798 Part VI: Measuring and Trading Volatility 
model, for example, uses a lognormal distribution. Personally, this author believes 
that the Black-Scholes model is an excellent tool for analyzing options and option 
strategies, but one must understand that it may not be affording enough probability 
to large moves by the underlying. 
Does this mean that most options are underpriced, since traders and market­
makers are using the Black-Scholes model (or similar models) to price them? 
Without getting too technical, the answer is that yes, some options - particularly out­
of-the-money options - are probably underpriced. However, one must understand 
that it is still a relatively rare occurrence to experience one of these big moves - ifs 
just not as rare as the lognormal distribution would indicate. So, an out-of-the-money 
option might be slightly underpriced, but often not enough to make any real differ­
ence. 
In fact, futures options in grains, gold, oil, and other markets that often experi­
ence large and sudden rallies display a distinct volatility skew. That is, out-of-the-money 
call options trade at significantly higher implied volatilities than do at-the-money 
options. Ironically, there is far less chance of one of these hyper-standard-deviation 
moves occurring in commodities than there is in stocks, at least if history is a guide. So, 
the fact that some out-of-the-money futures options are expensive is probably an incor­
rect overadjustment for the possibility of large moves. 
THE PROBABILITY OF STOCK PRICE MOVEMENT 
The distribution information introduced in this chapter can be incorporated into 
somewhat rigorous methods of determining probabilities. That is, one can attempt to 
assess the chances of a stock, futures contract, or index moving by a given distance, 
and those chances can incorporate the fat tails or other non-lognormal behavior of 
prices. 
The software that calculates such probabilities is typically named a "probability 
calculator." There are many such software programs available in the marketplace. 
They range from free calculators to completely overpriced ones selling for more than 
$1,000. In reality, high-level probability calculation software can be created by some­
one with a good understanding of statistics, or a program can be purchased for a 
rather nominal fee - perhaps $100 or so. 
Before getting into these various methods of probability estimation, it should be 
noted that all of them require the trader to input a volatility estimate. There are only 
a few other inputs, usually the stock price, target price(s), and length of time of the 
study. The volatility one inputs is, of course, an estimate of future volatility - some-