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Delta-Gamma Approximation: Estimating Nonlinear Option P/L

For educational purposes only; not investment advice.

Direct answer

Section titled “Direct answer”

The Delta-Gamma approximation estimates an option or portfolio value change from a move in the underlying while temporarily holding time, implied volatility, rates, and dividends constant:

ΔV ≈ Delta × ΔS + 0.5 × Gamma × (ΔS)²

Delta supplies the linear, directional term. Gamma supplies the curvature correction because Delta itself changes as spot moves. This is a local second-order Taylor approximation, useful for quick small-move scenarios and risk checks—not a replacement for repricing the options.

Why the Gamma term matters

Section titled “Why the Gamma term matters”

Delta is the first derivative of option value with respect to spot; Gamma is the second derivative, or the rate at which Delta changes. The factor 0.5 comes from the second-order Taylor term and must not be omitted.

Because (ΔS)² is positive for both an up and down move, positive Gamma adds a positive curvature term in both directions; negative Gamma subtracts it. That does not mean a long option profits from every move: the Delta term, premium paid, Theta, Vega, and trading costs still matter.

For a portfolio, first scale each option’s per-share Greeks by position sign, contract count, and multiplier, then add them:

Portfolio Delta = Σ(position sign × contracts × multiplier × Delta_i)

Portfolio Gamma = Σ(position sign × contracts × multiplier × Gamma_i)

Standard U.S. equity options commonly use a 100 multiplier. Stock contributes Delta of one per share and Gamma of zero. Multi-asset portfolios require a separate price shock for each risk factor and may need cross-Gamma terms; one common ΔS cannot be applied to unrelated underlyings.

A call with Delta 0.45 and Gamma 0.08

Section titled “A call with Delta 0.45 and Gamma 0.08”

Suppose a call is worth 5.00, Delta is 0.45, Gamma is 0.08, and the underlying moves from 100 to 103, so ΔS=3.

  • Delta term: 0.45×3=1.35
  • Gamma term: 0.5×0.08×3²=0.36
  • Estimated change: 1.35+0.36=1.71
  • Estimated option value: 5.00+1.71=6.71

Delta alone would estimate 6.35, understating the positive curvature by 0.36. If spot instead falls from 100 to 97, ΔS=−3; the Delta term is −1.35, while the Gamma term remains +0.36. Estimated change is −0.99, giving an estimated value of 4.01.

For 10 long contracts with a 100 multiplier, multiply the per-share change by 1,000: the estimated up-move gain is $1,710. For 10 short contracts, both position Delta and Gamma reverse sign, so the same up move estimates a $1,710 loss before other effects.

Proper-use checklist

Section titled “Proper-use checklist”
  • Confirm whether Greeks are per share, per contract, or already portfolio-scaled before applying the multiplier.
  • Use the signed position Greeks. Long standard calls and puts generally have positive Gamma; short positions have negative Gamma.
  • Test both positive and negative price moves. A one-sided scenario hides curvature and directional asymmetry.
  • Keep the shock local. Gamma changes with spot, time, and volatility, especially near the money and close to expiration.
  • For a large move, split the path into smaller steps and refresh Greeks, or fully reprice each contract under the new scenario.
  • Add Vega×ΔIV and Theta×Δt only after confirming units. Vega may be quoted per one volatility point or per 100% change; Theta may be daily or annual.
  • Include volatility-surface reshaping, dividends, rates, jumps, liquidity, and transaction costs in broader stress tests.
  • Compare the approximation with independent full repricing and record the residual; a larger residual signals model, unit, or omitted-factor risk.

Common misconceptions

Section titled “Common misconceptions”
  • “The Gamma correction is Gamma×(ΔS)².” Taylor expansion requires the 0.5 coefficient.
  • “A positive-Gamma position profits in both directions.” Curvature helps relative to the tangent, but total P/L includes Delta, Theta, Vega, premium, and costs.
  • “Delta-neutral means price-neutral.” Delta is neutral only at the current point; Gamma recreates Delta after spot moves.
  • “The approximation is a forecast.” It is conditional arithmetic using current model sensitivities, not a probability or target price.
  • “A bigger shock gives a more informative estimate.” Fixed starting Gamma becomes less reliable as the move travels through different moneyness regions.
  • “Greeks from different systems can be combined directly.” Models, timestamps, volatility inputs, dividends, and units may differ.
Section titled “Related topics”

Authoritative sources

Section titled “Authoritative sources”