Put-Call Parity

For European options on a non-dividend-paying stock, with the same strike $K$ and expiry $T$ , write down the relationship between the call price $C$ and the put price $P$ , and prove it.

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#Two portfolios with the same payoff

Build portfolio A from one call plus cash worth $Ke^{-rT}$ today, which grows to exactly $K$ by maturity. Build portfolio B from one put plus one share. At maturity each is worth $\max(S_T, K)$ .

In portfolio A, if $S_T > K$ you exercise the call and hold the share worth $S_T$ , and otherwise you let it lapse and keep the cash $K$ . In portfolio B, if $S_T < K$ you exercise the put and sell the share for $K$ , and otherwise you keep the share worth $S_T$ . Both land on $\max(S_T, K)$ in every state.

#No arbitrage equates them today

Two portfolios with identical payoffs in every future state must cost the same today, or you could buy the cheaper and sell the dearer for a riskless profit. Hence

C + Ke^{-rT} = P + S \quad\Longrightarrow\quad C - P = S - Ke^{-rT}. \tag{1}

A call plus a bond worth K at maturity, and a put plus a share, both pay the same kinked amount, the larger of the final price and K. Identical payoffs in every state force their prices to agree today.

#Read it off

C - P = S - Ke^{-rT}. \tag{2}

The relation needs no model of how the stock moves. It rests only on the two maturity payoffs matching and on the absence of arbitrage, so it holds whatever the volatility or drift.