Advanced Financial Management · Derivatives Analysis and Valuation

Put-Call Parity and Option Relationships for CA Final AFM

Q: What is put-call parity in simple words?

It is the link between the price of a European call, a European put, the share and the discounted strike. A call plus cash equal to PV of the strike gives the same payoff as a put plus the share. So they must cost the same.

Q: How do I calculate the put price using put-call parity?

Use P = C − S + PV(X). Discount the strike at the given rate, then add it to the call price and subtract the share price. If there is a dividend, reduce S by the present value of the dividend first.

Q: Does put-call parity work for American options?

Not as an equality. Early exercise adds value, so only an inequality holds. For a non-dividend share, S − X ≤ C − P ≤ S − PV(X).

Q: Why is an American call on a non-dividend share never exercised early?

Exercising early gives only S − X, which is less than the European call's lower bound S − PV(X). Waiting also keeps the downside protection. So the American call is worth the same as the European call.

Updated 5 October 2026 · Fact-checked

Put-call parity links the price of a European call and put with the same strike and expiry: C + X × e^(−rT) = P + S. In exams, use it to find a missing price. If the two sides differ, buy the cheaper package, sell the dearer one and lock in a risk-free profit.

Understand Put-Call Parity and Option Relationships

A call gives you the right to buy at the strike price. A put gives you the right to sell at the strike price. Both depend on the same share, so their prices cannot move independently.

Take two portfolios. Portfolio A: buy one European call and hold cash equal to the present value of the strike. Portfolio B: buy one European put and buy one share. Both must have the same strike X and expiry T. At expiry, if the share price is above X, A is worth the share price (you exercise the call using the cash). B is also worth the share price (the put is useless). If the share price is below X, A is worth X (call lapses, cash grows to X). B is also worth X (you exercise the put and sell the share at X).

Both portfolios give the same payoff in every state, so they must cost the same today. That gives put-call parity: C + PV(X) = P + S. It holds for European options on a share with no dividends. If there is a dividend during the option life, subtract its present value from S, because the option holder does not receive it.

If parity fails, there is arbitrage. Buy the cheaper portfolio, sell the dearer one, and the payoffs cancel at expiry. You keep the price difference as a risk-free gain.

Option prices also have bounds. A call can never be worth more than the share. For a European call on a non-dividend share, the lower bound is C ≥ max(0, S − PV(X)). American options can be exercised early, so they are worth at least as much as the equivalent European option. For American options parity becomes an inequality, not an equality.

Key rules to remember

Put-call parity (European, no dividend): C + X × e^(−rT) = P + S; Same strike, same expiry, same underlying. With discrete compounding use X ÷ (1 + r)^T. Use whichever the question gives.
Put price from parity: P = C − S + X × e^(−rT); Rearranged for the missing put. Check the sign of each term before substituting.
Call price from parity: C = P + S − X × e^(−rT); Use when the put price is given.
Parity with known dividend: C + PV(X) = P + S − PV(D); PV(D) is the present value of dividends paid before expiry. The share price is reduced by it.
European call bounds: max(0, S − PV(X)) ≤ C ≤ S; For a no-dividend share. A price below the lower bound gives arbitrage.
European put bounds: max(0, PV(X) − S) ≤ P ≤ PV(X); The maximum put value is PV(X) because the share price cannot fall below zero.
American call on non-dividend share: C(American) = C(European); Early exercise is never optimal when no dividend is paid, so the two have the same value. Exercising early gives only S − X, which is less than the lower bound S − PV(X).
American option vs European: C(Am) ≥ C(Eu); P(Am) ≥ P(Eu); The right to exercise early cannot reduce value.
American parity inequality (no dividend): S − X ≤ C − P ≤ S − PV(X); Parity is an inequality for American options.

How to solve Put-Call Parity and Option Relationships questions

Use this method for any question on parity, bounds or arbitrage. Work with the same strike and expiry throughout.

1Note the option type (European or American), the strike, the expiry, the interest rate and any dividend.
2Choose the discounting basis the question gives: continuous (e^(−rT)) or discrete (÷ (1 + r)^T). Compute PV(X).
3If a dividend is due before expiry, compute its PV and subtract it from S.
4Compute the left side C + PV(X) and the right side P + S (adjusted). If one price is missing, solve for it.
5If all prices are given, compare the two sides. Equal means no arbitrage. Unequal means arbitrage.
6For arbitrage, buy the cheaper side and sell the dearer side. Write each leg: buy or sell, call or put or share, and borrow or lend PV(X).
7Compute the net cash flow today. It is the arbitrage profit, and the payoffs cancel at expiry. Verify by testing the payoff at one price above X and one below.
8State the conclusion in a line, and mention the assumptions (European, no transaction costs).

Quickest way: Which side is dearer? Two-line check

When to use it: Use it when the question gives all four prices and asks if arbitrage exists or what to do.

Compute A = C + PV(X) and B = P + S.
If A > B: sell the call, borrow PV(X), buy the put, buy the share. If A < B: buy the call, lend PV(X), sell the put, short the share.
Profit today = |A − B| per share. Multiply by the lot size if given.
Spot-check one expiry price. Total should be zero.

Common mistakes in Put-Call Parity and Option Relationships

Using the undiscounted strike price X in the parity equation.
Students remember C + X = P + S from payoff diagrams at expiry.
Fix: Always discount X to today. Compute PV(X) before writing the equation.
Applying parity to American options as an equality.
The formula looks the same in every question.
Fix: Check the option type first. Parity is exact only for European options. For American options use the inequality.
Ignoring the dividend expected before expiry.
Students focus on the headline prices and skip the data about dividends.
Fix: Subtract PV(D) from S before using parity.
Taking the wrong direction in arbitrage, such as buying the dearer side.
Confusion about which side of the equation is which.
Fix: Buy the cheaper package and sell the dearer one. Then check the expiry payoff of the whole position at two prices.
Mixing strikes or expiries between the call and the put.
The question lists several options and the students pick the wrong pair.
Fix: Match the strike and expiry before using parity. Parity does not work across different strikes.
Forgetting that the borrowing or lending leg is part of the arbitrage.
Students list only the option and share trades.
Fix: Include borrowing or lending of PV(X) in every arbitrage table, so the net cash flow ties to the price difference.

Worked examples

Example 1

A European call on the shares of Kavya Ltd, strike ₹100, expiring in one year, trades at ₹12. The share price is ₹95. The risk-free rate is 10% per annum, discrete compounding. No dividend is expected. Find the fair price of the European put with the same strike and expiry.

Show the solution

PV(X) = 100 ÷ 1.10 = ₹90.91.
Parity: C + PV(X) = P + S.
P = C + PV(X) − S = 12 + 90.91 − 95.
P = ₹7.91.

Answer: The fair price of the put is about ₹7.91.

Example 2

A European call and a European put on a share, both with strike ₹50 and expiry one year, are priced at ₹8 and ₹3. The share price is ₹52 and the risk-free rate is 10% per annum, discrete compounding. There is no dividend. Check for arbitrage and state the strategy and profit per share.

Show the solution

PV(X) = 50 ÷ 1.10 = ₹45.45.
A = C + PV(X) = 8 + 45.45 = ₹53.45.
B = P + S = 3 + 52 = ₹55.
B is greater than A, so parity fails. The call package is cheaper.
Buy the call (−₹8), lend ₹45.45 (−₹45.45), sell the put (+₹3), short sell the share (+₹52).
Net cash flow today = −8 − 45.45 + 3 + 52 = +₹1.55.
Check at expiry if the share price is ₹60: call pays 10, lending returns 50, short share costs −60, put sold pays 0. Net = 0.
Check at expiry if the share price is ₹40: call pays 0, lending returns 50, short share costs −40, put sold costs −10. Net = 0.

Answer: Arbitrage exists. Buy the call, lend ₹45.45, sell the put and short the share. The risk-free profit is ₹1.55 per share.

Exam tips

Read the option type first. If the question says American, do not use parity as an equality.
Write PV(X) as a separate line. Examiners award marks for it.
In arbitrage answers, give a table of cash flows today and at expiry. Show that the expiry total is zero.
If the question gives a dividend, adjust S before anything else.
Add one line of interpretation. Say what the mispricing means and which assumptions (no costs, same strike and expiry) are needed.

Practice questions from Derivatives Analysis and Valuation

Put-Call Parity and Option Relationships 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 and Option Relationships: frequently asked questions

What is put-call parity in simple words?

It is the link between the price of a European call, a European put, the share and the discounted strike. A call plus cash equal to PV of the strike gives the same payoff as a put plus the share. So they must cost the same.

How do I calculate the put price using put-call parity?

Use P = C − S + PV(X). Discount the strike at the given rate, then add it to the call price and subtract the share price. If there is a dividend, reduce S by the present value of the dividend first.

Does put-call parity work for American options?

Not as an equality. Early exercise adds value, so only an inequality holds. For a non-dividend share, S − X ≤ C − P ≤ S − PV(X).

Why is an American call on a non-dividend share never exercised early?

Exercising early gives only S − X, which is less than the European call's lower bound S − PV(X). Waiting also keeps the downside protection. So the American call is worth the same as the European call.