Options Delta and Gamma: Why Risk Changes Faster Near Expiry

Delta and gamma in plain English

Options risk changes as the underlying price moves. Delta estimates how much an option's price may change for a small move in the underlying. Gamma estimates how much that delta may change when the underlying moves again.

Learning objective: use delta to estimate immediate directional exposure, use gamma to understand how that exposure can accelerate, and recognise why near-expiry options require tighter risk control.

বাংলায় পড়ুন: Options Delta ও Gamma: Expiry কাছে Risk কেন দ্রুত বদলায়

1. What delta measures

A call option usually has delta between 0 and 1. A put normally has delta between -1 and 0. If a call has delta 0.40, a £1 rise in the underlying may add roughly £0.40 to the option price, all else equal.

That estimate is local, not permanent. Time, implied volatility, interest rates and the size of the price move also affect the option. Delta itself changes continuously.

For standard equity options with a contract multiplier of 100, one call with delta 0.40 has roughly the same immediate directional exposure as 40 shares:

Share-equivalent exposure = delta × contract multiplier × number of contracts

This does not mean the option will copy 40 shares over a large move. Gamma causes the exposure to change.

2. What gamma measures

Gamma is the rate of change of delta for a one-unit move in the underlying. If the call's delta is 0.40 and gamma is 0.05, a £1 rise might increase delta to approximately 0.45. A £1 fall might reduce it to about 0.35, before accounting for other changing inputs.

Long options generally have positive gamma. As price rises, a long call's delta becomes more positive; as price falls, its delta becomes less positive. Short options have negative gamma, so directional exposure can move against the seller as the underlying moves.

Gamma tends to be highest for at-the-money options near expiry. That is why a position that looked modest in the morning can develop much larger directional sensitivity after a sharp price move.

3. A worked example

Suppose one share trades at £50. A trader owns two call contracts with:

• delta 0.35;
• gamma 0.06; and
• contract multiplier 100.

Initial share-equivalent exposure is:

0.35 × 100 × 2 = 70 shares

If the share rises by £1, the approximate new delta is:

0.35 + (0.06 × £1) = 0.41

The new share-equivalent exposure becomes approximately:

0.41 × 100 × 2 = 82 shares

The position gained about 12 share-equivalents of directional exposure because of gamma. This is a simplified estimate: gamma, volatility and time decay also change during the move.

4. Why expiry changes the risk

Near expiry, there is less time for an at-the-money option to finish on either side of the strike. A small underlying move can sharply change the probability of exercise, so delta may swing quickly from near zero towards one for a call, or towards minus one for a put.

This creates pin risk around the strike and makes hedging more demanding. A short-gamma trader may need to buy as price rises and sell as it falls to keep delta controlled—the opposite of the preferred execution direction.

Long-gamma positions can benefit from large realised moves, but they usually pay for that convexity through option premium and time decay. Positive gamma is not free profit.

5. Use delta and gamma in position planning

1. Record delta and gamma before entry.
2. Convert delta into share-equivalent or notional exposure.
3. Stress the position for several underlying moves, not just £1.
4. Recalculate after a large move, volatility change or passage of time.
5. Check how close the option is to expiry and the strike.
6. Set a maximum loss and exit rule that account for gaps.

A portfolio can look delta-neutral at one price and become directional after a move. Delta neutrality is a point-in-time estimate, not permanent protection.

6. Delta is not exercise probability

Some traders use call delta as a rough shorthand for the probability of expiring in the money. The two can be numerically similar under particular assumptions, but they are not the same concept. Delta is a sensitivity derived from an option-pricing model. Exercise probability depends on the model, distribution assumptions and whether the measure is risk-neutral or real-world.

Use the correct probability output when the decision requires probability, rather than silently replacing it with delta.

Common mistakes

1. Treating delta as fixed. Gamma, time and volatility change it.
2. Ignoring the contract multiplier. Small-looking Greeks can represent large exposure.
3. Assuming delta-neutral means risk-free. Gamma, vega, theta, gaps and basis risk remain.
4. Selling near-expiry gamma without a gap plan. Exposure can change faster than hedges can be placed.
5. Using a linear estimate for a large move. Delta approximation becomes less reliable as the move grows.
6. Forgetting liquidity. Wide spreads can make repeated hedging expensive.

Practical checklist

• Confirm contract multiplier and expiry.
• Record delta and gamma per contract.
• Calculate total share-equivalent exposure.
• Stress several up and down moves.
• Review implied volatility, theta and spread alongside the Greeks.
• Recalculate after material price changes.
• Plan for overnight gaps and partial fills.
• Define maximum loss before entry.

Key takeaway

Delta describes immediate directional sensitivity; gamma describes how quickly that sensitivity changes. Near expiry and near the strike, gamma can make exposure accelerate. Use both Greeks with contract size, liquidity, volatility and a complete loss plan.

Educational information only. Options are complex and can lose the entire premium; some short-option strategies can create losses beyond the initial credit. This is not personalised advice or a recommendation to trade.