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Put-Call Parity: A No-Arbitrage Derivation of Option Relationships

Exploring the cinematic intuition of Put-Call Parity: A No-Arbitrage Derivation of Option Relationships.

The Formal Theorem

Consider a non-dividend-paying stock with price St S_t at time t t . Let C(St,K,T) C(S_t, K, T) and P(St,K,T) P(S_t, K, T) be the prices of European call and put options with strike price K K and expiration T T . Under the assumption of a frictionless market and a constant risk-free rate r r , the portfolio of a long call and a short put must replicate the discounted stock price and strike. The relationship is defined as:
Ct−Pt=St−Ke−r(T−t) C_t - P_t = S_t - K e^{-r(T-t)}

Analytical Intuition.

Picture the financial markets as a vast, interconnected ecosystem governed by the iron law of no-arbitrage. Imagine you are a structural engineer of capital. To understand why a call and a put share a rigid parity, we construct two portfolios that yield the identical terminal payoff of ST−K S_T - K at expiration T T . Portfolio A consists of a long call and a short put; Portfolio B consists of the underlying stock St S_t and a zero-coupon bond with face value K K . At time T T , regardless of whether the market crashes or booms, the terminal values are perfectly synchronized. Because the terminal cash flows are deterministic and identical, the law of one price mandates that their present values must be equivalent. If the market prices deviated from this equality, an arbitrageur could instantaneously lock in a risk-free profit by shorting the expensive portfolio and going long on the cheap one. This parity is not merely an equation; it is the fundamental equilibrium constraint that binds options to the underlying asset, revealing that a call is effectively a leveraged stock position funded by debt.
CAUTION

Institutional Warning.

Students often struggle with the sign convention of the put, failing to realize that a short put is a liability. Additionally, confusion arises when attempting to incorporate dividends, where the stock price St S_t must be replaced by St−PV(D) S_t - PV(D) .

Academic Inquiries.

01

Does Put-Call Parity hold for American options?

No. Because American options can be exercised early, the parity becomes an inequality relationship: 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

Why is this considered a 'no-arbitrage' derivation?

It relies on the 'Law of One Price'. If the parity were violated, one could construct a self-financing portfolio with zero initial cost that yields a strictly positive terminal value, which is prohibited in an efficient market.

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). Put-Call Parity: A No-Arbitrage Derivation of Option Relationships: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/advanced-stochastic-processes/put-call-parity--a-no-arbitrage-derivation-of-option-relationships

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