, speed is typically negative — gamma peaks at-the-money and declines as the option moves in or out of the money.
- Speed is critical for managing gamma scalping strategies near barrier levels in knock-in/knock-out options, where delta and gamma can change discontinuously as the barrier is approached.
- Options market makers with large books use speed to understand how their delta hedging costs will evolve as markets move; a portfolio with high absolute speed requires more frequent rebalancing to maintain delta neutrality.
- In practice, speed is primarily relevant for exotic options with path-dependent payoffs, near-expiry standard options with high gamma, and for mathematical completeness in analytical risk frameworks.
Explanation
Options pricing theory generates an entire calculus of sensitivities — the Greeks — that describe how an option's value and risk characteristics respond to changes in market variables. The first-order Greeks (delta, vega, theta, rho) are universally monitored by options practitioners. The second-order Greeks (gamma, vanna, volga) provide insight into the convexity of first-order sensitivities. Third-order Greeks, of which speed is the primary example for the price dimension, extend this analysis to the curvature of curvature — the change in second-order sensitivity with respect to the underlying.
Mathematically, speed is defined as the third partial derivative of the option's price (V) with respect to the underlying price (S): Speed = ∂³V / ∂S³. Since gamma (Γ) = ∂²V / ∂S², speed can equivalently be written as ∂Γ / ∂S. Under the Black-Scholes framework, the speed of a European call option has an analytical closed form: Speed = -Γ(d₁ + σ√T) / (S × σ√T), where d₁ is the Black-Scholes d₁ parameter, σ is volatility, and T is time to expiry.
The practical relevance of speed arises in the context of large, sudden market moves. A delta-hedged options position is continuously rebalanced as the underlying moves to maintain delta neutrality (delta hedging or 'gamma scalping'). The amount of rebalancing required for a given underlying move is determined by gamma. But gamma itself changes as the underlying moves — and speed tells us how fast. An options portfolio with high speed (strongly positive or negative) will require substantially more hedging activity as markets move, generating higher transaction costs and path-dependency in P&L.
For exotic options, speed takes on added importance near discontinuities. Barrier options (knock-in/knock-out) have payoffs that depend on whether the underlying reaches a specified barrier price. Near the barrier, delta and gamma can become extremely large (for options near expiry approaching an in-the-money barrier) or discontinuous. Speed near these barriers describes the rate at which gamma is building or collapsing as the underlying approaches the barrier — critical information for risk managers hedging barrier option books who need to understand not just their current gamma exposure but how rapidly it will change.
In a broader analytical context, speed belongs to a family of higher-order Greeks used in precision risk management for sophisticated derivatives portfolios. 'Color' (or 'gamma bleed') is the rate of change of gamma with respect to time; 'zomma' is the rate of change of gamma with respect to volatility. Together, these sensitivities provide a complete local polynomial approximation to the value surface of an options portfolio, enabling risk managers to predict how their portfolio's P&L will evolve under various market scenarios without resorting to full repricing. This analytical tractability is valuable for intraday risk management in fast-moving markets.
Formula
Speed = ∂³V / ∂S³ = ∂Γ / ∂S = -Γ × (d₁ + σ√T) / (S × σ√T)
Example
A market maker holds a delta-hedged position in near-expiry S&P 500 options. At the current underlying price of $4,500, the option's gamma is 0.003 and speed is -0.000004. This means that if the S&P 500 rises by 50 points to $4,550, gamma changes by approximately -0.000004 × 50 = -0.0002 per unit, falling to approximately 0.0028. For a 500-contract position ($450 million notional), this means the effective gamma exposure decreases from 1.5 (0.003 × 500) to 1.4 (0.0028 × 500) as the market rises. A trader managing the delta hedge must recalibrate their expectations of future rebalancing frequency and size as the market moves, using speed to project gamma evolution without requiring full repricing. In a market-making context where the desk holds hundreds of different strikes and expiries, aggregated speed across the book determines whether the portfolio's hedging burden increases or decreases as markets trend in a given direction.
Related terms
Barrier Option Box Spread Call Option Color Convergence Convexity Delta Delta Hedge Exotic Options Gamma Gamma Scalping Greeks