Chapter 38: The Distribution of Stock Prices 805 
ing price of the naked put being sold is 550. The resulting probabilities might be 
something like this: 
Scenario Actual Probability of Occurrence 
1 . $OEX never falls below 550 
2. $OEX falls below 550 and remains there 
3. $OEX falls below 550 but rallies later 
67% 
19% 
14% 
The probabilities stated above are the "real" probabilities of the three various 
scenarios occurring. However, if one were using the simple probability calculator 
presented above, he would only have the following information: 
Probability of $OEX being above 550 at expiration: 81 % 
Probability of $OEX being below 550 at expiration: 19% 
So, with the simple calculator, it looks like there's an 81 % chance of a worry-free 
trade. Just sit back and relax and let the option expire worthless. However, in real 
life - as shown by the previous set of probabilities, there's only a 67% chance of a 
worry-free trade. The difference - the other 14% - is the probability of the third 
scenario occurring ($OEX falls below 550, but rallies back above it by expiration). 
The simple probability calculator doesn't account for that scenario at all. 
Hence, most serious traders don't use the simple model. Does that mean it's not 
useful at all? No, it is certainly viable as a comparative tool; for example, to compare 
the chances of the $OEX put expiring worthless versus those of another put sale 
being considered, perhaps something in a stock option. However, better analyses can 
be undertaken. 
Before leaving the scenario of the simple probability calculator, one more point 
should be made. It has been mentioned earlier in this book that the delta of an option 
is actually a fairly good estimate of the probability of the option being in-the-money 
at its expiration date. Thus, the delta and the simple endpoint probability calculator 
shown above attempt to convey the same information to a trader. In reality, because 
of the fact that implied volatility might be different for various strikes (a volatility 
skew), especially in index options, the delta of the option might not agree exactly with 
the probability calculator. Even so, the delta is a quick and dirty way of estimating the 
probability of the stock being above the strike price (in the case of call options) or 
below the strike price (in the case of put options) at expiration. 
THE nEVER" CALCULATOR 
Having seen the frailties of the endpoint calculator, the next step is to try to design a 
calculator that can estimate the probability of the stock ever hitting the target price(s)