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Delta and Gamma: Direction, Curvature, and Position Risk

For educational purposes only; not investment advice.

Direct answer

Section titled “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.

Signs, scale, and position exposure

Section titled “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.

Worked price and hedge estimate

Section titled “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.

Risks and controls

Section titled “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.

Common misconceptions

Section titled “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.
Section titled “Related topics”

Authoritative sources

Section titled “Authoritative sources”