# Delta-Gamma Approximation: Estimating Nonlinear Option P/L

Use Delta and Gamma to estimate option value changes, scale portfolio Greeks correctly, and understand when a local second-order approximation breaks down.

Canonical: https://wiki.fcontext.com/options/delta-gamma-approximation/
Fact checked: 2026-07-22

> For educational purposes only; not investment advice.

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## 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.

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## 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.

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## 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.

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## 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.

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## 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.

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## Related topics

- [Delta and Gamma](/options/delta-gamma/)
- [Net Greeks](/options/net-greeks/)
- [Gamma Risk](/options/gamma-risk/)
- [Vega](/options/vega/)

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## Authoritative sources

- [Understanding Options Greeks](https://www.optionseducation.org/advancedconcepts/understanding-options-greeks) - Options Industry Council
- [Volatility and the Greeks](https://www.optionseducation.org/advancedconcepts/volatility-the-greeks) - Options Industry Council
- [Options Calculator and Greeks Tools](https://www.cboe.com/oi/tools/) - Cboe Options Institute
- [Value at Risk for Linear and Non-Linear Derivatives](https://openscholar.uga.edu/record/7803) - University of Georgia