CFA Program Level I

Call and put payoffs and put-call parity: CFA Level I

At expiration a long call is worth max(0, S - X) and a long put is worth max(0, X - S); profit is that value minus the premium paid. Put-call parity links a European call and put with the same strike and expiration: c = S + p - X e^(-rT). With zero interest and dividends, stock = call - put + strike.

Payoff and profit at expiration

A call gives the buyer the right to buy at the strike price. A put gives the buyer the right to sell at the strike price. The writer of each has the matching obligation if assigned.

Long call value at expiration = max(0, S - X). Profit = value - premium. Break-even = strike + premium.

Long put value at expiration = max(0, X - S). Profit = value - premium. Break-even = strike - premium.

If the option expires out of the money, it is worthless and the buyer loses the premium paid, which is the buyer's maximum loss.

Worked examples

Call: strike USD50, premium USD2.50. At expiration the stock is USD56. Value = 56 - 50 = USD6.00. Profit = 6.00 - 2.50 = USD3.50 a share. Break-even = USD52.50. If the stock ends at USD48, the call expires worthless and the loss is the USD2.50 premium.

Put: strike USD40, premium USD1.80. At expiration the stock is USD35. Value = 40 - 35 = USD5.00. Profit = 5.00 - 1.80 = USD3.20 a share, or USD320 on a 100-share contract. Break-even = USD38.20.

Investor.gov's example: a December 70 call bought for USD2.20 breaks even at USD72.20, and a December 70 put bought for USD2.20 breaks even at USD67.80.

Put-call parity

The relationship holds for a call and put on the same underlying with the same strike and expiration. It follows from the no-arbitrage principle: a rational price must leave no riskless profit.

With zero interest rates and dividends: stock = call - put + strike. So long call + short put is a synthetic long stock, and long stock + long put has the same profile as a long call.

With interest, for a non-dividend-paying stock: c = S + p - X e^(-r(T-t)), and p = c - S + X e^(-r(T-t)).

Worked example: S = USD47, X = USD50, put = USD4.10, six months to expiration, continuous rate 4%. PV of strike = 50 x e^(-0.04 x 0.5) = 49.01. Call = 47 + 4.10 - 49.01 = USD2.09.

Spotting the arbitrage

Zero rates. Stock USD49, call and put both struck at USD50, call USD1.00, put USD2.00. Synthetic stock = 50 + 1.00 - 2.00 = USD49, equal to the stock price, so parity holds. This is the OIC's example.

If the stock moves to USD49.50 with the options unchanged, the synthetic stock is USD0.50 cheaper. Buy the call, sell the put and short the stock (a reversal) to lock in USD0.50.

If the stock is cheaper than the synthetic, do the reverse: buy the stock, buy the put and sell the call (a conversion). Transaction costs can remove these profits in practice.

Where the exam catches people

Believing the payoff is the profit: profit subtracts the premium paid.
Believing a put's break-even is strike plus premium: for a put it is strike minus premium.
Believing put-call parity applies to options with different strikes or expirations: both options must share strike, expiration and underlying.
Forgetting to discount the strike when interest rates are not zero: parity uses the present value of X, not X.
Believing a long call plus a short put replicates a long put: it replicates a long stock position.

Written from CFA Institute's documentation

Try 3 questions on this

Pick an answer to see why each option is right or wrong.

An investor buys one put option contract on 100 shares of a stock, with a strike price of USD45.00, for a premium of USD3.40 per share. At expiration the stock trades at USD39.50. Ignoring commissions, the investor's profit on the contract is closest to:
Right. In the game that’s Renown for your Derivatives building.
Not that one: B is right. In the game your Derivatives building would lose a little Renown, and this question would come back later.
Your answerA.−USD340A put is in the money when the strike is above the stock price, as here, so it does not expire worthless. The −USD340 loss of the whole premium (3.40 × 100) applies only if the stock ends at or above USD45.00.The answerB.USD210A long put's value at expiration is max(0, X − S) = 45.00 − 39.50 = USD5.50 per share. Profit = value − premium = 5.50 − 3.40 = USD2.10 per share, or 2.10 × 100 = USD210 on the contract.Your answerC.USD550USD550 = (45.00 − 39.50) × 100 is the put's value at expiration. The premium of 3.40 × 100 = USD340 must be subtracted, leaving USD210.
Written from www.investor.gov/introduction-investing/general-resources/news-alerts/alerts-bulletins/investor-bulletins-63
A non-dividend-paying stock trades at USD63.80. European call and put options on the stock, both with a strike price of USD65.00 and the same expiration, trade at USD1.40 and USD2.20, respectively. Assume zero interest rates and no transaction costs. The profit per share that an arbitrage exploiting put-call parity locks in is closest to:
Right. In the game that’s Renown for your Derivatives building.
Not that one: A is right. In the game your Derivatives building would lose a little Renown, and this question would come back later.
The answerA.USD0.40With zero interest and dividends, put-call parity gives stock = call - put + strike. Synthetic long stock (long call, short put) is worth 65.00 + 1.40 - 2.20 = USD64.20, but the stock trades at USD63.80.Your answerB.USD1.20USD1.20 is 65.00 - 63.80, the strike less the stock price. It leaves out the options: the put costs USD2.20 and the call brings in USD1.40, a net debit of USD0.80, so the profit is 1.20 - 0.80 = USD0.40.Your answerC.USD2.00USD2.00 comes from 65.00 - 1.40 + 2.20 - 63.80, which treats synthetic stock as a long put and a short call. Synthetic long stock is a long call and a short put, worth 65.00 + 1.40 - 2.20 = USD64.20, which is USD0.40 above the stock.
Show the full explanation and doc quotes
Buying the stock, buying the put and selling the call (a conversion) costs 63.80 + 2.20 - 1.40 = USD64.60 and delivers USD65.00 at expiration whatever the stock price, so the locked-in profit is 65.00 - 64.60 = USD0.40 per share.
Written from www.optionseducation.org/advancedconcepts/put-call-parity
A non-dividend-paying stock trades at USD52.00. A European put on the stock with a strike price of USD50.00 and nine months to expiration trades at USD3.10. The continuously compounded risk-free rate is 5%. Using put-call parity, the value of a European call with the same strike and expiration is closest to:
Right. In the game that’s Renown for your Derivatives building.
Not that one: C is right. In the game your Derivatives building would lose a little Renown, and this question would come back later.
Your answerA.USD0.74USD0.74 is 52.00 - 3.10 - 48.16, with the put's sign reversed. Parity adds the put to the stock: c = S + p - Xe^(-r(T-t)) = 52.00 + 3.10 - 48.16 = USD6.94.Your answerB.USD5.10USD5.10 is 52.00 + 3.10 - 50.00, which uses the undiscounted strike. The strike is paid at expiration in nine months, so it enters at its present value, 50.00 x e^(-0.0375) = USD48.16.The answerC.USD6.94Put-call parity for a non-dividend-paying stock: c = S + p - Xe^(-r(T-t)). The present value of the strike is 50.00 x e^(-0.05 x 0.75) = 50.00 x 0.96319 = USD48.16. So c = 52.00 + 3.10 - 48.16 = USD6.94.
Written from www.optionseducation.org/advancedconcepts/put-call-parity