Option Greek Gamma

Gamma is closely related to delta – both measure an option’s sensitivity to underlying price, although each
in a different way.
While delta indicates how much option premium will change if underlying price increases by $1, gamma
measures how much the delta itself will change if underlying price increases by $1.
While delta is the speed of option price change, gamma is the acceleration.

Consider a $35 strike call option on a stock that is currently trading at $35 (the option is at the money). With 20 days to expiration, implied volatility of 30% and interest rate at 2.50%, the option’s premium is $1.00, delta is 0.52 and gamma is 0.16.
One characteristic of delta is that it is not constant.
As underlying price changes, not only the option premium will change, but also the delta.
In our example, the delta was 0.52 when the stock was at $35, but it gradually increased as the stock was going up.
With the stock at $35.50, the delta was already 0.60.
With the stock at $36, the delta got to 0.68. As the underlying price increased by $1 from $35 to $36, the option’s delta increased by 0.16 from 0.52 to 0.68. This is the gamma of 0.16.

Gamma is highest (delta changes fastest) when an option is near or at the money.
When Underlying price close to the option’s strike price, delta is close to the middle of its possible range (near 0.50 for calls or -0.50 for puts) and even a small change in underlying price can cause a significant
change in delta.
Gamma is close to zero for far out of the money options.
Gamma is close to zero for deep in the money options.
When you draw a chart of gamma with underlying price in the X-axis, it often looks like the familiar bell curve: it peaks around the middle (at the money) and approaches zero on both ends (out of the money, in
the money).

One interpretation of Gamma is that its absolute value indicates the approximate probability of the option expiring in the money.
For instance, a deep in the money call option ($30 strike with underlying price at $40) is under normal circumstances (I am using 30% volatility and 50 days to expiration) almost certain to expire in the money, as its Gamma of 0.996 also suggests.
A $35 strike call on the same underlying is still very likely to expire in the money, although slightly less likely that the $30 strike call. It has Gamma of 0.90, indicating a 90% probability.
On the contrary, a $50 strike call is far out of the money, has Gamma of 0.025, and is unlikely to expire in the money (the underlying would have to increase by more than 25% from its current level).
It works the same with puts – you just need to ignore the minus sign. A deep in the money put with Gamma of -0.95 has approximately 95% probability of expiring in the money.

Gamma is also affected by passing time.
As expiration nears, gamma of at-the-money options increases and the bell-curve-shaped chart of gamma becomes more peaked.
If we think of gamma as a measure of option’s instability, it is no surprise that those options which are at the money and with very little time to expiration are the mostinstable, with highest gamma.
Conversely, as expiration approaches, both out-of-the-money and
in-the-money options lose gamma. Both ends of the bell curve are pushed even closer to zero.

Volatility affects gamma quite similarly as time.
Higher volatility is like more time to expiration; lower volatility is like less time.
Rising volatility increases out of the money and in the money gamma, while at the money gamma falls.
Decreasing volatility increases at the money gamma, while out of the money and in the money gamma decline.

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