# Delta and Gamma: Measuring Direction and Changing Exposure
Use Delta and Gamma to estimate option price changes, convert contracts to share-equivalent exposure, understand hedge drift, and manage near-expiration convexity risk.
Canonical: https://wiki.fcontext.com/options/delta-gamma/
Fact checked: 2026-07-22
> For educational purposes only; not investment advice.
## Direct answer
**Delta** estimates how much an option's value changes for a small change in the underlying price, holding other model inputs approximately constant. **Gamma** estimates how much Delta changes for a one-unit move in the underlying.
In calculus notation:
`Delta = ∂ option value / ∂ underlying price`
`Gamma = ∂ Delta / ∂ underlying price = ∂² option value / ∂ underlying price²`
Delta is the current slope; Gamma measures curvature. Delta is therefore not fixed. A portfolio that is Delta neutral now can become directionally exposed after the underlying moves.
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Delta describes the tangent at a point; Gamma describes how Delta changes as the underlying moves.
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## Signs, units, and position exposure
For ordinary long options, Call Delta is generally between `0` and `+1`, while Put Delta is between `-1` and `0`. Long Calls and long Puts normally have positive Gamma. Selling either option reverses both Delta and Gamma signs.
| Position | Delta sign | Gamma sign | Basic interpretation |
| --- | ---: | ---: | --- |
| Long Call | Positive | Positive | Gains upward exposure as price rises |
| Short Call | Negative | Negative | Loses more rapidly as price rises |
| Long Put | Negative | Positive | Becomes less negative as price rises |
| Short Put | Positive | Negative | Gains downside exposure as price falls |
Convert per-share Greeks to position units:
`position Delta in shares = option Delta × multiplier × contracts`
`position Gamma in shares per $1 = option Gamma × multiplier × contracts`
Ten standard Calls with Delta `0.50` have about `0.50 × 100 × 10 = 500` share-equivalent Delta. If Gamma is `0.04`, position Gamma is `0.04 × 100 × 10 = 40` share-equivalent Delta units for each `$1` underlying move.
This is a local model estimate, not actual stock ownership. Delta and Gamma change with price, time, implied volatility, rates, dividends, and model conventions.
### Small-move approximation
For an underlying move `ΔS`, holding other inputs constant:
`option value change ≈ Delta × ΔS + 0.5 × Gamma × (ΔS)²`
`new Delta ≈ old Delta + Gamma × ΔS`
The Gamma term improves on a linear Delta estimate by adding curvature. The approximation becomes less reliable for large jumps, high Gamma, changing IV, elapsed time, or discontinuous markets; reprice the option for material scenarios.
### Why Gamma concentrates near the strike and expiration
Gamma tends to be highest near at-the-money because a small price move most strongly changes whether the option may finish in or out of the money. As expiration approaches, that transition occurs over less time and Delta can move rapidly toward its expiration endpoints. Deep ITM and deep OTM options usually have lower Gamma.
Long Gamma provides favorable curvature but usually comes with negative Theta: the holder pays for convexity through time decay. Short Gamma often receives Theta but can require increasingly adverse rehedging during a large move. Neither side receives a free advantage.
## Example 1: one contract price approximation
A Call is worth `$4.00`, with Delta `0.50`, Gamma `0.04`, and multiplier100. If the stock rises `$2`, ignoring changes in time and IV:
`price change ≈ 0.50 × 2 + 0.5 × 0.04 × 2² = $1.08 per share`
`contract change ≈ $1.08 × 100 = $108`
`new Delta ≈ 0.50 + 0.04 × 2 = 0.58`
The linear Delta-only estimate was `$1.00`; Gamma adds `$0.08`. If the stock falls `$2`, the same local calculation gives `0.50 × (-2) + 0.5 × 0.04 × 4 = -$0.92`, and new Delta about `0.42`. Positive Gamma makes the upward gain larger than the equal downward loss in this frozen-input approximation.
## Example 2: Delta-neutral hedge drift
An account owns 10 Calls, each Delta `0.50` and Gamma `0.04`. It shorts 500 shares to offset the initial `+500` option Delta.
If the stock rises `$3`:
`estimated new option Delta = (0.50 + 0.04 × 3) × 100 × 10 = +620 shares`
With the existing `-500` stock hedge, net Delta becomes about `+120`; returning to neutral requires selling about 120 more shares.
If the stock instead falls `$3`, estimated option Delta becomes `+380` shares. Net Delta becomes `-120`, so returning to neutral requires buying about 120 shares. A long-Gamma rehedging pattern sells after rises and buys after falls, but realized benefit must exceed Theta, spreads, commissions, slippage, IV changes, jumps, and imperfect rehedging.
## Portfolio workflow
- Verify whether displayed Greeks are per share or per contract.
- Multiply each leg by sign, multiplier, ratio, and quantity.
- Aggregate Delta and Gamma only across compatible underlying units.
- Convert index or ETF hedges by notional, beta, and basis rather than raw share count.
- Calculate exposure after several upward and downward moves, not only at spot.
- Include Theta and Vega scenarios; Delta-Gamma is not a full revaluation.
- Short-Gamma positions need gap and near-expiration stress, not just one-dollar moves.
- Set rehedging thresholds by risk and cost rather than assuming continuous trading.
- Recalculate after time passes, IV changes, dividends update, or contracts are rolled.
- Compare approximations with independent full repricing for material decisions.
## Risks and boundaries
- **Local approximation:** Greeks describe small changes near current inputs.
- **Gamma drift:** Delta can change quickly, especially ATM near expiration.
- **Jump risk:** markets can cross many hedge thresholds before execution.
- **IV and skew movement:** volatility changes can overwhelm the frozen-IV estimate.
- **Theta-Gamma trade-off:** long convexity normally costs time value.
- **Short-Gamma loss:** adverse rehedging can compound during trends and gaps.
- **Unit error:** per-share, per-point, per-contract, and portfolio Greeks differ.
- **Multiplier error:** adjusted and index contracts may not use standard units.
- **Basis risk:** an ETF, futures, or stock hedge can differ from the option reference.
- **Liquidity and costs:** theoretical hedge adjustments may not be executable economically.
- **Model risk:** platforms can use different rates, dividends, surfaces, and conventions.
## Common misconceptions
- “Delta is fixed until expiration.” Gamma and changing inputs continually alter it.
- “Delta is exact shares owned.” It is a local sensitivity expressed in share-equivalent units.
- “A `0.50` Delta guarantees a `$0.50` move.” Other variables change and the estimate is local.
- “Delta is always the probability of expiring ITM.” Some models relate quantities, but Delta is primarily price sensitivity.
- “Gamma directly states option-dollar P&L.” It states the rate of Delta change and needs move, multiplier, and quantity.
- “Only Calls have positive Gamma.” Long Calls and long Puts normally do.
- “Positive Gamma guarantees profit.” Premium, Theta, IV, and execution determine results.
- “Delta neutral means risk free.” Gamma, Vega, Theta, jumps, and basis remain.
- “Short Gamma is safe when current Delta is small.” A move can rapidly enlarge Delta.
- “Greeks from different platforms can always be added.” Units and model assumptions must match.
## Related topics
- [Delta and Gamma basics](/options/delta-and-gamma/)
- [Delta-Gamma approximation](/options/delta-gamma-approximation/)
- [Gamma risk](/options/gamma-risk/)
- [Delta hedging](/options/delta-hedging/)
## Primary sources
- [OCC: Characteristics and Risks of Standardized Options](https://www.theocc.com/company-information/documents-and-archives/options-disclosure-document)
- [Cboe Options Institute: Options Basics](https://www.cboe.com/optionsinstitute/options_basics/)
- [Black and Scholes (1973): The Pricing of Options and Corporate Liabilities](https://doi.org/10.1086/260062)
- [Merton (1973): Theory of Rational Option Pricing](https://doi.org/10.2307/3003143)
