CM2A September 2019 Q6ii American Put option

[Rose95]

Let pt denote the value at time t of a European put option on a non-dividend paying
share St with maturity at time T and a strike price K. The risk-free rate of interest is r.
(i) Derive the lower bound for pt in terms of St and K.
(ii) Explain how the lower bound would change if pt were an American put option.

The lower bound of the European put is p_t + S_t >= K*exp[-r*(T-t)].
In the notes, we have the lower bound of an American Put as P_t >= K -S_t

However, the examiner's report states: "Because early exercise is always possible, we have p_t + S_t >= K*exp[-r*(T-t)]."

Please explain why the given solution differs from that in the Notes.
Thank you.

 

[Steve Hales]

From the examiner's report there doesn't appear to be any difference between the lower bound for the European and American version of the option, which could indicate an incomplete answer.
Also, lower bounds aren't unique (for example, zero is also a lower bound). For completeness, for an American option we have:
p_t + S_t >= K >= K*exp[-r*(T-t)]