Put-call parity: the no-arbitrage relationship that anchors option pricing
The price of a call, the price of a put, the price of the underlying, and the price of a zero-coupon bond on the same expiry are not independent. Put-call parity is the no-arbitrage relationship that ties them together, and it is the foundation on which option pricing is built.
What put-call parity is
For European options on a non-dividend-paying underlying, put-call parity states that buying a call and selling a put with the same strike and expiry is equivalent to holding the underlying minus a bond that pays the strike at expiry. Algebraically: C − P = S − K · e^(−rT), where C is the call price, P is the put price, S is the spot price, K the strike, r the risk-free rate, and T the time to expiry.
The relationship was formalised by Stoll (1969). It does not require any assumption about the distribution of the underlying or about volatility—it follows entirely from the absence of arbitrage. Whatever model is used to price the call, the put price is then pinned by the parity equation.
How it works
The intuition is portfolio replication. A long call plus a short put with the same strike has exactly the same payoff at expiry as holding the underlying and shorting a zero-coupon bond worth the strike: in both cases, the holder ends up long the underlying with a fixed cash outflow of the strike. If the two portfolios have different prices today, an arbitrageur can buy the cheaper one and sell the more expensive one, locking in a riskless profit.
For dividend-paying underlyings, the parity adjusts for the present value of dividends: C − P = S − D − K · e^(−rT), where D is the present value of dividends paid before expiry. For American options, parity becomes an inequality rather than an equality because of early-exercise optionality, but it still bounds the relationship tightly.
What the evidence shows
Empirical tests of put-call parity in liquid markets (Klemkosky & Resnick, 1979; Kamara & Miller, 1995) find that violations are typically small—within transaction costs—and rarely persistent. When parity does break down, it is usually due to friction: shorting costs on the underlying, dividend uncertainty, or borrowing constraints. In illiquid options or stressed markets, larger violations are observed but they typically arise alongside other market dislocations.
The relationship is so tight that practitioners use it routinely to imply unobservable quantities. The implied dividend yield in equity options markets is computed from put-call parity. The synthetic short stock position (long put, short call) is constructed via parity. Convertible-bond arbitrage (covered separately) relies on parity holding to extract the embedded option value.
Limitations and trade-offs
Put-call parity assumes frictionless markets: no transaction costs, no shorting costs, free borrowing and lending at the risk-free rate, and no exercise restrictions. Each assumption is violated in practice. Bid-ask spreads on options can swallow apparent parity violations entirely. Borrow costs on hard-to-borrow stocks introduce a wedge that can persist. American-style options on dividend-paying stocks can be optimally exercised early, breaking the equality.
The relationship is also a tautology in a sense: it does not tell anyone what the call should cost, only that the put price is determined once the call price is set (or vice versa). The pricing problem is pushed onto the model that determines either the call or the put—Black-Scholes, binomial, Monte Carlo, or another framework.
Put-call parity in pfolio
Options are not currently part of pfolio's investable universe; put-call parity is therefore relevant for investors who use options separately through their broker. The platform's analytics describe positions in the underlying instruments rather than option-specific risk and return.
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