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.
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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.
