Question on early exercise

[kittens]

for early exercise of American options,

I understand how it is rational to exercise a call if PV(Divs)>PV(strike)

in a few examples, it showed that I had to include PV(divs)>PV(strike)+Implicit Put

sometimes it includes the implicit option and sometimes it doesn't, so I am a little confused there

[pjc48]

I'm confused about that, too! When to include implicit put (or implicit call if you're considering a put) and when not to...

[Academic Actuary]

If you have a tree, and the asset value is at a point where the option remains in the money for all possible future paths then you can ignore the implicit put as it has 0 value at that node.

[Sherrbear]

My understanding is that it depends how the question is worded.

If it asks what the lowest dividend is that exercise MIGHT be or COULD be rational, you ignore the implicit put (value = 0). This is the minimum value a dividend could be because you're assuming that the put is worthless.

However, if you're given a scenario that asks if early exercise IS rational, you need to consider the value of the put. You're no longer solving for a minimum bound on the dividend, you're solving the inequality to see if it holds.