Color
Color is a third-order options Greek that measures the rate of change of Gamma with respect to the passage of time, effectively quantifying how quickly an option's curvature (Gamma) evolves as expiration approaches. It is one of several higher-order sensitivity measures used by sophisticated derivatives desks to manage the dynamic rehedging cost of options portfolios.
Key takeaways
- Color is the third derivative of option value — with respect to the underlying price twice and time once — making it a third-order Greek.
- It helps traders anticipate how their Gamma exposure will shift over the next trading day, enabling more accurate hedging schedules.
- Color is most significant for near-expiration options and during periods of high implied volatility.
- Positive Color means Gamma will increase as time passes; negative Color means Gamma will decrease.
- In practice, Color is used alongside Charm (Delta decay) and Vanna to construct fully time-adjusted hedge ratios.
Explanation
Color, sometimes called Gamma decay or DgammaDtime, sits within the constellation of higher-order Greeks that extend beyond the first-order sensitivities (Delta, Vega, Theta, Rho) and second-order sensitivities (Gamma, Vanna, Volga) to third-order behavior. Formally, Color = ∂Γ/∂t, where Γ is the option's Gamma and t is time. Because Gamma itself measures the convexity of option value with respect to the underlying price, Color tells the trader how that convexity will change overnight or over a given time horizon without any movement in spot price or implied volatility.
For a standard European call or put under Black-Scholes, Color has a closed-form expression involving the option's moneyness, time to expiry, dividend yield, and volatility. The formula is:
Color = -∂Γ/∂T = -(N'(d₁) / (2S·σ·√T)) · [2r·T + 1 + d₁·(2(r-q)·T - d₂·σ·√T) / (σ·√T)]
where d₁ and d₂ are the standard Black-Scholes parameters, r is the risk-free rate, q is the dividend yield, S is the spot price, σ is implied volatility, and T is time to expiry.
Practitioners on derivatives trading desks use Color to answer a critical operational question: if I hedge my Delta today using the current Gamma, by how much will my hedge ratio be wrong tomorrow simply due to the passage of time? Without accounting for Color, a delta-hedged book will drift out of hedge as Gamma evolves, creating unwanted P&L noise. For market-makers running large short-gamma books — common in equity variance swaps and index options — Color helps forecast the acceleration of rehedging costs as expiration draws near.
Color is also informative in the context of the volatility smile. Near-the-money options close to expiration exhibit dramatically accelerating Gamma (and thus large Color magnitudes), while deep in-the-money or out-of-the-money options show more muted Color. Exotic option structures, particularly barrier options and digital options, can exhibit path-dependent Color profiles that require numerical methods rather than closed-form solutions to compute.
Formula
Color = -(N'(d₁) / (2Sσ√T)) × [2rT + 1 + d₁(2(r−q)T − d₂σ√T) / (σ√T)]
Example
A derivatives desk holds a short position in 1,000 at-the-money S&P 500 call options with 5 days to expiry, each with a Gamma of 0.025. The desk computes Color = −0.004 per calendar day. This means tomorrow, with no change in the underlying, each option's Gamma will be approximately 0.025 + (0.004 × 1) = 0.029. The desk currently holds a delta hedge sized for Gamma of 0.025; by tomorrow that hedge will be short by 0.004 × 1,000 × contract multiplier in Gamma units. Knowing this in advance, the trader can pre-schedule a hedge adjustment to execute at the open the following morning rather than reacting to intraday drift.
Related terms
At The Money Convexity Credit Default Swap Delta Delta Hedge Dividend Dividend Yield Equity Gamma Greeks Hedge Ratio Implied Volatility