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Put-Call Parity and Synthetic Positions

Put-call parity
C−P=S−Ke−rTC - P = S - Ke^{-rT}

Model-free. It follows from replication alone, so it holds whatever volatility turns out to do.

For European options on the same underlying, strike and expiry. With a dividend yield:

C−P=Se−qT−Ke−rTC - P = Se^{-qT} - Ke^{-rT}

Why it must hold

Consider two portfolios:

A: long a call, short a put, both struck at KK. B: long the underlying, short Ke−rTKe^{-rT} of cash.

At expiry, if ST>KS_T > K the call is exercised and you buy at KK; if ST<KS_T < K the put is exercised against you and you buy at KK. Either way you end up owning the asset having paid KK. That is exactly portfolio B.

Identical payoffs in every state, so identical prices today. That is the law of one price applied directly.

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