Derivatives and advanced market conceptsModule 3, lesson 14 of 21
Options Greeks: delta and gamma
Delta as an option's sensitivity to the underlying and a rough probability, gamma as the change in delta, and why gamma risk grows sharply near expiry.
In this lesson
- Explain delta as an option's sensitivity to moves in the underlying.
- Convert an option position's delta into an equivalent number of shares.
- Explain gamma as the rate at which delta changes, and estimate an option's price change using both.
- Explain why gamma risk grows near expiry, especially for option sellers.
Builds on
- Historical and implied volatilityDelta and gamma depend on volatility and time, explained there.
On this page
In simple terms
Before expiry, an option's price moves when the share price moves, but not one for one. The Greeks are measures of how an option's price responds to changes in its inputs. This lesson covers the two that describe price sensitivity: delta and gamma. The time decay and vega lesson covers the others.
Delta
Delta is roughly the change in an option's price for a ₹1 change in the underlying.
| Option | Typical delta |
|---|---|
| Deep in-the-money call | close to 1 |
| At-the-money call | about 0.5 |
| Far out-of-the-money call | close to 0 |
| At-the-money put | about −0.5 |
| Deep in-the-money put | close to −1 |
A put's delta is negative because it gains when the price falls.
Delta is also often read as a rough indication of the chance that the option ends in the money. A delta of 0.2 suggests roughly a one-in-five chance. This is an approximation, not a precise probability.
Share-equivalent exposure
Multiplying delta by the number of shares in the position gives its share-equivalent exposure.
| Call delta | 0.5 |
|---|---|
| Lot size (made up) | 500 shares |
| Share-equivalent: 0.5 × 500 | like holding 250 shares |
| Gain if the share rises ₹10: 0.5 × 10 × 500 | about ₹2,500 |
| Loss if the share falls ₹10 | about ₹2,500 |
This is a quick way to see how large an options position really is, and is the basis for hedging, as the hedging lesson explains.
Gamma: delta does not stay still
As the share price moves, delta changes. Gamma measures how fast: the change in delta for a ₹1 move in the underlying.
Worked example: A made-up call with delta 0.5 and gamma 0.01
| Share move | New delta, roughly | Approximate option price change |
|---|---|---|
| +₹20 | 0.7 | 0.5 × 20 + ½ × 0.01 × 20² = +₹12 |
| −₹20 | 0.3 | 0.5 × (−20) + ½ × 0.01 × 20² = −₹8 |
For an option buyer, gamma helps: gains accelerate as the price moves their way and losses slow as it moves against them. For an option seller, gamma hurts: losses accelerate as the price moves against them.
Gamma near expiry
Gamma is highest for at-the-money options close to expiry. In the final days, an at-the-money option's delta can swing from near 0 to near 1 with a small move, because it is nearly all about which side of the strike the price ends on, as the expiry risks lesson explains.
For sellers of short-dated, at-the-money options, this means:
- Small price moves cause large, fast changes in profit and loss.
- Hedges based on delta need constant adjustment, often at bad prices.
- Losses can grow faster than expected from the delta alone.
Why it matters
Delta tells you how large an option position really is; gamma tells you how quickly that size changes. Together, they explain why option positions can behave very differently from what a simple payoff diagram suggests, especially near expiry.
Limitations
Delta and gamma come from pricing models and change continuously with price, time and volatility. The approximations work for small moves; large moves and gaps can produce very different results. Values shown by brokers differ with model assumptions.
Common mistakes
- Treating delta as fixed as the price moves.
- Ignoring position size in share-equivalent terms.
- Selling short-dated at-the-money options without understanding gamma.
- Reading delta as an exact probability.
Try it yourself
- For made-up calls with deltas of 0.2, 0.5 and 0.8 on a lot of 500, calculate each share-equivalent exposure.
- For the 0.5-delta call with gamma 0.01, estimate the option price change for share moves of +₹10 and −₹10.
- Explain why gamma is good news for an option buyer and bad news for a seller.
Key takeaways
- Delta is roughly how much an option's price moves for a ₹1 move in the underlying. Calls have delta between 0 and 1; puts between −1 and 0.
- At-the-money options have a delta near 0.5 (or −0.5 for puts); deep in-the-money options near 1; far out-of-the-money options near 0.
- Position delta times lot size gives a share-equivalent exposure, a quick measure of how large a position really is.
- Gamma is how much delta changes as the price moves. It makes option values change faster than delta alone suggests.
- Near expiry, at-the-money options have very high gamma. Small moves can swing values and deltas sharply, a serious risk for sellers.
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Terms used in this lesson
- Delta
- Roughly how much an option's price changes for a ₹1 change in the underlying. Calls have delta from 0 to 1, puts from −1 to 0.
- Gamma
- How much an option's delta changes for a ₹1 move in the underlying.
- Call option
- A contract giving the buyer the right, but not the obligation, to buy the underlying at the strike price.
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This lesson is general education, not investment advice. Examples use made-up or clearly dated figures to explain ideas; they are not recommendations. Read the disclaimer.