Economic Modelling · Principles of option pricing

Put-Call Parity: Derivation, Dividends and Arbitrage

Updated 11 October 2026 · Fact-checked

Put-call parity links the prices of a European call and put with the same strike K and expiry T on the same underlying. Without dividends: C − P = S₀ − K e^(−rT). Solve by building two portfolios with equal payoffs at T. If prices break the equation, buy the cheap side, sell the expensive side and lock in a risk-free profit.

Understand Put-Call Parity

A European call pays max(S_T − K, 0) at time T. A European put pays max(K − S_T, 0). Both have the same strike K and expiry T. Put-call parity says their prices are tied together. You cannot price them independently.

The idea is simple. Compare two portfolios. Portfolio A: one call plus cash of K e^(−rT), which grows to K at time T. Portfolio B: one put plus one share. At time T, if S_T > K, A is worth (S_T − K) + K = S_T. B is worth 0 + S_T = S_T. If S_T ≤ K, A is worth 0 + K = K. B is worth (K − S_T) + S_T = K. So both are worth max(S_T, K) in every state.

Two portfolios with the same payoff in every state at T must have the same price today, otherwise there is an arbitrage. This is the no-arbitrage principle. So C + K e^(−rT) = P + S₀. No model of share prices is needed, and no volatility assumption. This is why parity is exact, unlike Black-Scholes.

With dividends, the holder of the share in the put-plus-share portfolio receives the dividends paid before T, and the call-side portfolio does not. So the share is worth S₀ − D for comparison with the call portfolio, where D is the present value of dividends expected during the option's life. For a continuous dividend yield q, you replace S₀ by S₀ e^(−qT).

Parity is for European options. For American options it gives only inequalities, because early exercise adds value.

Key rules to remember

Put-call parity, no dividends: C + K e^(−rT) = P + S₀; European options, same K and T, same underlying. r is the continuously compounded risk-free rate.
Rearranged form: C − P = S₀ − K e^(−rT); Use this to find a missing call or put price.
Discrete dividends: C + K e^(−rT) = P + S₀ − D; D is the present value at time 0 of dividends paid before T, discounted at the risk-free rate.
Continuous dividend yield q: C + K e^(−rT) = P + S₀ e^(−qT); q is the continuously compounded dividend yield.
Annual effective rate form: C − P = S₀ − K (1 + i)^(−T); Use when the rate is given as an effective annual rate i. Same as e^(−rT) with r = ln(1 + i).

How to solve Put-Call Parity questions

Use this method for any parity question: find a price, test for arbitrage, or derive the relation.

1Check the options are European with the same strike, expiry and underlying. If not, parity gives only bounds.
2Write down S₀, K, T, r and any dividend details. Convert the rate to the form the question uses (continuous or effective).
3Find the present value of dividends D paid before T, or the yield q. Adjust the share price: S₀ − D or S₀ e^(−qT).
4Write parity: C − P = adjusted S₀ − K e^(−rT). Substitute the numbers.
5If finding a price, solve for the unknown. If testing arbitrage, compare the two sides.
6For arbitrage, buy the cheaper portfolio and sell the dearer one. Compute the cash flow at time 0 and show the positions cancel at T.
7State the profit at time 0 (or its value at T) and check the sign.

Quickest way: Compare the two sides and read off the trade

When to use it: Use in multiple-choice questions and when a written question asks only for the arbitrage direction.

Compute L = C + K e^(−rT) and R = P + adjusted S₀.
If L > R, the call side is too dear: sell the call, borrow K e^(−rT), buy the put and the share. If dividends are paid, borrow an extra D as well, because the dividends repay it. Profit at time 0 is L − R.
If R > L, reverse: buy the call, lend K e^(−rT) (plus D if dividends are paid), sell the put and short the share, paying out the dividends. Profit at time 0 is R − L.
The arbitrage profit at time 0 is |L − R|.
Sanity check: the profit must be positive and the final payoffs must cancel.

Common mistakes in Put-Call Parity

Using parity for American options as an equality.
The formula looks general, and the question may not stress the exercise style.
Fix: Check the exercise style first. Parity is an equality only for European options. American options give inequalities.
Forgetting to discount the strike.
Students write C − P = S₀ − K.
Fix: Always use K e^(−rT) or K(1 + i)^(−T). The strike is paid at T, not today.
Subtracting dividends at face value instead of present value.
Students copy the dividend amount straight from the question.
Fix: Discount each dividend from its payment date to time 0 at the risk-free rate, then subtract the total from S₀.
Mixing up the sign of the arbitrage trade.
It is easy to buy the expensive side by accident.
Fix: Work out which side is cheaper. Buy that side, sell the other, and check that the final payoffs cancel.
Using a wrong rate convention.
The question gives an effective annual rate but the student uses e^(−rT) with the same number.
Fix: If the rate is effective, use (1 + i)^(−T). If it is a force of interest or continuous rate, use e^(−rT).

Worked examples

Example 1

A share trades at ₹500. A 1-year European call with strike ₹520 costs ₹45. The continuously compounded risk-free rate is 6% per year and no dividends are paid. Find the price of the European put with the same strike and expiry.

Show the solution

Parity: C − P = S₀ − K e^(−rT).
K e^(−rT) = 520 × e^(−0.06) = 520 × 0.941765 = 489.718.
S₀ − K e^(−rT) = 500 − 489.718 = 10.282.
P = C − 10.282 = 45 − 10.282 = 34.718.

Answer: The put price is about ₹34.72.

Example 2

A share trades at ₹200 and pays a dividend of ₹10 in 6 months. European options expire in 1 year with strike ₹210. The call costs ₹12 and the put costs ₹20. The continuously compounded risk-free rate is 5% per year. Is there an arbitrage? If so, describe it.

Show the solution

PV of dividend: D = 10 × e^(−0.05 × 0.5) = 10 × 0.975310 = 9.7531.
Adjusted share price: 200 − 9.7531 = 190.2469.
K e^(−rT) = 210 × e^(−0.05) = 210 × 0.951229 = 199.758.
Left side: C + K e^(−rT) = 12 + 199.758 = 211.758.
Right side: P + adjusted S₀ = 20 + 190.2469 = 210.247.
Left > right by 1.511, so the call side is too dear.
Trade at time 0: sell the call (+12), buy the put (−20), buy the share (−200), and borrow 199.758 + 9.7531 = 209.511, which is the PV of the strike plus the PV of the dividend.
Net cash at time 0 = 12 − 20 − 200 + 209.511 = +1.511.
At 6 months the dividend of 10 repays part of the loan. At T, the put and share together are worth max(S_T, K), and the short call costs max(S_T − K, 0), so these net to K = 210. The loan balance at T is also 210, so the final position is zero in every state and the 1.511 is locked in.

Answer: Yes. The call side exceeds the put side by about ₹1.51. Sell the call, buy the put and the share, and borrow ₹209.51 at the risk-free rate. This gives a net inflow of about ₹1.51 at time 0 and a zero payoff at T.

Exam tips

Write the parity equation first and then substitute. Examiners give marks for the correct relation even if the arithmetic slips.
In derivation questions, show both portfolios and their payoffs in the two cases S_T > K and S_T ≤ K.
State that parity needs European options and no-arbitrage, and say that it needs no model for the share price.
For dividend questions, show the discounting of each dividend separately and say whether the rate is continuous.
In arbitrage questions, give the trades, the cash flow at time 0 and a line showing the payoffs cancel at T.

Practice questions from Principles of option pricing

Put-Call Parity in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Put-Call Parity: frequently asked questions

What is the put-call parity formula?

For European options with the same strike K and expiry T on a non-dividend share, C + K e^(−rT) = P + S₀. Here r is the continuously compounded risk-free rate. It follows from the no-arbitrage principle.

How do you change put-call parity for dividends?

Subtract the present value of dividends paid before expiry from S₀. For a continuous yield q, use S₀ e^(−qT) instead of S₀. The strike term stays the same.

Does put-call parity hold for American options?

Not as an equality. Early exercise adds value, so you only get bounds on C − P. The exact relation holds only for European options.

Do I need Black-Scholes to use put-call parity?

No. Parity needs only no-arbitrage and the payoffs of the options. It holds whatever model the share price follows.