Payoffs and Put–Call Parity

What options pay, and the no-arbitrage relationship that ties calls and puts.

A European call gives the holder the right to buy at strike $K$ at expiry $T$ , paying $\max(S_T - K, 0)$ . A European put gives the right to sell, paying $\max(K - S_T, 0)$ .

Put-call parity ties them together. For a stock with no dividends:

C - P = S - K e^{-rT}

The argument is no-arbitrage. A long call plus short put pays $S_T - K$ at expiry; a long stock plus short bond paying $K$ at expiry pays $S_T - K$ too. They must cost the same now, or you can arbitrage.

Put-call parity is one of the few model-free results in derivative pricing. It always holds (for European options on non-dividend-paying stock with constant rate), regardless of any model assumption about the dynamics of $S$ .