Chapter 38: The Distribution of Stock Prices 797 
models predict; but it certainly means that the buyer of underpriced options stands 
to benefit in a couple of ways. Conversely, an option seller must certainly concentrate 
his efforts where options are expensive, and even then should be acutely aware that 
he may experience larger-than-expected stock price movements while the option 
position is in place. 
So what does this mean for option strategies? On the surface, it means that if 
one uses the normal (or lognormal) distribution for estimating the probability of a 
strategy's success, he may get a big move in the stock that he didn't originally view as 
possible. If one were long straddles, that's great. However, if he is short naked 
options, then there could be a nasty surprise in store. That's one reason why extreme 
caution should be used regarding selling naked options on stocks; they can make 
moves of this sort too often. At least with indices, such moves are far less frequent, 
although the Dow drop of over 550 points in October 1997 was a move of seven stan­
dard deviations, and the crash of '87 was about a 16-standard deviation move - which 
Professor Mark Rubenstein of the University of California at Berkeley says was some­
thing that should occur about once in ten times the life of our current universe! That's 
according to lognormal distribution, of course, which we know understates things 
somewhat, but it's still a big number under any distribution. 
There are two approaches that one can take, then, regarding option strategies. 
One is to invent another method for estimating stock price distributions. Suffice it to 
say that that is not an easy task, or someone would have made it well-known already. 
There have been many attempts, including some in which a large history of stock 
price movements is observed and then a distribution is fitted to them. The problem 
with accounting for these occasional large price moves is that it is perhaps an even 
more grievous error to overestimate the probabilities of such moves than to underes­
timate them. 
The second approach is to continue to use the normal distribution, because it's 
fast and accessible in a lot of places. Then, either rely on option buying strategies 
( straddles, for example) where implied volatility appears to be low - knowing that you 
have a chance at better results than the statistics might indicate - or adjust your cal­
culations mentally for these large potential movements if you are using option selling 
strategies. 
THE PRICING OF OPTIONS 
The extreme movements of the fat tail distribution should be figured into the pricing 
of an option, but they really are not, at least not by most models. The Black-Scholes