Chapter 40: Advanced Concepts 857 
delta will be more dramatic than it would be for a volatile stock. Out-of-the-money 
options are an entirely different story. Since the nonvolatile stock will have difficulty 
moving fast enough to reach an out-of-the-money striking price, the delta of the out­
of-the-money option is small and it will not change quickly (that is, the gamma is 
small also). 
These concepts are summarized in Figure 40-5 (see Table 40-5), which depicts 
the gammas for similar options on stocks with differing volatilities. For the purposes 
of these graphs, XYZ is equal to 50 and there are three months remaining until expi­
ration. 
Notice that for a very volatile stock, the gamma is quite stable over nearly all 
striking prices when there are 3 months remaining until expiration. This means that 
the deltas of all options on such a volatile stock will be changing quite a bit for even 
a 1-point move in the underlying stock. This is an important point for neutral strate­
gists to note, because a position that starts out as delta neutral may quickly change if 
the underlying stock is very volatile. As this table implies, the deltas of the options in 
that "neutral" spread may be altered quickly, thereby rendering the spread quite un­
neutral. This concept will be discussed in greater detail later in this chapter. 
As delta was used to construct the equivalent stock position of an entire option 
position or portfolio, gamma can be used in a similar manner. An example of this fol­
lows, using the same securities from the preceding example on the delta of a posi­
tion. An important point to note is that the gamma of the underlying security itself is 
zero. This is true because the delta of the underlying security (which is always 1.0) 
never changes - hence the gamma is zero. The gamma is measuring the amount of 
change of the delta; if the delta of the underlying security never changes, the gamma 
of the underlying security must be zero. 
Example: The following position exists when XYZ is at 31.75. Recall that it resem­
bles a long straddle ( or backspread) in that there is increased profit potential in either 
direction if the stock moves far enough by expiration. In addition to the delta previ­
ously listed, the gamma is now shown as well. Note that since gamma is a small 
absolute number, it is sometimes calculated out to three or four decimal places. 
Option Position Option Position 
Position Delta Delta Gamma Gamma 
Short 4,500 XYZ 1.00 - 4,500 0.0000 0 
Short l 00 XYZ April 25 calls 0.89 - 8,900 0.0100 -100 
Long 50 XYZ April 30 calls 0.76 + 3,800 0.0300 +150 
Long 139 XYZ July 30 calls 0.74 +10,286 0.0200 +278 
Totals: + 686 +328