Gamma
The strike price is the only constant in the pricing model. When the stock
price moves relative to this constant, the option in question becomes more
in-the-money or out-of-the-money. This means the delta changes. This
isolated change is measured by the option’s gamma, sometimes called
curvature .
Gamma (Γ) is the rate of change of an option’s delta given a change in
the price of the underlying security . Gamma is conventionally stated in
terms of deltas per dollar move. The simplified examples above under
Definition 1 of delta, used to describe the effect of delta, had one important
piece of the puzzle missing: gamma. As the stock price moved higher in
those examples, the delta would not remain constant. It would change due
to the effect of gamma. The following example shows how the delta would
change given a 0.04 gamma attributed to the call option.
The call in this example starts as a 0.50-delta option. When the stock
price increases by $1, the delta increases by the amount of the gamma. In
this example, delta increases from 0.50 to 0.54, adding 0.04 deltas. As the
stock price continues to rise, the delta continues to move higher. At $62, the
call’s delta is 0.58.
This increase in delta will affect the value of the call. When the stock
price first begins to rise from $60, the option value is increasing at a rate of
50 percent—the call’s delta at that stock price. But by the time the stock is
at $61, the option value is increasing at a rate of 54 percent of the stock
price. To estimate the theoretical value of the call at $61, we must first
estimate the average change in the delta between $60 and $61. The average
delta between $60 and $61 is roughly 0.52. It’s difficult to calculate the
average delta exactly because gamma is not constant; this is discussed in
more detail later in the chapter. A more realistic example of call values in
relation to the stock price would be as follows: