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+Chapter 28: Mathematical Applications 471 
+Thus, once the low starting point is chosen and the probability of being below that 
+price is determined, one can compute the probability of being at prices that are suc­
+cessively higher merely by iterating with the preceding formula. In reality, one is 
+using this information to integrate the distribution curve. Any method of approxi­
+mating the integral that is used in basic calculus, such as the Trapezoidal Rule or 
+Simpson's Rule, would be applicable here for more accurate results, if they are 
+desired. 
+A partial example of an expected return calculation follows. 
+Example: XYZ is currently at 33 and has an annual volatility of 25%. The previous 
+bull spread is being established- buy the February 30 and sell the February 35 for a 
+2-point debit - and these are 6-month options. Table 28-7 gives the necessary com­
+ponents for computing the expected return. Column (A), the probability of being 
+below price q, is computed according to the previously given formula, where p = 33 
+and vt = .177 (t = .25-V ½). The first stock price that needs to be looked at is 30, since 
+all results for the bull spread are equal below that price - a 100% loss on the spread. 
+The calculations would be performed for each eighth (or tenth) of a point up through 
+a price of 35. The expected return is compute<l by multiplying the two right-hand 
+columns, (B) and (C), and summing the results. Note that column (B) is determined 
+by subtracting successive numbers in column (A). It would not be particularly 
+enlightening to carry this example to completion, since the rest of the computations 
+are similar and there is a large number of them. 
+In theory, if one had the data and the computer power, he could evaluate a wide 
+range of strategies every day and come up with the best positions on an expected 
+return basis. He would probably get a few option buys (puts or calls), some bull 
+TABLE 28-7. 
+Calculation of expected returns. 
+Price at Expiration (A) (B) (() 
+(q) P (below q) P (of being at q) Profit on Spread 
+30 .295 .295 -$200 
+30 1/s .301 .006 - 187.50 
+30 1/4 .308 .007 - 175 
+303/s .316 .008 - 162.50