Put–call parity

Build two portfolios today, at time $t$. Portfolio A holds one call option plus an amount of cash $Ke^{-r(T-t)}$ invested at the risk-free rate, so that it grows to exactly $K$ by time $T$. Portfolio B holds one put option plus one share of the stock.

Compare their values at maturity $T$ in the two possible cases. If $S_T \ge K$: the call is exercised and worth $S_T - K$, and the cash has grown to $K$, so Portfolio A is worth $(S_T - K) + K = S_T$. The put expires worthless, and the stock is worth $S_T$, so Portfolio B is worth $0 + S_T = S_T$. The two portfolios agree.

If $S_T < K$: the call expires worthless, and the cash is still worth $K$, so Portfolio A is worth $0 + K = K$. The put is exercised and worth $K - S_T$, and the stock is worth $S_T$, so Portfolio B is worth $(K - S_T) + S_T = K$. The two portfolios again agree, this time both equal to $K$.

So in every possible outcome, Portfolio A and Portfolio B have exactly the same value at $T$. If their values at $t$ differed, an arbitrageur could sell the more expensive portfolio, buy the cheaper one, invest the difference at the risk-free rate, and at $T$ collect a risk-free profit regardless of what the stock does. Since real markets do not allow such riskless profit to persist, the two portfolios must have the same price at $t$: $C + Ke^{-r(T-t)} = P + S$, which rearranges to $C - P = S - Ke^{-r(T-t)}$.