a faster rate and lose value at a slower rate, (positive) gamma helps the
option buyer. A trader buying one call or put in these examples would have
+0.04 gamma. Buying 10 of these options would give the trader a +0.4
gamma.
When traders sell options, gamma works against them. When options lose
value, they move toward zero at a slower rate. When the underlying moves
adversely, gamma speeds up losses. Selling options yields a negative
gamma position. A trader selling one of the above calls or puts would have
−0.04 gamma per option.
The effect of gamma is less significant for small moves in the underlying
than it is for bigger moves. On proportionately large moves, the delta can
change quite a bit, making a big difference in the position’s P&(L). In
Exhibit 2.1 , the left side of the diagram showed the call price not
increasing at all with advances in the stock—a 0 delta. The right side
showed the option advancing in price 1-to-1 with the stock—a 1.00 delta.
Between the two extremes, the delta changes. From this diagram another
definition for gamma can be inferred: gamma is the second derivative of the
graph of the option price relative to the stock price. Put another way,
gamma is the first derivative of a graph of the delta relative to the stock
price. Exhibit 2.5 illustrates the delta of a call relative to the stock price.
EXHIBIT 2.5 Call delta compared with stock price.
Not only does the delta change, but it changes at a changing rate. Gamma
is not constant. Moneyness, time to expiration, and volatility each have an
effect on the gamma of an option.