Chapter 38: The Distribution of Stock Prices 809 
a deeply out-of-the-money put credit spread usually destroys most or all of its prof­
itability, so an accurate initial assessment of the probabilities of having to make such 
an adjustment is important. 
Option buyers, too, would benefit from the use of a more accurate probability 
estimate. This is especially true for neutral strategies, such as straddle or strangle 
buying, when the trader is interested in the chances of the stock being able to move 
far enough to hit one or the other of the straddle's break-even points at some time 
during the life of the straddle. 
The Monte Carlo probability calculation can be expanded to include other sorts 
of distributions. In the world of statistics, there are many distributions that define ran­
dom patterns. The lognormal distribution is but one of them (although it is the one 
that most closely follows stock prices movements in general). Also, there is a school of 
thought that says that each stock's individual price distribution patterns should be ana­
lyzed when looking at strategies on that stock, as opposed to using a general stock 
price distribution accumulated over the entire market. There is much debate about 
that, because an individual stock's trading pattern can change abruptly just consider 
any of the Internet stocks in the late 1990s and early 2000s. Thus, a probability esti­
mate based on a single stock's behavior, even if that behavior extends back several 
years, might be too unreliable a statistic upon which to base a probability estimate. 
In summary, then, one should use a probability calculator before taking an 
option position, even an outright option buy. Perhaps straight stock traders should 
use a probability calcutor as well. In doing so, though, one should be aware of the 
limitations of the estimate: It is heavily biased by the volatility estimate that is input 
and by the assumption of what distribution the underlying instrument will adhere to 
during the life of the position. While neither of those limitations can be overcome 
completely, one can mitigate the problems by using a conservative volatility estimate. 
Also, he can look at the results of the probability calculation under several distribu­
tions (perhaps lognormal, fat tail, and the distribution using only the past price 
behavior of the underlying instrument in question) and see how they differ. In that 
case, he would at least have a feeling for what could happen during the life of the 
option position. 
EXPECTED RETURN 
The concept of expected return was described in the chapter on mathematical appli­
cations. In short, expected return is a position's expected profit divided by its invest­
ment ( or expected investment if the investment varies with stock price, as in a naked 
option position or a futures position). The crucial component, though, is expected 
profit.