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+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: