Quiz 6-1 in ASM Manual

For an American call option on a stock:

The stock price is 50

The strike price is 45

There are 3 months to expiry

The stock is about to pay a dividend of D immediately and will pay another dividend of D in 3 months

r=.04



Determine the highest value of D such that early exercise is definitely not optimal.



I first solve for the present value of the dividend:

D+De^(-.04/4) = 1.990050D



Then according to the previous text on the prior page, I would solve for the present value of the interest on the strike price plus the value of the put

However, they do not give the value of the put, and in the solution, this isn't used. They only compare the present value of the dividend vs the present value of the interest on the strike price:



1.990050D <= 45*(1-e^(-.04/4))

1.990050D <= .447757

D <= .2250



Why do they not require the value of the put in this example?

Also, should it be less than or equal to, or just less than?

In example 6-A, when checking if early exercise was optimal, they used

PV future dividends >= PV of interest of strike price + put value



What if PV future dividends = PV of interest?

Quiz 6-1 in ASM Manual