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FRM Exam Part I · Properties of Options

Put-Call Parity Formula and Arbitrage for FRM Part 1

Updated 11 October 2026 · Fact-checked

Put-call parity links a European call, a European put with the same strike and expiry, the underlying and the present value of the strike: c + K e^(−rT) = p + S0. If prices break it, you can lock in risk-free profit by buying the cheap side and selling the expensive side.

Understand Put-Call Parity

A European call and put on the same stock, with the same strike K and expiry T, are linked by a no-arbitrage rule. You can see why with two portfolios.

Portfolio A: one European call plus cash equal to the present value of the strike, K e^(−rT). Portfolio B: one European put plus one share. At expiry, both are worth the same. If the stock ends above K, A exercises the call and pays K with the cash, so it holds one share. B holds the share and the put expires worthless. If the stock ends below K, the call expires worthless and A holds K in cash. B exercises the put and gets K. Same payoff in every state, so same price today.

That gives c + K e^(−rT) = p + S0. Rearrange it to price a call from a put or the reverse. It holds only for European options, because the proof needs both options to be exercised only at expiry.

If the stock pays dividends, the share in Portfolio B is worth less in the future by the dividends you give up. You subtract the present value D of the dividends during the option's life from S0. That gives c + K e^(−rT) = p + S0 − D, which you can also write as c + D + K e^(−rT) = p + S0. Both forms say the same thing. For a continuous dividend yield q, replace S0 with S0 e^(−qT).

American options can be exercised early, so parity becomes a pair of bounds, not an equality. For a non-dividend-paying stock: S0 − K ≤ C − P ≤ S0 − K e^(−rT).

Key formulas to remember

Put-call parity (European, no dividends)
c + K e^(−rT) = p + S0
Same strike K and expiry T. Use continuous compounding unless the question gives another convention.
Parity with discrete dividends
c + K e^(−rT) = p + S0 − D (equivalently c + D + K e^(−rT) = p + S0)
D is the present value of dividends paid during the option's life. The two forms are the same equation. This page uses the first form: subtract D from S0.
Parity with continuous dividend yield q
c + K e^(−rT) = p + S0 e^(−qT)
Use when the question gives a yield, such as an index.
American options, no dividends
S0 − K ≤ C − P ≤ S0 − K e^(−rT)
Bounds, not an equality. C and P are American prices.
American options, with dividends
S0 − D − K ≤ C − P ≤ S0 − K e^(−rT)
D is the present value of dividends during the life.
Synthetic positions
c = p + S0 − K e^(−rT); p = c − S0 + K e^(−rT)
Long call = long put + long stock + borrowing PV of K. Use to build synthetics.

How to solve Put-Call Parity questions

Use this routine for any put-call parity question, whether it asks for a missing price, an arbitrage check or a synthetic position.

  1. 1Check the option type. If both are European, use equality. If American, use the bounds.
  2. 2Confirm same strike and same expiry for the call and the put.
  3. 3Find the present value of the strike, K e^(−rT), using the given rate and time in years.
  4. 4Adjust the stock leg: subtract PV of dividends, or use S0 e^(−qT) for a yield.
  5. 5Write both sides: c + PV(K) on the left, p + S0 (adjusted) on the right.
  6. 6Solve for the missing value, or compare the two sides if checking for arbitrage.
  7. 7If sides differ, buy the cheaper portfolio and sell the dearer one, then state the profit as the difference at time 0.
  8. 8Sanity-check: the answer should be positive and within basic option bounds.

Quickest way: Left-minus-right check

When to use it: Use when the question gives all four prices and asks whether arbitrage exists or which trade to make.

  1. Compute c + PV(K) and p + S0 (adjusted for dividends).
  2. Whichever is larger is overpriced. Sell it and buy the other.
  3. The gap between them is the profit today, per share.
  4. To exploit c + PV(K) < p + S0, buy the call, lend PV(K), sell the put and short the stock. In the reverse case, do the opposite: sell the call, borrow PV(K), buy the put and buy the stock.
  5. For a missing price, just rearrange the equality. Do not rebuild the proof.

Common mistakes in Put-Call Parity

  • Using K instead of K e^(−rT).

    Students remember the payoff at expiry and forget parity compares values today.

    Fix: Always discount the strike to time 0 before adding it to the call.

  • Applying the equality to American options.

    The formula looks the same, and the question may drop the word European.

    Fix: Check the style. For American options use the bounds only.

  • Forgetting to subtract dividends from the stock price.

    Basic version has no dividend term.

    Fix: Scan the question for dividends or yield and adjust S0 before anything else.

  • Using the wrong time in years.

    Expiry is quoted in months, such as 6 months.

    Fix: Convert to years first: 6 months = 0.5.

  • Placing the arbitrage trade in the wrong direction.

    Students do not tie each side to a portfolio.

    Fix: Sell the portfolio with the higher value and buy the lower one. Then list every leg.

  • Mixing American and European dividend bounds.

    Both bounds look similar.

    Fix: Remember the upper bound is S0 − K e^(−rT) in both cases; the dividend only changes the lower bound, which becomes S0 − D − K.

Worked examples

Example 1

A European call on a non-dividend stock trades at $3.00. The stock is $50, the strike is $52, expiry is 6 months and the continuously compounded risk-free rate is 4%. What is the price of the European put with the same strike and expiry?

Show the solution
  1. Use c + K e^(−rT) = p + S0, so p = c + K e^(−rT) − S0.
  2. T = 6 months = 0.5 years. rT = 0.04 × 0.5 = 0.02.
  3. e^(−0.02) = 0.980199, so PV of strike = 52 × 0.980199 = 50.9703.
  4. p = 3.00 + 50.9703 − 50.00 = 3.9703.

Answer: The put is worth about $3.97.

Example 2

A stock is $80. A European call with strike $80 and 1 year to expiry trades at $6.00. The stock pays one dividend of $2 in 6 months. The continuously compounded rate is 5%. What is the price of the European put with the same strike and expiry?

Show the solution
  1. With discrete dividends, c + K e^(−rT) = p + S0 − D, so p = c + K e^(−rT) − S0 + D.
  2. PV of dividend: D = 2 × e^(−0.05 × 0.5) = 2 × e^(−0.025) = 2 × 0.975310 = 1.9506.
  3. PV of strike: 80 × e^(−0.05) = 80 × 0.951229 = 76.0983.
  4. p = 6.00 + 76.0983 − 80.00 + 1.9506 = 4.0489.

Answer: The put is worth about $4.05.

Example 3

A non-dividend stock is $40. A 3-month European call with strike $40 trades at $3.00 and the matching European put trades at $2.00. The continuously compounded rate is 6%. Is there an arbitrage? If so, what trade locks in a profit and how much per share?

Show the solution
  1. T = 3 months = 0.25 years. rT = 0.06 × 0.25 = 0.015.
  2. PV of strike = 40 × e^(−0.015) = 40 × 0.985112 = 39.4045.
  3. Left side: c + PV(K) = 3.00 + 39.4045 = 42.4045.
  4. Right side: p + S0 = 2.00 + 40.00 = 42.00.
  5. The sides differ, so there is arbitrage. The left side (call plus cash) is larger, so it is overpriced.
  6. Trade: sell the call (+3.00), borrow 39.4045 for 3 months, buy the put (−2.00) and buy the stock (−40.00).
  7. Cash at time 0: 3.00 + 39.4045 − 2.00 − 40.00 = 0.4045.
  8. Check at expiry: repay the loan of $40. If the stock is above 40, the call is exercised against you, you deliver the share and receive 40, which repays the loan. If the stock is below 40, you exercise the put, sell the share for 40 and repay the loan. Net at expiry is zero either way.

Answer: Yes. Sell the call, borrow the PV of the strike, buy the put and buy the stock. The risk-free profit is about $0.40 per share today.

Exam tips

  • Read the first line for the option style. If it says American, parity is only a pair of bounds.
  • Convert months to years before you do anything else.
  • Compute K e^(−rT) once and reuse it. A financial calculator with an e^x key (use +/− on the exponent) saves time.
  • Adjust S0 for dividends before you write the equation. Subtract the PV of discrete dividends, or use S0 e^(−qT) for a yield.
  • Sell the larger side and buy the smaller side. Then write every leg, including the borrowing or lending.
  • The arbitrage profit today is the gap between the two sides. Check that your answer is a sensible size, usually small relative to the stock price.
  • If an answer option mixes up c and p or uses K undiscounted, discard it quickly.

Practice questions from Properties of 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 FRM Part I?

For European options on a non-dividend stock it is c + K e^(−rT) = p + S0. Same strike and expiry apply to both options. It follows from two portfolios that pay the same at expiry.

How do I use put-call parity to find arbitrage?

Compute c + PV(K) and p + S0, adjusted for dividends. Sell the larger side and buy the smaller. The difference is your profit today, per share.

How does put-call parity change with dividends?

Subtract the present value of dividends from the stock price. With discrete dividends the equation is c + K e^(−rT) = p + S0 − D, which can also be written c + D + K e^(−rT) = p + S0. With a yield q, use S0 e^(−qT).

Does put-call parity hold for American options?

Not as an equality. You get bounds: S0 − K ≤ C − P ≤ S0 − K e^(−rT) with no dividends. With dividends the lower bound becomes S0 − D − K.