The Parity Equation

Put-call parity — C + PV(K) = P + S — is the arbitrage-free pricing relationship between European calls and puts. A portfolio of a call plus a zero-coupon bond paying the strike replicates a portfolio of a put plus the underlying stock. Since both have identical payoffs at expiry, they must trade at identical prices.

Reading the Discrepancy

When the call side exceeds the put side, the call is relatively cheap or the put expensive — sell the put, buy the call, buy the stock, and lend PV(K) to capture the difference risklessly. In efficient markets, transaction costs and bid-ask spreads keep discrepancies under a few cents. This calculator flags gaps above a $0.05 tolerance.

Beyond the Basics

Real-world adjustments: American options replace the equality with inequalities, dividends subtract PV(D) from the right side, and interest rates use continuous compounding for precision. Parity is also the foundation of synthetic positions — a synthetic long stock is call minus put — used by traders to build exposures without trading the underlying.