CAS 2007 #12

[xandermtl]

I did a search, but couldn't find anything on this question, so sorry if someone has already responded to it.
This Question seems really just wrong to me.
It goes as follows:
Which of the following are correct on the price of a stock option ?
I. The premiums would not decrease if the options were american rather than european
II. For European put, the premiums increase when the stock price increases
III. For American call, the premims increase when the strike price increases.

Clearly II, and III are false. I is apparently the right answer, but it is not strictly true. It seems to me the premium of an american option would always be greater than a european option, but the premium should still decrease ! Could someone explain to me why the premium of an american option wouldn't decrease ?

Thanks Xand