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Delta-Gamma Approximation: Local P/L, Units, and Repricing

Estimate local option and portfolio value changes with signed Delta, Gamma, cross-Gamma, compatible units, finite-difference checks, and full-repricing controls.

Updated

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

Direct answer

For a model value V_i(S, x) at a fixed timestamp and model state, the one-factor Delta-Gamma approximation is ΔV_i,DG ≈ Delta_i × ΔS + ½ × Gamma_i × (ΔS)². Here Delta_i = ∂V_i / ∂S, Gamma_i = ∂²V_i / ∂S², and ΔS is an absolute move in the same underlying-price coordinate used by the derivatives.

The formula is a local second-order Taylor estimate of model-value change. It is not a forecast, probability, executable P/L, or substitute for full repricing. Its interpretation requires the exact claim, signed position, multiplier, currency, coordinate, volatility surface, time, rates, dividends, borrow, exercise model, settlement, and vendor scaling.

Curved option value line with a Delta tangent and a Gamma curvature correction Curved option value line with a Delta tangent and a Gamma curvature correction
The Delta-Gamma estimate is local; full repricing controls a finite scenario.

A signed, unit-consistent approximation ledger

Let q_i be signed option contracts, positive when long and negative when short; M_i the compatible premium multiplier; and h signed underlying units. For claims on one compatible price coordinate, position Delta is D_pos = h + Σ_i q_i × M_i × Delta_i, position Gamma is G_pos = Σ_i q_i × M_i × Gamma_i, and the local position estimate is ΔΠ_DG ≈ D_pos × ΔS + ½ × G_pos × (ΔS)². Stock contributes Delta +1 per long unit and Gamma 0.

  1. Lock every exact series, timestamp, position sign, quantity, multiplier, deliverable, currency, exercise style, settlement method, price coordinate, and model version. Similar symbols do not establish compatible claims.
  2. Freeze the model state and shock convention: spot or forward, rates, dividends, borrow, exact time to expiration, volatility surface, sticky-strike or sticky-delta rule, and discrete-event treatment. A simultaneous time or volatility move is not a pure spot Delta-Gamma scenario.
  3. Normalize each leg to compatible per-unit Greeks. Distinguish spot, forward, futures, premium-adjusted, and cash Delta; Gamma per $1, index point, futures point, or 1% spot move; and vendor values that may already include quantity or multiplier.
  4. Aggregate with signed q_i × M_i, then add the underlying hedge separately. Do not net unlike currencies, coordinates, expiries, settlement types, or independently calibrated surfaces merely because a dashboard displays one number.
  5. For several risk factors x, use the gradient and Hessian form ΔV ≈ gᵀΔx + ½ × ΔxᵀHΔx. In a symmetric Hessian, the off-diagonal pair is counted twice inside the quadratic form, so its simplified contribution is Gamma_AB × ΔA × ΔB, not zero and not twice that amount again.
  6. Validate smooth-model Greeks with centered bumps: Delta_FD ≈ [V(S + ε) − V(S − ε)] / (2ε) and Gamma_FD ≈ [V(S + ε) − 2V(S) + V(S − ε)] / ε². Vary ε to reveal truncation, cancellation, grid, and surface-recalibration effects.
  7. Fully reprice every leg for finite scenarios and reconcile the residual. For trading, replace model marks with executable bid-ask prices and include fills, fees, hedge slippage, borrow, dividends, assignment, exercise, settlement cash flows, taxes, and post-shock Greeks.

Adding Vega × ΔIV or Theta × Δt is valid only after defining volatility and time units and the direction of the time derivative. It can still omit Vanna, Vomma, Charm, skew reshaping, cross terms, and higher orders. Splitting a large shock into steps helps only if the entire state and Greeks are refreshed; it remains an approximation and can become path-dependent.

Worked examples

  • One call versus full repricing. A European call has 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 Delta term is 2.8296615855, the Gamma term is 0.2765072125, and the estimate is ΔV_DG = 3.1061687980, so estimated value is 10.8277210283. Full same-IV repricing gives V_1 = 10.8172471731, change 3.0956949428, and residual full − DG = −0.0104738552 per share or −$1.047386 per M = 100. If endpoint IV is instead 30%, full value is 12.1628459929 and total change 4.4412937626; the original spot-only estimate omits 1.3351249646 of state change and interaction.
  • Signed short position. Six short calls have q = −6, M = 100, Delta_i = 0.55, and Gamma_i = 0.04, so D_pos = −330 shares and G_pos = −24 shares per $1. For ΔS = +2, the Delta and Gamma terms are −$660 and −$48, total −$708, and updated Delta is about −378 shares. For ΔS = −2, they are +$660 and −$48, total +$612, and updated Delta is about −282 shares. Negative Gamma makes the curvature contribution negative in both directions; it does not make total P/L symmetric.
  • Options plus stock hedge. Hold four calls with Delta_i = 0.35 and Gamma_i = 0.025, short three puts with Delta_i = −0.40 and Gamma_i = 0.030, use M = 100, and short 200 shares. Option Delta is +140 + 120 = +260 shares; net D_pos = +60 shares. Gamma is +10 − 9 = +1 share per $1. For ΔS = +5, the position estimate is 60 × 5 + ½ × 1 × 5² = +$312.50, and updated Delta is about +65 shares. Leg attribution is +$825.00 from the calls, +$487.50 from the puts, and −$1,000 from stock, which reconciles to +$312.50.
  • Two-factor cross-Gamma. Suppose ΔA = +2, ΔB = −3, Delta_A = 120, Delta_B = −80, Gamma_AA = 4, Gamma_BB = 2, and symmetric Gamma_AB = Gamma_BA = 1.5, all already in compatible position and currency units. The linear term is 120 × 2 + (−80) × (−3) = 480; diagonal curvature is ½ × 4 × 2² + ½ × 2 × (−3)² = 17; and the simplified cross term is 1.5 × 2 × (−3) = −9. Total is 488. Omitting cross-Gamma gives 497; multiplying the simplified cross term by two again gives 479.

Risks and controls

  • Wrong series, root, expiration, strike, option type, or deliverable invalidates the approximation.
  • Omitting the position sign reverses Delta, Gamma, hedge direction, and attribution.
  • Contract count and premium multiplier can be double-counted or omitted.
  • Adjusted contracts, futures options, and vendor-aggregated Greeks can use nonstandard units.
  • Spot, forward, futures, log-price, premium-adjusted, and cash coordinates are not interchangeable.
  • An absolute price shock differs from a percentage or standard-deviation shock.
  • Currency and FX conversion can make apparently comparable Greeks inconsistent.
  • Different timestamps or stale underlying, surface, and option marks create false netting.
  • Sticky-strike, sticky-delta, skew, and term-structure rules change the measured sensitivity.
  • Rates, dividends, borrow, financing, and exact expiry time affect values and Greeks.
  • Time and volatility terms can have opposite signs or factors of 100 under different vendor units.
  • American exercise and discrete dividends create boundaries absent from a smooth European formula.
  • Digital, barrier, and other discontinuous claims may have unstable or distribution-like Gamma.
  • Near expiration, Gamma and finite differences can become highly strike- and grid-dependent.
  • Large shocks expose third and higher orders that fixed starting Greeks omit.
  • Jumps, gaps, and halts bypass continuous local rebalancing assumptions.
  • Cross-Gamma, correlation, and factor mapping can be omitted or double-counted in multi-asset books.
  • Finite-difference bumps that are too large add truncation error; tiny bumps add cancellation noise.
  • Model marks and Greeks do not guarantee executable prices, size, or hedge liquidity.
  • Dynamic hedging, fees, impact, borrow, assignment, settlement, model versions, and broker records require final reconciliation.

Common misconceptions

  • “The Gamma term is Gamma_i × (ΔS)².” The Taylor coefficient is ½.
  • “Positive Gamma guarantees a profit in either direction.” It improves value relative to the tangent locally; Delta, premium, time, volatility, costs, and path still matter.
  • “Delta-neutral means risk-free.” It removes one current first-order term, not Gamma, Vega, Theta, jumps, basis, liquidity, or model risk.
  • “Adding any displayed Vega and Theta completes the P/L estimate.” Units, derivative direction, cross sensitivities, surface changes, and higher orders must also match.
  • “Precise Greeks or repeated small steps replace full repricing.” They improve attribution, but finite scenarios, execution, and lifecycle events still require complete valuation and reconciliation.

Authoritative sources

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