Chapter 28: Mathematical Applications 481 
action point (recall that the collateral requirement increases for naked options on an 
adverse stock movement), and the break-even points themselves. The probabilities of 
being above the upper break-even point at expiration or below the lower break-even 
point should be computed as well. Moreover, an expected return analysis could be 
performed on the position to determine the general level of profitability of the posi­
tion with respect to all other positions of the same type on other stocks. Such an 
expected return analysis need not assume that the position is held to expiration. Firm 
traders, paying little or no commissions, might be interested in seeing the expected 
reswts for a holding period as short as 30 days or less. Public customers might use a 
longer holding period, on the assumption that they would not trade the position as 
readily because of commission costs. Ratio positions should be ranked either by 
return if unchanged or by expected return. 
The analyses described for calendar spreads and ratio positions should not be 
relied upon as gospel. In the proposed forms of analysis, one is projecting future 
option prices and stock prices under the assumption that the volatility of the under­
lying stock will remain the same. Although this may be true in some cases, there will 
also be many times when the volatility of the underlying stock will change during the 
life of the position. If the volatility decreases, the projected break-even points for a 
calendar spread will be too far away from the striking price. Thus, a loss would result 
at some prices where the spreader expected to make money. If the volatility increas­
es, the expected return of a ratio position will drop, because the probabilities of the 
stock moving outside the profit range will increase, thereby increasing the probabil­
ity ofloss. 
The effect of a changing volatility can be counteracted, in theory, by continuing 
to monitor the position daily after it has been established. In a straddle write, for 
example, if the stock begins to move dramatically, the expected return may become 
very low. If this happens, adjustments could be made to the position to improve it. 
Such monitoring is difficult to apply in practice for the public customer, because the 
commission costs involved in constant position adjustments would mount rapidly. 
There is no exact method that would allow for infrequent, periodic adjustments, but 
by using a follow-up analysis the public customer may be able to get a better feeling 
for the timing of adjusting a position. For example, suppose that one initially wrote a 
5-point straddle when the stock was at 30. Sometime later, the stock is at 34. The 
expected return of writing a 5-point straddle with a strike of 30 when the stock is at 
34 could be computed for the shorter time period remaining until expiration. If the 
expected return is negative, an adjustment needs to be made. Adopting this form of 
adjusting would keep the number of trades to a minimum, but would still allow the 
strategist to determine when his position has become improperly balanced. Of 
course, the current volatility would be used in making these determinations. Another