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Put-call parity is the no-arbitrage relationship that links the prices of a European call and put with the same strike and expiry. The formula is: C − P = S − K · e^(−rT). It says that a long call and a short put is equivalent to holding the stock and borrowing the present value of the strike. If this relationship breaks, free money is available.
Buy a call and sell a put at the same strike. If the stock ends above K, the call pays S − K and the put expires worthless. If below, the call expires worthless and the short put costs K − S. Either way, your payoff is S − K at expiry. This is identical to owning the stock and owing K at expiry (a forward contract). The cost of this synthetic forward must equal S − K · e^(−rT), which gives us the parity formula.
Stock at $100, strike $100, risk-free rate 5%, 1 year to expiry. The discounted strike is $100 × e^(−0.05) = $95.12. If the call costs $12 and the put costs $7: C − P = $12 − $7 = $5. And S − K · e^(−rT) = $100 − $95.12 = $4.88. The parity is violated by $0.12. An arbitrageur buys the put, sells the call, and buys the stock, locking in a $0.12 risk-free profit per share.
Put-call parity is not just a pricing identity. It is how dealers hedge. Need a put but the put market is illiquid? Synthesise one from the call: P = C − S + K · e^(−rT). This is called a synthetic put. The same logic creates synthetic calls, synthetic stock, and synthetic bonds. Every complex strategy can be decomposed into simpler parts using parity.
Put-call parity holds exactly for European options only. American options can be exercised early, which adds an early-exercise premium to the put (or call with dividends). Dividends also require an adjustment: replace S with S − PV(dividends). Transaction costs, bid-ask spreads, and borrowing costs mean the parity holds within a band, not exactly.
Put-call parity says that the price of a call option minus the price of a put option (same strike and expiry) must equal the stock price minus the present value of the strike. If this relationship is violated, traders can construct a risk-free arbitrage profit by buying the cheap side and selling the expensive side.
A synthetic position uses put-call parity to replicate one instrument using others. A synthetic stock is created by buying a call and selling a put at the same strike: the combined payoff matches owning the stock. A synthetic put is created by buying a call, selling the stock, and lending the present value of the strike. These are used when the direct instrument is illiquid or expensive.
Not exactly. Put-call parity holds precisely only for European options. American options can be exercised early, which adds an early-exercise premium that breaks the exact equality. For American options, put-call parity becomes an inequality: S minus K is less than or equal to C minus P, which is less than or equal to S minus K times the discount factor.
FinLingo covers put-call parity in Level 2 — part of the 7-unit options mechanics module. Level 1 is free.
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