Each $1 increase in the stock shows an increase in the call value about
equal to the average delta value between the two stock prices. If the stock
were to decline, the delta would get smaller at a decreasing rate.
As the stock price declines from $60 to $59, the option delta decreases
from 0.50 to 0.46. There is an average delta of about 0.48 between the two
stock prices. At $59 the new theoretical value of the call is 2.52. The
gamma continues to affect the option’s delta and thereby its theoretical
value as the stock continues its decline to $58 and beyond.
Puts work the same way, but because they have a negative delta, when
there is a positive stock-price movement the gamma makes the put delta
less negative, moving closer to 0. The following example clarifies this.
As the stock price rises, this put moves more and more out-of-the-money.
Its theoretical value is decreasing by the rate of the changing delta. At $60,
the delta is −0.40. As the stock rises to $61, the delta changes to −0.36. The
average delta during that move is about −0.38, which is reflected in the
change in the value of the put.
If the stock price declines and the put moves more toward being in-the-
money, the delta becomes more negative—that is, the put acts more like a
short stock position.
Here, the put value rises by the average delta value between each
incremental change in the stock price.
These examples illustrate the effect of gamma on an option without
discussing the impact on the trader’s position. When traders buy options,
they acquire positive gamma. Since gamma causes options to gain value at