Options Delta and Gamma

Delta: first-order sensitivity

For one unit of underlying exposure, a call delta is commonly positive and a put delta negative under standard conventions. A delta of 0.40 means a small £1 rise in the underlying is associated with roughly a £0.40 rise in option value, all else equal and locally.

Delta can also be interpreted in hedging terms: an option position with +0.40 delta has roughly the local directional sensitivity of +0.40 units of underlying, subject to multiplier and contract conventions.

Gamma: how delta bends

Gamma measures the change in delta for a change in the underlying. Long vanilla calls and puts generally have positive gamma; short options have negative gamma.

• Positive gamma means directional exposure tends to increase as a long option moves favourably and decrease as it moves adversely.
• Negative gamma means the opposite curvature for a short option and can force increasingly adverse hedging during large moves.
• Gamma is often highest for near-the-money options close to expiry, though exact shape depends on model inputs.

Why delta hedging is dynamic

A trader who delta-hedges offsets an option’s current directional sensitivity with spot/futures. But gamma means the hedge ratio changes when spot changes. Volatility and time also alter Greeks.

Transaction costs, spreads, slippage and discrete hedge intervals mean theoretical hedging behaviour is not frictionless.

Worked example

A hypothetical call has delta = 0.40 and gamma = 0.00002 per £1. Spot is £80,000.

For an initial +£500 spot move, a first-order delta approximation suggests about +£200 of option value change (0.40 × £500), before other Greeks. Gamma suggests delta itself rises by roughly 0.00002 × 500 = 0.01, toward 0.41.

For a much larger move, using the original 0.40 delta for the entire path becomes increasingly inaccurate because delta is changing. That is exactly what gamma is telling you.

Common mistakes and misunderstandings

• Treating delta as a fixed probability rather than primarily a sensitivity measure.
• Using one delta estimate for a very large underlying move.
• Ignoring contract multiplier when converting Greeks into portfolio exposure.
• Assuming delta hedging removes volatility, gamma, liquidity or gap risk.
• Comparing gamma numbers from analytics systems with different unit conventions.

Knowledge checkpoint

Q1. Why does a +0.40 delta option not remain equivalent to +0.40 spot units after a large price move?

Q2. What does positive gamma do to a long option’s delta as spot rises?

Q3. Why can short gamma be dangerous in a fast market?

Q4. Why is delta × price move only a local approximation?

FAQ

Summary

• Delta is first-order directional sensitivity; gamma is the curvature of that sensitivity.
• Greeks are dynamic and model-dependent.
• Near-expiry gamma can make exposure change very quickly.
• Hedging must include multiplier, transaction costs and discrete execution.

Use this lesson as one component of a wider risk and execution process. Derivative specifications, margin formulas, settlement and loss-allocation rules can differ substantially between venues.