904 Part VI: Measuring and Trading Volatility 
Recall that, in the same example used to describe gamma, the position was delta 
long 686 shares and had a positive gamma of 328 shares. Furthermore, we now see 
that the gamma itself is going to decrease as the stock moves up ( it is negative) or will 
increase as the stock moves down. In fact, it is expected to increase or decrease by 
22 shares for each point XYZ moves. 
So, if XYZ moves up by 1 point, the following should happen: 
a. Delta increases from 686 to 1,014, increasing by the amount of the gamma. 
b. Gamma decreases from 328 to 306, indicating that a further upward move by 
XYZ will result in a smaller increase in delta. 
One can build a general picture of how the gamma of the gamma changes over 
different situations - in- or out-of-the-money, or with more or less time remaining 
until expiration. The following table of two index calls, the January 350 with one 
month of life remaining and the December 350 with eleven months of life remain­
ing, shows the delta, gamma, and gamma of the gamma for various stock prices. 
Index January 350 call December 350 call 
Price Delta Gamma Gamma/Gamma Delta Gamma Gamma/Gamma 
310 .0006 .0001 .0000 .3203 .0083 .0000 
320 .0087 .0020 .0004 .3971 .0082 .0000 
330 .0618 .0100 .0013 .4787 .0080 -.0000 
340 .2333 .0744 .0013 .5626 .0078 -.0001 
350 .5241 .0309 -.0003 .6360 .0073 -.0001 
360 .7957 .0215 -.0014 .6984 .0067 -.0001 
370 .9420 .0086 -.0010 .7653 .0060 -.0001 
380 .9892 .0021 -.0003 .8213 .0052 -.0001 
Several conclusions can be drawn, not all of which are obvious at first glance. 
First of all, the gamma of the gamma for long-term options is very small. This should 
be expected, since the delta of a long-term option changes very slowly. The next fact 
can best be observed while looking at the shorter-term January 350 table. The 
gamma of the gamma is near zero for deeply out-of-the-money options. But, as the 
option comes closer to being in-the-money, the gamma of the gamma becomes a pos­
itive number, reaching its maximum while the option is still out-of-the-money. By the 
time the option is at-the-money, the gamma of the gamma has turned negative. It 
then remains negative, reaching its most negative point when slightly in-the-money. 
From there on, as the option goes even deeper into-the-money, the gamma of the 
gamma remains negative but gets closer and closer to zero, eventually reaching 
(minus) zero when the option is very far in-the-money.