Put-Call Parityintermediate

Put-Call Parity Equation

Put-call parity is a no-arbitrage condition that pins the prices of European call and put options with identical strike and expiry to the current stock price and the present value of the strike. The relation is $C + K e^{-rT} = P + S_0$ , first formalised by Stoll (1969) and extended by Merton (1973). Any deviation from this equality allows a riskless profit, so in liquid markets violations are fleeting. The relation applies strictly to European options. For American options on non-dividend-paying stocks, only an inequality holds.

Why it matters

Consider two portfolios, both priced today and both worth $\max(S_T, K)$ at expiry $T$ . Portfolio A holds a European call plus a zero-coupon bond paying $K$ at $T$ . If $S_T > K$ the call pays off. If not, the bond pays $K$ . Portfolio B holds a European put plus one share. If $S_T < K$ the put pays off, otherwise you keep the share worth $S_T$ . Because the payoffs are identical in every scenario, the two portfolios must cost the same today. If Portfolio A traded cheaper than Portfolio B, you could buy A, sell B, and collect the difference as a riskless profit. Arbitrageurs operating on ASX or any liquid exchange would close that gap immediately.

Formulas

Put-call parity

C + K e^{-rT} = P + S_0

Hull (2022) eq. 11.6, p. 264. Here

C

and

P

are European call and put prices,

K

is the common strike,

r

is the continuously compounded risk-free rate,

T

is time to expiry in years, and

S_0

is the current stock price. The term

K e^{-rT}

is the present value of the strike (the price of a zero-coupon bond maturing at

K

Rearranged (call minus put)

C - P = S_0 - K e^{-rT}

The right-hand side equals the value of a long forward contract on the stock with delivery price

K

and maturity

T

. This makes the forward-parity interpretation explicit: a long call and short put with the same strike and expiry replicates a long forward.

American option inequality (no dividends)

S_0 - K \leq C - P \leq S_0 - K e^{-rT}

Hull (2022) eq. 11.7, p. 264. For American calls

C

and puts

P

on non-dividend-paying stocks, put-call parity becomes a pair of bounds rather than an equality, because early exercise introduces optionality that breaks the exact equivalence.

Worked examples

Scenario

Spot A$50, strike A$50, $r = 4.35\%$ (RBA cash rate proxy, continuously compounded), $T = 1$ year, European call $C$ priced at A$5. What should the European put $P$ be?

Solution

$P = C + K e^{-rT} - S_0 = 5 + 50 \, e^{-0.0435} - 50 = 5 + 50 \times 0.9575 - 50 = 5 + 47.87 - 50 = 2.87$ , i.e. A$2.87. The original entry used $r = 5\%$ , giving $P = 5 + 47.56 - 50 = 2.56$ , which is arithmetically correct. Only the rate assumption changes with the RBA context.

NoteVerify with the rearranged form.

C - P = S_0 - K e^{-rT} = 50 - 47.87 = 2.13

, and from the prices

C - P = 5 - 2.87 = 2.13

✓

Scenario

Parity is violated and the call is overpriced. Spot A$50, strike A$50, $r = 5\%$ , $T = 1$ year, call $C$ priced at A$6 (fair value A$5), put $P$ priced at A$2.56 (fairly priced).

Solution

The left side of the parity equation gives $C + K e^{-rT} = 6 + 47.56 = 53.56$ . The right side gives $P + S_0 = 2.56 + 50 = 52.56$ . The gap is A$1.00. To capture it, sell the call (receive 6), buy the put (pay 2.56), buy the stock (pay 50), and borrow $K e^{-rT} = 47.56$ at 5\% for one year. Net cash flow today $= 6 - 2.56 - 50 + 47.56 = +1.00$ , a riskless profit of A$1.00. At expiry all positions offset regardless of $S_T$ .

Common mistakes

✗It's tempting to apply put-call parity to American options. The equality holds strictly only for European options. For American options on non-dividend-paying stocks, Hull (2022) shows only the inequality $S_0 - K \leq C - P \leq S_0 - K e^{-rT}$ holds.
✗The risk-free rate $r$ is often treated as a single constant. In practice, traders use the OIS rate (in Australia, the RBA cash rate as proxied by overnight index swaps) rather than BBSW, because BBSW carries bank credit risk. For longer maturities, the choice of discount rate materially affects the parity calculation.
✗Parity holds regardless of dividends. It does not. If the stock pays dividends with present value $D$ during $[0, T]$ , the relation becomes $C + K e^{-rT} = P + S_0 - D$ , or equivalently replace $S_0$ with the forward price $F_0 e^{-rT}$ . Ignoring dividends produces a systematic mispricing.

Revision bullets

• $C + K e^{-rT} = P + S_0$ (European only)
•Two portfolios with identical payoffs → identical prices
•Violation yields a riskless arbitrage profit
•American options give an inequality, not equality
•Dividends shift $S_0$ down by PV(dividends)
• $C - P$ equals value of long forward at strike $K$
• $r$ is OIS or risk-free, not a lending rate

Quick check

Put-call parity states that:

Which condition is required for put-call parity $C + K e^{-rT} = P + S_0$ to hold as an equality?

Connected topics

Synth. Stock Synth. Call Parity Arb.

In learning paths

Options Path Exam Revision Path

Sources

Hull (2022), §11.4
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson Education, 2022. ISBN 978-0-13-699518-4.
Primary textbook treatment of put-call parity (eq. 11.6, p. 264) for European options and the American-option inequality (eq. 11.7, p. 264), including the portfolio proof and arbitrage implications.
Stoll (1969)
Stoll, Hans R. 'The Relationship between Put and Call Option Prices.' Journal of Finance 24, no. 5 (December 1969): 801–824.
Seminal paper that first formalised put-call parity as a no-arbitrage pricing relationship; the DOI is https://doi.org/10.1111/j.1540-6261.1969.tb01694.x.
Merton (1973)
Merton, Robert C. 'Theory of Rational Option Pricing.' Bell Journal of Economics and Management Science 4, no. 1 (Spring 1973): 141–183.
Extended Stoll's parity to a general continuous-time framework, derived the American-option inequality, and provided the theoretical scaffolding on which Black-Scholes rests.
ASX — Options Contract Specifications
ASX Limited. 'Equity Options and Index Options Contract Specifications.' Australian Securities Exchange, 2023.
Confirms that ASX equity options are American-style and ASX XJO index options are European-style — directly relevant to which form of put-call parity applies to each product type.
CAIA — 50 Years of Put-Call Parity (2018)
CAIA Association. '50 Years of Put-Call Parity.' Portfolio for the Future Blog, November 2018.
Concise retrospective tracing parity from Stoll (1969) through modern derivatives markets, useful for contextualising the historical development of the concept.

How to cite this page

Dr. Phil's Quant Lab. (2026). Put-Call Parity Equation. Derivatives Atlas. https://phucnguyenvan.com/concept/put-call-parity-equation

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