Skip to content
Financial Context Wiki

Delta and Gamma: Local Direction, Curvature, and Position Units

Measure local option direction and curvature with explicit model coordinates, signed quantities, compatible multipliers, second-order estimates, hedge units, finite-difference checks, and full repricing.

Updated

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

Direct answer

For a model value V(S, t, σ, ...), Delta is the local first derivative Delta = ∂V / ∂S, and Gamma is the local curvature Gamma = ∂²V / ∂S² = ∂Delta / ∂S. Delta estimates the value change for a small one-unit move in the chosen underlying coordinate; Gamma estimates how Delta changes as that coordinate moves.

Both are timestamped model outputs, not forecasts, probabilities, executable quotes, or permanent hedge ratios. Their meaning depends on the exact claim, position sign, price coordinate, volatility surface, time, rates, distributions, borrow, exercise model, settlement, quote unit, multiplier, and vendor scaling.

Curved option value line with a tangent for Delta and changing slopes for Gamma Curved option value line with a tangent for Delta and changing slopes for Gamma
Delta is the local tangent; Gamma is the local rate at which that tangent changes.

A signed and unit-consistent Greek ledger

Let q be signed option contracts, positive when long and negative when short; M the compatible premium multiplier; h signed underlying units; Delta_i model Delta per option unit; and Gamma_i model Gamma per option unit per one price-unit move. Position Delta in underlying-equivalent units is Δ_pos = q × M × Delta_i + h, and position Gamma is Γ_pos = q × M × Gamma_i. Stock has Delta +1 per long unit and Gamma 0.

  1. Lock the exact series, timestamp, long-short sign, quantity, multiplier, deliverable, quote currency, underlying coordinate, exercise style, settlement, and model version. Do not net Greeks from economically different claims merely because symbols look similar.
  2. Declare the derivative coordinate and scale. Distinguish spot Delta from forward, futures, premium-adjusted, or cash Delta; distinguish Gamma per $1, index point, futures point, or 1% spot move; and record whether the vendor already multiplied by contracts or multiplier.
  3. Freeze the model state: spot or forward, rates, dividends, borrow, time to exact expiration, volatility surface, sticky-strike or sticky-delta shock rule, discrete-event treatment, and numerical method. A displayed Greek without this state is not independently reproducible.
  4. For a small move ΔS, calculate per-unit Taylor change ΔV_Taylor ≈ Delta_i × ΔS + ½ × Gamma_i × (ΔS)² and local updated Delta Delta_new ≈ Delta_i + Gamma_i × ΔS. Multiply the price estimate by signed q × M; do not multiply an already aggregated vendor value again.
  5. Validate analytic or vendor Greeks with centered bumps where the pricing function is smooth: Delta_FD ≈ [V(S + h_S) − V(S − h_S)] / (2h_S) and Gamma_FD ≈ [V(S + h_S) − 2V(S) + V(S − h_S)] / h_S². Shrink and enlarge h_S to expose truncation, cancellation, grid, or surface-recalibration effects.
  6. Aggregate only after normalizing every leg to the same underlying-equivalent, price-unit, currency, timestamp, and sign convention. Then add stock or futures hedge units and stress spot, time, volatility, skew, rates, dividends, borrow, jumps, and exercise boundaries.
  7. Use full repricing as the control for finite scenarios and executable bid-ask prices for trading. Reconcile fills, hedge quantity, fees, borrow, assignment, settlement, model version, and post-shock Greeks rather than treating a local Taylor estimate as realized P/L.

For ordinary long vanilla calls and puts in a smooth European model, Gamma is positive; shorting reverses the position Gamma. That statement is not a universal claim about digital payoffs, barriers, American exercise boundaries, adjusted claims, portfolios, or vendor conventions. Near expiry, Gamma can concentrate near a strike while remaining small far in or out of the money, and jumps make the continuous local description incomplete.

Worked examples

  • Long-call local price and hedge estimate. A long call has value V_0 = $3.10, Delta_i = 0.42, Gamma_i = 0.06, q = +1, and M = 100. For ΔS = +$1.50, the Delta term is 0.42 × $1.50 = $0.6300 per share and the curvature term is ½ × 0.06 × ($1.50)² = $0.0675, so ΔV_Taylor = $0.6975 per share or +$69.75 per contract. The estimated value is $3.7975, and Delta_new ≈ 0.42 + 0.06 × 1.50 = 0.51. Position Delta moves from +42 shares to about +51 shares; a static hedge of 42 short shares is no longer neutral.
  • Short Gamma reverses position signs. Six short calls have q = −6, M = 100, Delta_i = 0.55, and Gamma_i = 0.04. Initial position Delta is −330 shares and position Gamma −24 shares per $1. For ΔS = +$2, estimated position Delta changes by −48 shares to −378 shares; option P/L is approximately −6 × 100 × [0.55 × 2 + ½ × 0.04 × 2²] = −$708. For ΔS = −$2, the same local formula gives +$612. A Delta-neutral trader would buy about 48 more shares into the rise or sell about 48 into the decline, before costs and model changes.
  • Portfolio aggregation includes stock and signed option legs. Hold four calls with Delta_i = 0.35, Gamma_i = 0.025, short three puts with Delta_i = −0.40, Gamma_i = 0.030, use M = 100, and short 200 shares. Option Delta is +140 + 120 = +260 shares; after stock, net Delta is +60 shares. Option Gamma is +10 − 9 = +1 share per $1. For ΔS = +$5, local net Delta becomes about +65 shares. The two option legs contribute +$825.00 and +$487.50, while stock contributes −$1,000, giving a second-order portfolio estimate of +$312.50 before fees, borrow, IV, and time changes.
  • Local Taylor attribution versus full repricing. For a European call with S = 100, K = 100, r = 4%, continuous yield y = 1%, σ = 25%, and τ = 0.5, a Black-Scholes benchmark gives V_0 = 7.7215522303, Delta_i = 0.5659323171, and Gamma_i = 0.0221205770. For ΔS = +5, the second-order estimate is 0.5659323171 × 5 + ½ × 0.0221205770 × 5² = 3.1061687980 per share. Full same-IV repricing gives V_1 = 10.8172471731, a change of 3.0956949428, so the local estimate differs by 0.0104738552 or about $1.047386 per 100 multiplier. If IV also rises to 30%, full value is 12.1628459929; the total change 4.4412937626 cannot be attributed by the original Delta and Gamma alone.

Risks and controls

  • Wrong series, root, expiration, strike, option type, or deliverable invalidates the Greek map.
  • Omitting the long-short sign reverses position Delta, Gamma, hedge direction, and P/L attribution.
  • Contract quantity and premium multiplier can be double-counted or omitted.
  • Adjusted contracts and futures options may use units that are not ordinary shares.
  • Spot, forward, futures, premium-adjusted, and cash Delta are not interchangeable.
  • Gamma per price unit differs from Gamma scaled to a 1% spot move or cash Gamma.
  • Currency and FX conversion can make apparently comparable Greeks inconsistent.
  • Different timestamps or stale underlying and option quotes create false netting.
  • Midpoint or model Greeks do not guarantee executable option or hedge prices.
  • Volatility surface, sticky rule, skew, and term-structure choices change both Greeks.
  • Rates, dividends, borrow, and financing assumptions affect model values and hedges.
  • American exercise and discrete dividends create boundaries absent from a smooth European formula.
  • Cash settlement, physical delivery, and post-exercise inventory create different operational risks.
  • Near-expiry nonsmoothness can make Gamma large, unstable, or grid-dependent around a strike.
  • Large moves invalidate a fixed-Delta and fixed-Gamma extrapolation.
  • Jumps, halts, and gaps bypass continuous rebalancing assumptions.
  • Finite-difference bumps that are too large add truncation error; tiny bumps add cancellation noise.
  • Netting different expiries, surfaces, models, currencies, or settlement types can hide basis risk.
  • Discrete hedge size, bid-ask spread, fees, market impact, stock borrow, and liquidity alter realized results.
  • Model, vendor, rounding, assignment, settlement, and broker-record differences require final reconciliation.

Common misconceptions

  • “Delta is a universal probability.” It is a local derivative; probability interpretations require specific models, numeraires, and conventions.
  • “Delta stays fixed.” Gamma and every changing model input move it.
  • “A Delta-neutral portfolio has no risk.” Gamma, Vega, Theta, jumps, basis, liquidity, execution, and model risk remain.
  • “Gamma directly predicts price change.” It enters the local price expansion through ½ × Gamma_i × (ΔS)² and changes during the move.
  • “Precise Greeks eliminate the need for full repricing.” Local derivatives aid attribution; finite scenarios, execution, and lifecycle events still require complete valuation and reconciliation.

Authoritative sources

Navigation

Search the wiki...