Gamma

The option's gamma is a measure of the rate of change of its
delta. The gamma of an option is expressed as a percentage and reflects the change in
the delta in response to a one point movement of the underlying stock price.

Like the delta, the gamma is constantly changing, even with tiny movements of the
underlying stock price. It generally is at its peak value when the stock price is
near the strike price of the option and decreases as the option goes deeper into
or out of the money. Options that are very deeply
into or out of the money have gamma
values close to 0.

Example

Suppose for a stock XYZ, currently trading at $47, there is a FEB 50 call option
selling for $2 and let's assume it has a delta of 0.4 and a gamma of 0.1 or 10 percent.
If the stock price moves up by $1 to $48, then the delta will be adjusted upwards
by 10 percent from 0.4 to 0.5.

However, if the stock trades downwards by $1 to $46, then the delta will decrease
by 10 percent to 0.3.

Passage of time and its effects on the gamma

The chart above depicts the behaviour of the gamma of options at various strikes expiring
in 3 months, 6 months and 9 months when the stock is currently trading at $50.

Changes in volatility and its effects
on the gamma

When volatility is low, the gamma of
at-the-money options is high while the gamma for
deeply into or out-of-the-money options approaches 0. This phenomenon arises because
when volatility is low, the
time value of such options are low but it goes up dramatically
as the underlying stock price approaches the
strike price.

When volatility is high, gamma tends to be stable across all strike prices.
This is due to the fact that when volatility is high, the time value of deeply in/out-of-the-money
options are already quite substantial. Thus, the increase in the time value of these
options as they go nearer the money will be less dramatic and hence the low and
stable gamma.

The chart above illustrates the relationship between the option's gamma and the
volatility of the underlying security which is trading at $50 a share.

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