Chapter 40: Advanced Concepts 905 
Can one possibly reason this risk measurement out without making severe 
mathematical calculations? Well, possibly. Note that the delta of an option starts as a 
small number when the option is out-of-the-money. It then increases, slowly at first, 
then more quickly, until it is just below 0.60 for an at-the-money option. From there 
on, it will continue to increase, but much more slowly as the option becomes in-the­
money. This movement of the delta can be observed by looking at gamma: It is the 
change in the delta, so it starts slowly, increases as the stock nears the strike, and then 
begins to decrease as the option is in-the-money, always remaining a positive num­
ber, since delta can only change in the positive direction as the stock rises. Finally, 
the gamma of the gamma is the change in the gamma, so it in tum starts as a positive 
number as gamma grows larger; but then when gamma starts tapering off, this is 
reflected as a negative gamma of the gamma. 
In general, the gamma of the gamma is used by sophisticated traders on large 
option positions where it is not obvious what is going to happen to the gamma as the 
stock changes in price. Traders often have some feel for their delta. They may even 
have some feel for how that delta is going to change as the stock moves (i.e., they 
have a feel for gamma). However, sophisticated traders know that even positions that 
start out with zero delta and zero gamma may eventually acquire some delta. The 
gamma of the gamma tells the trader how much and how soon that eventual delta will 
be acquired. 
MEASURING THE DIFFERENCE OF IMPLIED VOLATILITIES 
Recall that when the topic of implied volatility was discussed, it was shown that if one 
could identify situations in which the various options on the same underlying securi­
ty had substantially different implied volatilities, then there might be an attractive 
neutral spread available. The strategist might ask how he is to determine if the dis­
crepancies between the individual options are significantly large to warrant attention. 
Furthermore, is there a quick way (using a computer, of course) to determine this? 
A logical way to approach this is to look at each individual implied volatility and 
compute the standard deviation of these numbers. This standard deviation can be 
converted to a percentage by dividing it by the overall implied volatility of the stock. 
This percentage, if it is large enough, alerts the strategist that there may be opportu­
nities to spread the options of this underlying security against each other. An exam­
ple should clarify this procedure. 
Example: XYZ is trading at 50, and the following options exist with the indicated 
implied volatilities. We can calculate a standard deviation of these implieds, called 
implied deviation, via the formula: