Put-Call Parity

A European call minus a put with the same strike replicates a forward, forcing a fixed algebraic relationship between call and put prices.

1. The idea

Consider a portfolio long one European call and short one European put, both with strike $K$ and maturity $T$ on the same underlying. At expiry, the payoff is $max (S_{T} - K, 0) - max (K - S_{T}, 0) = S_{T} - K$ regardless of the stock price — identical to a forward contract to buy the stock at $K$ .

By no-arbitrage, this portfolio must be priced identically to the replicating forward position: today's cost of the forward payoff $S_{T} - K$ is $S_{0} - K e^{- r T}$ . This gives put-call parity: $C - P = S_{0} - K e^{- r T}$ .

Put-call parity holds only for European options on the same underlying, strike and maturity (no early exercise), and must be adjusted for dividends by replacing $S_{0}$ with $S_{0} - P V (dividends)$ or using a dividend yield.

2. Why it matters

Parity lets you infer any one of call price, put price, or implied forward from the other two, and violations are a direct arbitrage signal tested frequently in interviews.

3. Core mathematics

Put-call parity

C - P = S_{0} - K e^{- r T}

With dividend yield q

C - P = S_{0} e^{- q T} - K e^{- r T}

Synthetic forward

Long call + short put \equiv Long forward

4. How to recognize it

Think of this when…

Given call and put prices on the same strike/maturity, asked to find one missing quantity
'Is there an arbitrage between this call and put price' style questions
Constructing a synthetic stock, forward, or bond from options

5. Worked example

Worked example

A stock trades at $40. A 1-year call with strike $40 costs $5. The risk-free rate is 5% continuous. Find the price of the corresponding put.

1
Write put-call parity.
$C - P = S_{0} - K e^{- r T}$
2
Compute the right side.
$40 - 40 e^{- 0.05} = 40 - 38.05 = 1.95$
3
Solve for P.
$P = C - 1.95 = 5 - 1.95 = 3.05$
Answer: Put price ≈ $3.05

6. Common mistakes

Applying parity to American options, where early exercise breaks the exact equality
Forgetting to discount the strike price $K$ to present value
Mixing up the sign convention (writing $P - C$ instead of $C - P$ )

7. Quick check

Put-call parity states:

Put-call parity as stated requires the options to be:

8. Practice

4 QuantSolve questions use this concept.

Practice this concept

9. Related methods

No-Arbitrage Replication

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