806 Part VI: Measuring and 1iading Vo/atillty 
at any time during the life of the probability study, usually the life of an option. It 
turns out that there are a couple of ways to approach this problem. One is with a 
Monte Carlo analysis, whereby one lets a computer run a large number of random­
ly-generated scenarios (say, 100,000 or so) and counts the number of times the tar­
get price is hit. A Monte Carlo analysis is a completely valid way of estimating the 
probability of an event, but it is a somewhat complicated approach. 
In reality, there is a way to create a single formula that can estimate the "ever" 
probability, although it is not any easy task either. In the following discussion, I am 
borrowing liberally from correspondence with Dr. Stewart Mayhew, Professor of 
Mathematics at the University of Georgia. For proprietary reasons, the exact formu­
la is not given here, but the following description should be sufficient for a mathe­
matics or statistics major to encode it. If one is not interested in implementing the 
actual formula, the calculation can be obtained through programs sold by McMillan 
Analysis Corp. at www.optionstrategist.com. 
This discussion is quite technical, so readers not interested in the description of 
the mathematics can skip the next paragraph and instead move ahead to the next sec­
tion on Monte Carlo studies. 
These are the steps necessary in determining the formula for the "ever" proba­
bility of a stock hitting an upside target at any time during its life. First, make the 
assumption that stock prices behave randomly, and perform at the risk-free rate, r. 
Mathematicians call random behavior "Brownian motion." There are a number of 
formulae available in statistics books regarding Brownian motion. If one is to esti­
mate the probability of reaching a maximum (upside target) point, what is needed is 
the known formula for the cumulative density function (CDP) for a running maxi­
mum of a Brownian motion. In that formula, it is necessary to use the lognormal 
function to describe the upside target. Thus, instead of using the actual target price 
in the CDF formula, one substitutes ln( qlp ), where q is the target price and p is the 
current stock price. 
The "ever" probability calculator provides much more useful information to a 
trader of options. Not only does a naked option seller have a much more realistic esti­
mate of the probability that he's going to have to make an adjustment during the life 
of an option, but the option buyer can find the information useful as well. For exam­
ple, if one is buying an option at a price of 10, say, then he could use the "ever" prob­
ability calculator to estimate the chances of the stock trading 10 points above the 
striking price at any time during the life of the option. That is, what are the chances 
that the option is going to at least break even? The option buyer can, cf course, deter­
mine other things too, such as the probability that the option doubles in price ( or 
reaches some other return on investment, such as he might deem appropriate for his 
analysis).