Put-Call Parity

The relationship rests on a payoff identity, not on any pricing model. At expiration, the underlying combined with a put worth the strike minus the terminal price when in the money pays whichever is greater, the terminal price or the strike. A call plus a zero-coupon bond redeeming at the strike pays exactly the same: the bond delivers the strike, and the call adds any excess of the terminal price over it. Two positions with identical payoffs in every state of the world must trade at identical prices today, otherwise a trader buys the cheap package, sells the dear one, and locks in a riskless spread.

Written out, the call premium plus the strike discounted at the risk-free rate over the option's life equals the put premium plus the current price of the underlying. Note what is absent: volatility, expected return, and any distributional assumption. Parity is a static replication argument, which is why it constrains any model rather than depending on one.

Because the equation contains four instruments, isolating any one of them describes a synthetic version built from the other three. Buying the call and selling the put at the same strike, then holding the discounted strike in bills, replicates a long position in the underlying; reversing the option legs gives a synthetic short. Holding the underlying and a put while borrowing the present value of the strike creates a synthetic call. Selling the underlying and the put while lending produces a synthetic put.

Combining the long and short synthetic underlying at two different strikes yields a box spread, whose payoff is the fixed difference between the strikes, a synthetic zero-coupon bond. Desks use boxes to lend or borrow through the options market, and the implied rate on a box is a direct read on financing conditions. The arbitrage trades that enforce parity carry the standard names conversion and reversal.

For a dividend-paying underlying, the holder of the stock receives cash the option holder does not, so the present value of dividends expected before expiration is subtracted from the spot price in the parity equation. Written against a forward or futures contract instead of the spot, the relationship becomes put-call forward parity: the call plus the discounted strike equals the put plus the discounted forward price, which conveniently sweeps carry, dividends and storage into the forward.

Run the logic backwards and option prices become an information source. Given liquid calls and puts, traders extract implied dividends, implied repo rates, and the cost of borrowing hard-to-locate shares. Because early exercise has value, American options satisfy only a pair of inequalities rather than a clean equality, and the width of that band grows with dividends and interest rates.

Three tasks recur. First, solve for a missing premium: given three of the four components, rearrange and compute, remembering to discount the strike over the exact time to expiry and to net out any dividends. Second, diagnose mispricing: compare the cost of the protective put package with the fiduciary call package, buy the cheaper and short the dearer, and state the riskless profit as the difference in present value terms. Third, construct the synthetic instrument a question asks for and identify which legs are long and which are short.

A further implication is worth carrying into the option-pricing readings. Since Black-Scholes-Merton prices satisfy parity by construction, a European call and put on the same underlying with the same strike and expiry must share one implied volatility. Any observed gap in quoted implied volatilities at a single strike reflects frictions, stale quotes, or borrow costs, not disagreement about volatility.