Strategic Financial Management · Options

Put-Call Parity Formula and Arbitrage for CMA Final

Updated 11 October 2026 · Fact-checked

Put-call parity links a European call and put with the same strike and expiry on the same asset. Formula: C + PV(X) = P + S. If the two sides differ, buy the cheaper side, sell the dearer side, and lock in a risk-free profit equal to the gap.

Understand Put-Call Parity

Start with two portfolios. Portfolio A holds one European call and cash equal to the present value of the strike price. Portfolio B holds one European put and one share. Both have the same strike X and the same expiry date T.

Check their value at expiry. If the share price ST is above X, the call pays ST − X. The cash has grown to X. Portfolio A is worth ST. Portfolio B holds the share worth ST, and the put is worthless. Both equal ST.

If ST is below X, the call is worthless and the cash is X, so A is worth X. In B, the put pays X − ST and the share is worth ST, so B is worth X. Again both are equal.

Two portfolios with the same payoff in every state must cost the same today. That is the put-call parity theorem: C + PV(X) = P + S. This is the proof you write in the exam: build the two portfolios, compare payoffs in both cases, conclude.

The rule holds only for European options on the same asset, with the same strike and expiry, in a market without frictions. If it breaks, you can arbitrage. Any mispricing can be exploited by buying the cheaper portfolio and selling the dearer one.

Key rules to remember

Put-call parity (no dividend): C + X ÷ (1 + r)^T = P + S; European options, same strike X and expiry T. Use X × e^(−rT) if the question gives a continuous rate.
Put-call parity with known dividends: C + PV(X) + PV(D) = P + S; PV(D) is the present value of dividends expected before expiry. Alternatively, use S − PV(D) in place of S.
Call from put: C = P + S − PV(X); Use it to find the fair price of a call when the put is known.
Put from call: P = C − S + PV(X); Use it to find the fair price of a put when the call is known.
Arbitrage gap: Profit today = |(C + PV(X)) − (P + S)|; Per unit of the asset. Multiply by the lot size.

How to solve Put-Call Parity questions

Use the same routine for every parity question, whether it asks for a missing price or for an arbitrage.

1Confirm the options are European with the same strike and expiry on the same asset.
2List C, P, S, X, r, T and any dividends before expiry with their timing.
3Compute PV(X) using the given rate. Compute PV(D) if dividends exist.
4Work out the left side C + PV(X) and the right side P + S (adjusted for dividends).
5If the question asks for a missing price, solve the equation. Stop there.
6If the sides differ, buy the cheaper portfolio and sell the dearer one. Show each leg and its cash flow.
7Add the cash flows today. This is the arbitrage profit, and the position is risk free because the expiry payoffs cancel.
8State the conclusion and, if asked, the profit for the whole lot.

Quickest way: Compare the two sides and read the trade

When to use it: Use this when the question gives all prices and asks whether an arbitrage exists, and what to do.

Compute L = C + PV(X) and R = P + S.
If L > R, the call side is dearer: sell the call, sell PV(X) worth of bonds (borrow), buy the put, buy the share.
If R > L, the put side is dearer: buy the call, lend PV(X), sell the put, short the share.
Profit today equals the difference L − R, in absolute terms.
Skip the expiry payoff table unless the question asks for it.

Common mistakes in Put-Call Parity

Using X instead of PV(X).
Students remember the payoff at expiry and forget that today's prices need discounting.
Fix: Always discount the strike for the time to expiry before comparing with today's prices.
Ignoring dividends paid before expiry.
The basic formula is memorised without its condition.
Fix: Read the question for dividends. Subtract PV(D) from S, or add PV(D) to the call side.
Taking the trade in the wrong direction.
Students are not sure which side is dearer.
Fix: Sell whichever portfolio is dearer and buy the cheaper one. Check that the net cash today is positive.
Applying parity to American options.
The formula looks general.
Fix: The exact equality holds for European options. Mention this condition in your answer.
Mixing strikes or expiries between the call and put.
Data in case questions is spread across tables.
Fix: Tick that both options share the same X and T before you start.
Forgetting the lot size in the profit.
Per-share working feels complete.
Fix: Multiply the per-share gap by the number of shares in the lot or contract.

Worked examples

Example 1

Shares of an Indian company trade at ₹100. A 1-year European call with strike ₹105 costs ₹8 and a European put with the same strike costs ₹9. The risk-free rate is 10% per year, with annual compounding. No dividends. Check whether put-call parity holds and find the arbitrage profit per share, if any.

Show the solution

PV(X) = 105 ÷ 1.10 = ₹95.4545.
Left side: C + PV(X) = 8 + 95.4545 = ₹103.4545.
Right side: P + S = 9 + 100 = ₹109.
Right side is higher by 109 − 103.4545 = ₹5.5455, so parity fails.
The put side is dearer. Sell the put and the share (short), buy the call, and lend ₹95.4545.
Cash today: +9 + 100 − 8 − 95.4545 = +₹5.5455.
At expiry, the lending returns ₹105. The call and put with short share combine to give a payoff of −105 in both price outcomes, which the ₹105 repays, so the net is zero.

Answer: Parity does not hold. Arbitrage profit is about ₹5.55 per share today, risk free.

Example 2

A stock is at ₹200. A 6-month European call with strike ₹210 is priced at ₹12. The risk-free rate is 8% per year (simple discounting for 6 months, i.e., PV = X ÷ (1 + 0.08 × 0.5)). A dividend of ₹4 is expected in 3 months, and you may ignore its discounting. Find the no-arbitrage price of the put with the same strike and expiry.

Show the solution

PV(X) = 210 ÷ 1.04 = ₹201.9231.
PV(D) = ₹4 as instructed.
Parity with dividends: C + PV(X) + PV(D) = P + S.
Left side = 12 + 201.9231 + 4 = ₹217.9231.
P = 217.9231 − 200 = ₹17.9231.

Answer: The put should be priced at about ₹17.92.

Exam tips

Write the condition (European, same strike and expiry) in your first line. Examiners look for it.
Show each arbitrage leg with its cash flow sign. Marks go to the full trade, not just the gap.
Check the dividend line in the question before you begin. Missing it changes the answer.
For MCQs, compute C + PV(X) and P + S and compare. Do not do the full trade.
In a proof question, present both portfolios and both expiry cases, in a small clean layout.

Practice questions from Options

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 and expiry, C + PV(X) = P + S. Here C is the call price, P the put price, S the spot price and PV(X) the present value of the strike.

How do I find arbitrage using put-call parity?

Compute both sides. Buy the cheaper portfolio and sell the dearer one. The difference is the risk-free profit today, and the positions offset at expiry.

How does put-call parity change with dividends?

Dividends paid before expiry lower the share price on ex-date. So you subtract PV of dividends from S, or add it to the call side: C + PV(X) + PV(D) = P + S.

Does put-call parity work for American options?

Not as an exact equality. It is derived for European options. For American options only an inequality range holds, so state the European condition in exams.