# Delta and Gamma: Direction, Curvature, and Position Risk

Use Delta and Gamma as local option-model sensitivities, calculate first- and second-order price estimates, and understand changing directional exposure.

Canonical: https://wiki.fcontext.com/options/delta-and-gamma/
Fact checked: 2026-07-13

> For educational purposes only; not investment advice.

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## Direct answer

**Delta** estimates how much an option's theoretical value changes for a small `$1` change in the underlying, holding other model inputs constant. **Gamma** estimates how much Delta changes for that same underlying move. Delta is the local slope of the option-value curve; Gamma measures its curvature.

They are model outputs at a particular underlying price, time, volatility surface, rate, and dividend assumption. They change as those inputs change and do not guarantee a quote, fill, hedge result, or probability.

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## Signs, scale, and position exposure

For a long standard equity call, Delta is normally between `0` and `+1`; for a long put, it is normally between `-1` and `0`. Selling reverses the position Delta. A 100-share contract with Delta `0.42` has approximately `+42` effective shares of local directional exposure when long; shorting it produces about `-42` effective shares. This is a local hedge ratio, not ownership of 42 shares.

Long plain-vanilla calls and puts generally have positive Gamma; short positions have negative Gamma. With positive Gamma, Delta becomes more positive as the stock rises and less positive—or more negative for a put—as it falls. Negative Gamma does the reverse, making adverse directional exposure grow as price moves.

For a small underlying change `ΔS`, a second-order approximation is:

`option-value change ≈ Delta × ΔS + ½ × Gamma × (ΔS)²`

and the updated Delta is roughly:

`new Delta ≈ old Delta + Gamma × ΔS`

Values are normally quoted per share. Multiply by the contract multiplier and number of contracts, then net all legs and any stock. A stock position has constant Delta of `+1` or `-1` per share and zero Gamma.

Gamma is often concentrated near at-the-money as expiration approaches. That does not mean every near-expiry option has high Gamma: deep in-the-money and far out-of-the-money contracts may have little remaining Delta transition. Higher implied volatility can also spread the transition across a wider price range.

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## Worked price and hedge estimate

Assume a long call has theoretical value `$3.10`, Delta `0.42`, Gamma `0.06`, and multiplier 100. The stock rises `$1.50`, while time, IV, rates, and dividends are assumed unchanged.

The Delta-only estimate is:

`0.42 × $1.50 = $0.6300` per share

The Gamma adjustment is:

`½ × 0.06 × ($1.50)² = $0.0675` per share

Together, the estimated option change is `$0.6975` per share, or `$69.75` per contract. Its estimated theoretical value becomes `$3.7975`, and its new Delta is:

`0.42 + 0.06 × 1.50 = 0.51`

For a `$1.50` decline, the same local approximation gives `-$0.5625` per share because the positive Gamma term remains `+$0.0675`; estimated Delta becomes `0.33`. This asymmetry illustrates convexity, but real repricing will differ as Gamma, IV, time, and the quote change.

At the starting point, one contract's `+42` effective-share Delta could be locally offset by selling about 42 shares. After the rise, the estimated Delta is `+51` shares, so the old hedge is no longer neutral. Rebalancing creates trading costs and gap risk; “Delta neutral” is a momentary estimate, not a permanent state.

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## Risks and controls

- **Local approximation:** large moves require full repricing because Delta and Gamma themselves change.
- **Input interaction:** passage of time and IV changes can move Delta and Gamma even when the stock is unchanged.
- **Short-Gamma risk:** adverse exposure can accelerate, especially near expiration and around gaps.
- **Hedge risk:** discrete rebalancing, spreads, commissions, stock borrow, and overnight jumps prevent perfect continuous hedging.
- **Quote risk:** Greeks based on a midpoint or stale IV need not describe executable prices.
- **Portfolio risk:** leg-level Greeks must be multiplied, signed, and netted with other options and stock.
- **Expiration risk:** Delta can move rapidly around the strike, while exercise, assignment, and after-hours movement create exposures not summarized by a displayed Greek.

Stress-test stock moves in both directions, time passage, and IV changes. Recalculate Greeks at each scenario rather than stretching one set of local values across a large range.

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## Common misconceptions

- **“Delta `0.42` means a guaranteed 42% chance of expiring in the money.”** Delta may be used as a rough probability proxy under particular models, but it is not a universal probability statement.
- **“Delta stays fixed.”** Gamma, time, and volatility make it change.
- **“A Delta-neutral portfolio has no risk.”** It retains Gamma, Vega, Theta, jump, basis, liquidity, and model risk.
- **“Gamma directly predicts the option-price change.”** Gamma first describes how Delta changes; it enters price approximation with the `½ × Gamma × (ΔS)²` term.
- **“High Gamma is always beneficial.”** It helps long-Gamma positions from movement but accompanies premium/time costs; it can be dangerous when short.
- **“Greeks from two platforms must match.”** Quotes, models, dividends, rates, and timestamps can differ.

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## Related topics

- [Theta](/options/theta/)
- [Vega](/options/vega/)

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## Authoritative sources

- [Understanding Options Greeks](https://www.optionseducation.org/advancedconcepts/understanding-options-greeks) — Options Industry Council (2026-07-13)
- [Gamma](https://prd-web.optionseducation.org/advancedconcepts/gamma) — Options Industry Council (2026-07-13)
- [Characteristics and Risks of Standardized Options](https://www.theocc.com/company-information/documents-and-archives/options-disclosure-document) — OCC (2026-07-13)