Example 2E from ASM Manual

Hi,

Can you guys help me with a question?



Example 2E (Page 47)

For three 6-month American call options on a stock:

(a) One with strike price 45 sells for 6.30

(b) One with strike price 44 sells for 7

(c) One with strike price 40 sells for 9.5



The option with strike price 44 is overpriced based on the convexity

property of option premiums. You therefore sell it.

Determine the maximum and the minimum amount of the other 2 options you should buy to guarantee a profit.



Answer: They come up with three conditions that must be satisfied:

(1) 9.5x + 6.3y <=7

(2) x>= 1/5

(3) x+y>=1



I got (1) and (2) but I am struggling how they got (3).

They get (3) from

x(S-40) + y(S-45)>= S-44

becomes

S(x+y-1) -40x -45y +44 >=0.



I understand how they got to S(x+y-1) -40x -45y +44 >=0.

Then they say If x+y -1 <0., then the left hand side is a decreasing linear function of S, so for S sufficiently high, the inequality is violated. Conversely, if x+y-1>=0, the left hand side is a non-decreasing linear function of S, so if it is greater than 0 at S=45, it is certainly greater than 0 for S>45. So we conclude that the inequality is satisfied for all S>=45 if and only if x+y>=1.



Can you explain the bolded part to me and how they got to x+y>=1?

Thank you!

Example 2E from ASM Manual