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The No-Arbitrage Derivation of Put-Call Parity

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The Formal Theorem

Consider a frictionless market with a non-dividend-paying stock price St S_t at time t t , a European call option Ct C_t and a European put option Pt P_t with identical strike price K K and maturity T T . Assuming a constant risk-free rate r r and no arbitrage opportunities, the relationship between the premiums is given by:
Ct−Pt=St−Ke−r(T−t) C_t - P_t = S_t - K e^{-r(T-t)}

Analytical Intuition.

Imagine you are standing at the threshold of two divergent financial realities at maturity T T . In the first, you hold a synthetic portfolio consisting of a long position in a call option Ct C_t and a short position in a put option Pt P_t . In the second, you hold the underlying stock St S_t financed by borrowing the present value of the strike price Ke−r(T−t) K e^{-r(T-t)} . As the clock strikes T T , both strategies converge with mathematical certainty to the identical payoff of ST−K S_T - K . Because these portfolios are clones, the Law of One Price dictates that they must command the same entry price today. If the prices were to deviate, a sophisticated arbitrageur would immediately purchase the cheaper portfolio and sell the expensive one, locking in a risk-free profit with zero net investment. Thus, the equilibrium state—the Put-Call Parity—is not merely a suggestion, but a necessary condition for a market devoid of free lunches.
CAUTION

Institutional Warning.

Students often struggle to see why the portfolios are identical at T T . They focus on the path of St S_t , whereas the proof relies entirely on the terminal payoff structure, rendering the stochastic volatility of the underlying irrelevant to the static parity relationship.

Academic Inquiries.

01

Does Put-Call Parity hold for American options?

No. Due to the early exercise feature, the parity becomes an inequality: St−K≤Ct−Pt≤St−Ke−r(T−t) S_t - K \leq C_t - P_t \leq S_t - K e^{-r(T-t)} .

02

How do dividends affect the parity?

If the stock pays a continuous dividend yield q q , the stock price is replaced by Ste−q(T−t) S_t e^{-q(T-t)} , modifying the equation to Ct−Pt=Ste−q(T−t)−Ke−r(T−t) C_t - P_t = S_t e^{-q(T-t)} - K e^{-r(T-t)} .

Standardized References.

  • Definitive Institutional SourceHull, J. C., Options, Futures, and Other Derivatives.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). The No-Arbitrage Derivation of Put-Call Parity: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/advanced-stochastic-processes/the-no-arbitrage-derivation-of-put-call-parity

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