Speed

The Speed of an option (also called “color” or “gamma of gamma”) is a third-order derivative in the options greeks hierarchy, measuring how the gamma itself changes as the underlying asset price moves. While delta measures the option’s price sensitivity and gamma measures how delta changes, speed quantifies the curvature of that curvature, making it a critical input for options traders managing convexity risk over multi-day holding periods and for algorithmic trading systems adjusting hedge ratios dynamically.

Why speed matters in multi-day options strategies

When an options trader holds a long call or put, the gamma grows as the stock rallies (for a long call) or falls (for a long put), toward the strike. This accelerating hedge-ratio change is not gradual — it compounds. Speed captures this compound effect. If a trader rebalances a delta-hedged position once per day, ignoring speed leads to underestimation of the cost of daily rebalancing, because each day’s rebalancing itself changes the size of tomorrow’s rebalancing. Speed quantifies this feedback loop mathematically.

Mathematical definition and computation

Speed is the third derivative of the option price with respect to the underlying stock price:

Speed = ∂Γ/∂S = ∂³C/∂S³

Under the Black-Scholes model, speed depends on the same variables as delta and gamma — stock price, strike, volatility, time to expiration, interest rate — but it has a different sign pattern and sensitivity landscape. For a long call:

• Speed is negative out-of-the-money (where gamma is declining as you move further away).
• Speed is positive in-the-money (where gamma is rising as you move further away).
• Speed peaks in magnitude near the strike and very close to expiration.

This contrasts with gamma, which is always positive for a long call and always negative for a short call.

How speed affects hedging costs and P&L

A trader managing a covered call or other option position must rebalance the underlying hedge as delta changes. The cost of rebalancing includes both:

1. The difference in delta between today and tomorrow (first order, captured by gamma).
2. The change in that change (second order, where speed enters).

In a market with large daily moves (high realized volatility), speed effects compound rebalancing costs quickly. A portfolio of short out-of-the-money calls, which have negative speed, becomes cheaper to hedge as the market moves further away from the strikes — a convex payoff. Conversely, a short straddle (long gamma risk on both sides) experiences positive speed effects that increase hedging costs when the market swings hard in either direction.

Portfolio applications and dimension reduction

Options desks typically monitor speed alongside the familiar greeks (delta, gamma, vega, theta, and rho) in the “Greeks matrix” for a portfolio. However, speed often takes a back seat to the first two greeks because:

• Its magnitude is usually small relative to delta and gamma.
• It only dominates portfolio risk in scenarios of extreme daily price swings or when holding period is very short.
• Most hedge-rebalancing algorithms can absorb speed effects through daily recalibration without explicit modeling.

Algorithmic options traders and high-frequency trading (HFT) firms building execution algorithms do monitor speed explicitly, because even small changes in gamma compounding accumulate into significant P&L surprises over thousands of microsecond-scale rebalances.

Speed, time decay, and the “color” relationship

Speed is sometimes called “color” because it captures the interaction between gamma (spacial curvature) and theta (time decay). An option losing time value (theta decay) while simultaneously experiencing gamma changes creates a “colored” or textured P&L surface that a static Black-Scholes model underestimates. Near expiration, when theta spikes, speed also becomes pronounced. This is why binary options and other highly levered, short-dated derivatives can exhibit violent speed effects.

Practical bounds and risk limits

Risk management frameworks rarely set explicit speed limits (unlike delta, gamma, and vega notional limits), but traders use speed to:

• Assess rebalancing frequency. High speed indicates frequent rebalancing is necessary.
• Flag correlation basis risks. If a hedge instrument’s speed differs from the position it hedges, basis risk emerges.
• Detect gamma scalping opportunities. A trader capturing gamma profit must account for how gamma itself is changing; speed quantifies that edge.