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+Chapter 28: Mathematical Applications 463 
+This is then the proper way to calculate historical volatility. Obviously, the 
+strategist can calculate 10-, 20-, and 50-day and annual volatilities if he wishes - or 
+any other number for that matter. In certain cases, one can discern valuable infor­
+mation about a stock or future and its options by seeing how the various volatilities 
+compare with one another. 
+There is, in fact, a way in which the strategist can let the market compute the 
+volatility for him. This is called using the implied volatility; that is, the volatility that 
+the market itself is implying. This concept makes the assumption that, for options 
+with striking prices close to the current stock price and for options with relatively 
+large trading volume, the market is fairly priced. This is something like an efficient 
+market hypothesis. If there is enough trading interest in an option that is close to the 
+money, that option will generally be fairly priced. Once this assumption has been 
+made, a corollary arises: If the actual price of an option is the fair price, it can be fixed 
+in the Black-Scholes equation while letting volatility be the unknown variable. The 
+volatility can be determined by iteration. In fact, this process of iterating to compute 
+the volatility can be done for each option on a particular underlying stock This might 
+result in several different volatilities for the stock If one weights these various results 
+by volume of trading and by distance in- or out-of-the-money, a single volatility can 
+be derived for the underlying stock This volatility is based on the closing price of all 
+the options on the underlying stock for that given day. 
+Example: XYZ is at 33 and the closing prices are given in Table 28-1. Each option 
+has a different implied volatility, as computed by determining what volatility in the 
+Black-Scholes model would result in the closing price for each option: That is, if .34 
+were used as the volatility, the model would give 4¼ as the price of the January 30 
+call. In order to rationally combine these volatilities, weighting factors must be 
+applied before a volatility for XYZ stock itself can be arrived at. 
+The weighting factors for volume are easy to compute. The factor for each 
+option is merely that option's daily volume divided by the total option volume on all 
+XYZ options (Table 28-2). The weighting functions for distance from the striking 
+price should probably not be linear. For example, if one option is 2 points out-of-the­
+money and another is 4 points out-of-the-money, the former option should not nec­
+essarily get twice as much weight as the latter. Once an option is too far in- or out-of­
+the-money, it should not be given much or any weight at all, regardless of its trading 
+volume. Any parabolic function of the following form should suffice: 
+{ 
+(x - a)2 if xis less than a 
+Weighting factor = -;;,r-
+= 0 if x is greater than a