Chapter 38: The Distribution of Stock Prices 799 
thing that cannot be predicted with certainty. Nevertheless, any probability calcula­
tor requires this input. So, one must understand that the results one obtains from any 
of these probability calculators is an estimate of what might happen. It should not be 
relied on as "gospel." 
Additionally, probability calculators make a second assumption: that the volatil­
ity one inputs will remain constant over the entire length of the study. We know this 
is incorrect, for volatility can change daily. However, there really isn't a good way of 
estimating how volatility might change in the course of the study, so we are pretty 
much forced to live with this incorrect assumption as well. 
There is no certain way to mitigate these volatility "problems" as far as the prob­
ability calculator is concerned, but one helpful technique is to bias the volatility pro­
jection against your objectives. That is, be overly conservative in your volatility pro­
jections. If things tum out to be better than you estimated, fine. However, at least 
you won't be overstating things initially. An example may help to demonstrate this 
technique. 
Example: Suppose that a trader is considering buying a straddle on XYZ. The five­
month straddle is selling for a price of 8, with the stock currently trading near 40. A 
probability calculator will help him to determine the chances that XYZ can rise to 48 
or fall to 32 (the break-even points) prior to the options' expiration. However, the 
probability calculator's answer will depend heavily on the volatility estimate that the 
trader plugs into the probability calculator. Suppose that the following information is 
know about the historical volatility of XYZ: 
l 0-day historical volatility: 
20-day historical volatility: 
50-day historical volatility: 
l 00-day historical volatility 
22% 
20% 
28% 
33% 
Which volatility should the trader use? Should he choose the 100-day historical 
volatility since this is a five-month straddle, which encompasses just about 100 trad­
ing days until expiration? Should he use the 20-day historical volatility, since that is 
the "generally accepted" measure that most traders refer to? Should he calculate a 
historical volatility based exactly on the number of days until expiration and use that? 
To be most conservative, none of those answers is right, at least not for the right 
reasons. Since one is buying options in this strategy, he should use the lowest of the 
above historical volatility measures as his volatility estimate. By doing so, he is taking 
a conservative approach. If the straddle buy looks good under this conservative 
assumption, then he can feel fairly certain that he has not overstated the possibilities