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