The relation linking European call and put prices

State the put-call parity relation for European options and explain why it holds.

$C-P=S-Ke^{-r(T-t)}$ for a call and put on a non-dividend stock with the same strike $K$ and expiry $T$. Long a call and short a put pays $S_T-K$ in every state, exactly like holding the stock and borrowing the present value of $K$. Equal payoffs require equal prices, so the relation is model-free; dividends replace $S$ by $S$ minus their present value, and it does not hold exactly for American options.