470 
where 
v = annual volatility 
t = time, in years 
vt = volatility for time, t. 
Part IV: Additional Considerations 
As an example, a 3-month volatility would be equal to one-half of the annual 
volatility. In this case, t would equal .25 (one fourth of a year), so v_25 = v65 = .5v. 
The necessary groundwork has been laid for the computation of the probabili­
ty necessary in the expected return calculation. The following formula gives the prob­
ability of a stock that is currently at price p being below some other price, q, at the 
end of the time period. The lognormal distribution is assumed. 
Probability of stock being below price q at end of time period t: 
P (below) = N (In~)) 
where 
N = cumulative normal distribution 
p = current price of the stock 
q = price in question 
In = natural logarithm for the time period in question. 
If one is interested in computing the probability of the stock being above the 
given price, the formula is 
P (above)= 1- P (below) 
With this formula, the computation of expected return is quickly accomplished 
with a computer. One merely has to start at some price - the lower strike in a bull 
spread, for example - and work his way up to a higher price - the high strike for a 
bull spread. At each price point in between, the outcome of the spread is multiplied 
by the probability of being at that price, and a running sum is kept. 
Simplistically, the following iterative equation would be used. 
P ( of being at price x) = P (below x) - P (below y) 
where y is close to but less than x in price. As an example: 
P (of being at 32.4) = P (below 32.4) - P (below 32.3)