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Color Greek: Gamma's Local Sensitivity to Time

Define Color with explicit calendar and remaining-time signs, derive a Black-Scholes benchmark, normalize units, validate finite differences, aggregate positions, and control model risk.

Updated

For educational purposes only; not investment advice. Investing may result in loss.

Direct answer

Color is practitioner terminology for the local time sensitivity of Gamma. It is a third derivative of one specified pricing function, not a standardized exchange field, forecast of tomorrow’s Gamma, hedge order, or P/L. Two vendors can report opposite signs or factors of 252, 365, the underlying price, or the contract multiplier while both are internally consistent.

Let t be calendar time, T the exact expiration timestamp, and τ=T−t time remaining. For option value V(S,σ,τ,…), Δ=∂V/∂S and Γ=∂²V/∂S²=∂Δ/∂S. A remaining-time convention defines Color_τ=∂Γ/∂τ; a calendar-time convention defines Color_t=∂Γ/∂t=−Color_τ while spot, volatility-surface coordinates, rates, dividends, borrow, and every other model input are held fixed.

Raw Gamma is Delta change per one unit of underlying price. A “Gamma per 1% spot move” display instead scales it to 0.01SΓ. Color inherits the Gamma unit per stated time unit. Per-share, per-point, per-contract, currency, cash-Gamma, annual, ACT/365 daily, and 252-trading-day conventions must be normalized before comparison or aggregation. If a vendor already reports daily Color, dividing by the day basis again is an error.

How to calculate and control it

  1. Lock the exact claim and state: series, call or put, long or short quantity, multiplier, deliverable, currency, S, K, exact T, exercise style, settlement, rates, continuous or discrete dividends, borrow, volatility surface, model, calibration timestamp, and version.
  2. Declare coordinates and units before the number: spot- or forward-based Delta, raw Gamma per price unit or scaled Gamma per percentage move, calendar t or remaining τ, year fraction and day type, per-share or per-contract exposure, and any cash-Gamma convention including whether it contains 1/2.
  3. For a Black-Scholes-Merton European benchmark with continuous dividend yield y, define d₁=[ln(S/K)+(r−y+σ²/2)τ]/(σ√τ) and Γ=e^(−yτ)φ(d₁)/(Sσ√τ).
  4. Under that benchmark, ∂d₁/∂τ=[(r−y+σ²/2)τ−ln(S/K)]/(2στ^(3/2)) and Color_τ=Γ[−y−d₁(∂d₁/∂τ)−1/(2τ)]; then change sign for Color_t. This closed form does not govern American exercise, discrete dividends, jumps, stochastic volatility, or proprietary surfaces.
  5. Validate with centered date bumps: estimate Color_τ≈[Γ(τ+h)−Γ(τ−h)]/(2h), repeat with smaller h, and compare with a true one-day clock advance. Control truncation, cancellation, grid, calibration, holiday, partial-day, and near-expiry errors.
  6. Normalize each series, then multiply by signed contract quantity and actual multiplier. Net only compatible currencies, coordinates, expiries, surfaces, models, styles, and settlements; stock itself has zero Gamma and Color even though it can hedge Delta.
  7. Full-reprice spot, IV level, skew, term structure, rates, dividends, borrow, time, jumps, exercise, and settlement scenarios. Reconcile model output with executable bid-ask, discrete hedge size, liquidity, fees, actual fills, lifecycle events, and post-shock Greeks rather than extrapolating Color.

Worked examples

  • Analytic sign and clock. For a European option with S=K=100, r=5%, y=2%, σ=20%, and τ=0.5, d₁=0.176776695297 and Γ=0.027495794412. The analytic results are Color_τ=−0.028904953876/year and Color_t=+0.028904953876/year; under ACT/365, local daily Color is +0.000079191654/day. The signs differ only because τ falls as t advances.
  • Centered difference and finite-step error. In the same benchmark with h=1/365, Γ(τ+h)=0.027416921734, Γ(τ)=0.027495794412, and Γ(τ−h)=0.027575307959. The centered estimate is [Γ(τ+h)−Γ(τ−h)]/(2h)=−0.028905485995/year, only −0.000000532119/year from analytic Color_τ. Advancing a full day instead changes Gamma by +0.000079513547, not exactly the local +0.000079191654; one-sided finite movement includes truncation and curvature.
  • Color can reverse sign and positions must be signed. With S=80, K=100, r=3%, y=1%, σ=25%, and τ=0.25, d₁=−1.682648410514, Γ=0.009660822757, Color_τ=+0.041952095723/year, and therefore Color_t=−0.041952095723/year or −0.000114937249/day. A complete one-day reprice gives Γ=0.009545008312, an actual change of −0.000115814445. Near-expiration Gamma does not always rise, and a short position reverses every signed contribution.
  • Linear extrapolation fails near expiry. For S=K=100, r=3%, y=1%, σ=25%, and τ=10/365, initial Γ₀=0.096326878660, Color_t=1.760952872868/year, and ACT/365 local Color is 0.004824528419/day. After one full day, Gamma is 0.101546073059, a +0.005219194399 change. After five days it is 0.136284663571, an actual +0.039957784911, while five times initial daily Color predicts only +0.024122642094. The local derivative is attribution at the starting state, not a multiday path forecast.

Model, unit, and execution checklist

  • Verify the exact series, valuation timestamp, expiration timestamp, and remaining partial day.
  • State position sign; long and short contracts reverse the same per-option Greek.
  • Distinguish t from τ and record which direction defines positive Color.
  • State ACT/365, ACT/252, trading-day, calendar-day, or other year-fraction convention.
  • Prevent annual-versus-daily double scaling and weekend or holiday misapplication.
  • Distinguish spot from forward coordinates and record any premium-adjusted Delta convention.
  • Separate Gamma per $1 move from 0.01SΓ per 1% spot move.
  • Define cash Gamma explicitly, including whether the display uses ΓS², ΓS²/100, ΓS²/10000, or a 1/2 factor.
  • Separate per-underlying-unit, per-point, per-contract, signed quantity, multiplier, and adjusted deliverable.
  • Normalize currency, FX conversion, quanto treatment, and contract quotation units.
  • Freeze or specify sticky-strike, sticky-delta, or another volatility-surface movement rule.
  • Stress IV level, skew, term structure, and calibration or extrapolation changes.
  • Reconcile rates, continuous or discrete dividends, distributions, borrow, and corporate actions.
  • Use an exercise-compatible model for American claims and test early-exercise boundaries.
  • Distinguish cash from physical settlement and model the post-exercise or assignment position.
  • Compare centered and one-sided bumps, vary h, and detect truncation and numerical cancellation.
  • Refine tree, PDE, quadrature, or simulation grids near expiry and nonsmooth payoff boundaries.
  • Do not net unlike expiries, surfaces, models, styles, settlements, currencies, or stale timestamps blindly.
  • Full-reprice jumps, gaps, event moves, path changes, and multiple days instead of extrapolating local Color.
  • Include bid-ask width, liquidity, discrete hedge contracts, fees, model reserves, execution, and lifecycle operations.

Common misconceptions

  • Positive Color guarantees that Gamma will be higher tomorrow.
  • Color is Theta or the time decay of the option price.
  • Every long call or put has the same Color sign, and vendor fields need no normalization.
  • Daily Color multiplied by days predicts future Gamma or hedge quantity.
  • A more precise higher-order Greek guarantees an executable hedge or certain P/L.

Authoritative sources

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