# Confusion about payoff for an option [closed]

My teacher said that the payoff of a put is $$\mathrm{max}(K-S_T, 0)$$, where $$K$$ is the strike price and $$S_T$$ is the spot price at maturity. Why isn't it $$K$$ if $$K-S_T > 0$$ and $$0$$ otherwise (i.e. $$K*\mathbf{I}_{K-S_T>0})$$? If you exercise the option when $$K-S_T>0$$, then you make $$K$$ by selling it and otherwise make $$0$$; doesn't the extra term of $$S_T$$ assume that you will rebuy the asset after selling it? Or put another way, I'm confused in that $$\mathrm{max}(K-S_T, 0)$$ seems to capture the "value" of the put, rather than its payoff.

A European put, which you're describing, gives the holder the right to sell the asset $$S_t$$ at time $$T$$ for price $$K$$. From the putholder's perspective, they receive $$K$$, but they have to part with an asset worth $$S_T$$.
It might also be worth noting that the "transaction" of "selling the stock $$S_t$$ for $$K$$ at time $$T$$" doesn't always literally happen when the putholder exercises their option. Sometimes, options are financially settled (aka cash settled), meaning that the putwriter just sends the putholder $$K-S_T$$ at time $$T$$, and never takes the stock $$S_T$$ or any money from the putholder.