According this answer, https://quant.stackexchange.com/a/39298/29108, the European put price (with maturity $T$) at time $t$ for a stock whose current price is $0$ should be the strike $K$ discounted from $T$ to $t$. So $P(t,T)K = p$.
Is this exactly true? I had thought that the value $P(t,T)K$ should be an upper bound on the put price, as there is still a chance the stock price increases in the time before maturity. And if the European put price is basically growing at the risk free rate, doesn't this remove the probability of stock price movements?
Any help would be appreciated!