Chapter 28: Mathematical Applications 487 
Finally, the computer can compute the expected return of a position already in 
place. This would give a more dynamic picture of the position, and this expected 
return is usually for a relatively short time period. That time period might be 30 days, 
or the time remaining until expiration, whichever is less. The expected return is cal­
culated in much the same manner as the expected return computation described ear­
lier in this chapter. First, one uses the stock's volatility to construct a range of prices 
over which to examine the position. Second, one uses the Black-Scholes model to 
calculate the values of the various options in the position at that future time and at 
the various stock prices. Some of the results should be displayed in table form by the 
computer program. The expected profit is computed, as described earlier, by sum­
ming the multiples of the probabilities of the stock being at each price by the result 
of the position at that price. The expected return is then computed by dividing the 
expected profit by the expected investment. Since margin computations can require 
involved computer programs, it is sufficient to omit this last step and merely observe 
the expected profit. The following example shows how a sample position might look 
as the computer displays the position itself, the ESP, the profit at expiration, and the 
expected profit in 30 days. A complex position is assumed, in order that the value of 
these analyses can be demonstrated. 
Example: The following position exists when XYZ is at 31 ¾. It is essentially a back­
spread combined with a reverse ratio write. It resembles a long straddle in that there 
is increased profit potential in either direction if the stock moves far enough by expi­
ration. Initially, the computer should display the position and the ESP. 
Position Delta ESP 
Short 4,500 XYZ 1.00 Short 4,500 shares 
Short 100 XYZ April 25 calls 0.89 Short 8, 900 shares 
Long 50 XYZ April 30 calls 0.76 Long 3,800 shares 
Long 139 XYZ July 30 calls 0.74 Long 10,286 shares 
Total ESP Long 686 shares 
Total money in position: $163,500 credit 
The advantage of using the ESP is that this fairly complex position is reduced to a sin­
gle number. The entire position is equivalent to being long 686 shares of the com­
mon stock. Essentially, this is close to delta-neutral for such a large position. The next 
item that the computer should display is the total credit or debit in the position to 
date. With this information, an expiration picture can be drawn if it is applicable. In 
this position, since there is a mixture of April and July options, a strict expiration pie-