466 Part IV: Additional Considerations 
taking a moving average of the last 20 or 30 days' implied volatilities. An alternative 
that does not require the saving of many previous days' worth of data is to use a 
momentum calculation on the implied volatility. For example, today's final volatility 
might be computed by adding 5% of today's implied volatility to 95% of yesterday's 
final volatility. This method requires saving only one previous piece of data - yester­
day's final volatility - and still preserves a "smoothing" effect. 
Once this implied volatility has been computed, it can then be used in the 
Black-Scholes model ( or any other model) as the volatility variable. Thus one could 
compute the theoretical value of each option according to the Black-Scholes formu­
la, utilizing the implied volatility for the stock. Since the implied volatility for the 
stock will most likely be somewhat different from the implied volatility of this par­
ticular option, there will be a discrepancy between the option's actual closing price 
and the theoretical price as computed by the model. This differential represents the 
amount by which the option is theoretically overpriced or underpriced, compared to 
other options on that same stock. 
EXPECTED RETURN 
Certain investors will enter positions only when the historical percentages are on 
their side. When one enters into a transaction, he normally has a belief as to the pos­
sibility of making a profit. For example, when he buys stock he may think that there 
is a "good chance" that there will be a rally or that earnings will increase. The investor 
may consciously or unconsciously evaluate the probabilities, but invariably, an invest­
ment is made based on a positive expectation of profit. Since options have fixed 
terms, they lend themselves to a more rigorous computation of expected profit than 
the aforementioned intuitive appraisal. This more rigorous approach consists of com­
puting the expected return. The expected retum is nothing more than the retum that 
the position should yield over a large number of cases. 
A simple example may help to explain the concept. The crucial variable in com­
puting expected return is to outline what the chances are of the stock being at a cer­
tain price at some future time. 
Example: XYZ is selling at 33, and an investor is interested in determining where 
XYZ will be in 6 months. Assume that there is a 20% chance of XYZ being below 30 
in 6 months, and that there is a 40% chance that XYZ will be above 35 in 6 months. 
Finally, assume that XYZ has an equal 10% chance of being at 31, 32, 33, or 34 in 6 
months. All other prices are ignored for simplification. Table 28-5 summarizes these 
assumptions.