Lesson 4 of 6 6–9% of the exam

By Daniil Tarakanov Updated Oct 20, 20262027 curriculum

Call and put options: payoff and profit at expiration

A call pays max(0, S_T − X) and a put pays max(0, X − S_T) to the buyer at expiration. Profit subtracts the premium paid. The seller's payoff and profit are the exact opposite, so the buyer's loss is capped at the premium and the seller's gain is capped at the premium.

8 min cram, 32 min full read. Based on the 2027 CFA® Program Level I curriculum, Derivatives, Learning Module 2. Also part of the 2026 curriculum, so it applies to the November 2026 exam too.

Tailor this to you: take the self-assessment

Before this: Forward contracts: payoff for the long and the short

On the exam

Define forward contracts, futures contracts, swaps, options (calls and puts), and credit derivatives and compare their basic characteristics
Determine the value at expiration and profit from a long or a short position in a call or put option

Define calls and puts and determine the value at expiration and the profit of a long or short position. Expect a numeric item nearly every sitting: profit vs payoff, the seller's sign, a breakeven, or the maximum gain or loss.

How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.

You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.

What are calls and puts?

A contingent claim lets one side decide whether the trade happens. The most common contingent claim is the option, and the buyer of an option has the right, but not the obligation, to transact at a price agreed today. The seller must perform if the buyer chooses.

A call option gives the right to buy the underlying.
A put option gives the right to sell it.
The buyer (long, holder) pays an option premium upfront: c₀ for a call, p₀ for a put. The seller (short, writer) receives it.
The price at which the underlying may be bought or sold is the exercise price , X, also called the strike price.

Using the right is called exercising. A buyer exercises only if that gives a positive payoff. If not, the option expires worthless and the buyer loses the premium, which is the most the buyer can lose.

The four option positions

Optionsthe buyer chooses; the seller must perform

Call (right to buy at X) Memorize
- Long call (buyer) Memorize pays c₀; gains when the underlying rises; loss capped at c₀
- Short call (seller) Memorize receives c₀; loses when the underlying rises; loss unlimited
Put (right to sell at X) Memorize
- Long put (buyer) Memorize pays p₀; gains when the underlying falls; loss capped at p₀
- Short put (seller) Memorize receives p₀; loses when the underlying falls; loss at most X − p₀

European options can be exercised only at expiration. American options can be exercised at any time up to expiration. The names refer to when, not where. The calculations in this lesson are for European options.

Only the buyer faces counterparty credit risk. Once the premium has been paid, the seller is owed nothing more, but the buyer depends on the seller to pay at expiration.

When is an option in the money?

Compare the underlying price S with the exercise price. If exercising now would give a positive payoff, the option is in-the-money . If it would give nothing, the option is out-of-the-money . When S equals X exactly, it is at-the-money , and exercising gives nothing either.

Moneyness: exercise now, would it pay?

Moneyness at time tcompare the underlying price S with the exercise price X

In the money Memorize call: S above X. Put: S below X. Positive exercise value
At the money Use it S equals X. Zero exercise value
Out of the money Memorize call: S below X. Put: S above X. Zero exercise value

A call is in the money when S is above X. A put is in the money when S is below X. Before expiration an option can still be worth more than its exercise value, because there is time for the price to move: that extra is the time value, and it falls to zero by expiration.

How do you find the payoff and profit of a call?

The payoff is what the option is worth at expiration, before counting what was paid for it. The profit subtracts the premium. These are different numbers, and the exam asks for each.

Call payoff at expiration Memorize

c_T = \max(0,\, S_T - X)

Long call profit Memorize

\Pi = \max(0,\, S_T - X) - c_0

The buyer’s payoff is zero or positive, never negative. A call buyer gains without limit as the underlying rises, but the loss is limited to c₀.

Rearrange itLong call: profit at expiration, solve for any input

The question asks for

An investor buys a call with an exercise price of 60.00 for a premium of 4.50. At expiration the underlying is 71.00. The buyer's profit per share is closest to:

Exercise price, XCall premium, c₀Underlying price at expiration, S_T

Π = max (0, S_{T} - X) - c_{0}

(71.00 − 60.00) − 4.50 = 6.50

Profit to the buyer, Π6.50

On your calculator

71−60−4.5=6.50

71ENTER60−4.5−6.50

ENTER enter (on the HP, puts the number on the stack)

The call seller has the mirror image: the payoff is −max(0, S_T − X) and the profit is the premium less the payoff. The most the seller can make is the premium, and because the underlying has no ceiling, the loss is unlimited.

Short call profit Memorize

\Pi = -\max(0,\, S_T - X) + c_0

Call breakeven price Memorize

S_T^{BE} = X + c_0

Rearrange itShort call: profit at expiration, solve for any input

The question asks for

A trader sells a call with an exercise price of 60.00 and receives a premium of 4.50. At expiration the underlying is 71.00. The seller's profit per share is closest to:

Exercise price, XCall premium, c₀Underlying price at expiration, S_T

Π = - max (0, S_{T} - X) + c_{0}

−(71.00 − 60.00) + 4.50 = -6.50

Profit to the seller, Π-6.50

On your calculator

4.5−71+60=-6.50

4.5ENTER71−60+-6.50

ENTER enter (on the HP, puts the number on the stack)

How do you find the payoff and profit of a put?

Same logic, with the roles of S_T and X reversed. A put gains when the underlying falls below X.

Put payoff at expiration Memorize

p_T = \max(0,\, X - S_T)

Long put profit Memorize

\Pi = \max(0,\, X - S_T) - p_0

Short put profit Memorize

\Pi = -\max(0,\, X - S_T) + p_0

Put breakeven price Memorize

S_T^{BE} = X - p_0

The put buyer’s best case is the underlying falling to zero, for a profit of X − p₀. The put seller’s worst case is the same number as a loss. This is why the put writer’s risk is large but not unlimited.

Rearrange itLong put: profit at expiration, solve for any input

The question asks for

An investor buys a put with an exercise price of 55.00 for a premium of 3.20. At expiration the underlying is 48.00. The buyer's profit per share is closest to:

Exercise price, XPut premium, p₀Underlying price at expiration, S_T

Π = max (0, X - S_{T}) - p_{0}

(55.00 − 48.00) − 3.20 = 3.80

Profit to the buyer, Π3.80

On your calculator

55−48−3.2=3.80

55ENTER48−3.2−3.80

ENTER enter (on the HP, puts the number on the stack)

Rearrange itShort put: profit at expiration, solve for any input

The question asks for

A trader sells a put with an exercise price of 55.00 and receives a premium of 3.20. At expiration the underlying is 48.00. The seller's profit per share is closest to:

Exercise price, XPut premium, p₀Underlying price at expiration, S_T

Π = - max (0, X - S_{T}) + p_{0}

−(55.00 − 48.00) + 3.20 = -3.80

Profit to the seller, Π-3.80

On your calculator

3.2−55+48=-3.80

3.2ENTER55−48+-3.80

ENTER enter (on the HP, puts the number on the stack)

InteractiveOption payoff and profit: the four positions

Try this 0 of 3

On the long call, move S_T to the breakeven price
Make the short put lose money even though the writer kept the premium
Load the put example and find the long put's breakeven

One option, four ways to hold it. Drag S_T on any chart and tap a chart title to read that position.

Payoff +11.00 · Profit +6.50

Long call, X = 60.00, premium 4.50: at S_T = 71.00 the payoff is +11.00 and the profit +6.50. Breakeven 64.50. Maximum loss is the premium, 4.50.

Payoff −11.00 · Profit −6.50

Short call, X = 60.00, premium 4.50: at S_T = 71.00 the payoff is −11.00 and the profit −6.50. Breakeven 64.50. Maximum gain is the premium, 4.50.

Payoff 0.00 · Profit −4.50

Long put, X = 60.00, premium 4.50: at S_T = 71.00 the payoff is 0.00 and the profit −4.50. Breakeven 55.50. Maximum loss is the premium, 4.50.

Payoff 0.00 · Profit +4.50

Short put, X = 60.00, premium 4.50: at S_T = 71.00 the payoff is 0.00 and the profit +4.50. Breakeven 55.50. Maximum gain is the premium, 4.50.

Long call: Payoff +11.00 · Profit +6.50

The call is in the money by 11.00. After the 4.50 premium the buyer makes 6.50.

Read the position

Price at expiry, S_TGBP 71.00

Exercise price, XGBP 60.00

PremiumGBP 4.50

Long call profitGBP 6.50

Payoff at expiryGBP 11.00

BreakevenGBP 64.50

Maximum gainUnlimited

Maximum lossGBP 4.50

Show working

Call payoff to the buyer = max(0, 71.00 − 60.00) = 11.00

Buyer's profit = payoff − premium = 11.00 − 4.50 = +6.50

Breakeven = X + c₀ = 60.00 + 4.50 = 64.50

Each short chart is the long chart flipped about the x-axis: whatever the buyer makes, the writer loses.

On your calculator

71−60=11.00, the payoff
−4.5=+6.50, the profit

71ENTER60−11.00, the payoff
4.5−+6.50, the profit

ENTER enter (on the HP, puts the number on the stack)

Display: +6.50

Profit is just the payoff shifted by the premium; the seller's chart is the buyer's turned upside down.

Exam tip Write the buyer’s profit first, then flip the sign for the seller. The buyer pays the premium and the seller keeps it, so the premium is a cost to one and a gain to the other.

Go deeper Intrinsic value and time value

Before expiration an option is worth its exercise value (intrinsic value) plus a time value. A call’s intrinsic value is max(0, S − X). Out-of-the-money and at-the-money options have zero intrinsic value, so their whole price is time value. The longer the time left, the more likely a favorable move is, so time value is positive and shrinks to zero at expiration. Option pricing is covered in a later module.

Go deeper Why the put writer's loss is capped but the call writer's is not

A share price can rise without limit, so a call writer who must deliver it faces unlimited loss. A share price cannot fall below zero, so a put writer who must buy it at X loses at most X, less the premium kept. That is the X − p₀ in the table.

Go deeper Other options

Options also exist on futures, on interest rates and on other underlyings, with the same payoff logic. American options add the question of early exercise, which Level I does not examine in detail.

Memorize this

MemorizeThe four positions at expiration

Position	Payoff	Profit	Max gain	Max loss	Breakeven
Long call	$\max(0, S_T - X)$	payoff − $c_0$	unlimited	$c_0$	$X + c_0$
Short call	$-\max(0, S_T - X)$	$c_0$ − payoff	$c_0$	unlimited	$X + c_0$
Long put	$\max(0, X - S_T)$	payoff − $p_0$	$X - p_0$	$p_0$	$X - p_0$
Short put	$-\max(0, X - S_T)$	$p_0$ − payoff	$p_0$	$X - p_0$	$X - p_0$

Memory hook Buyers pay the premium, sellers collect it. The seller's numbers are the buyer's with the sign flipped.

Know it cold Profit = payoff − premium for the buyer, premium − payoff for the seller. The call breaks even at X + premium, the put at X − premium.

MemorizeMoneyness and styles

	Call	Put
In the money	$S > X$	$S < X$
At the money	$S = X$	$S = X$
Out of the money	$S < X$	$S > X$

Memory hook A call wants a high S, a put wants a low S: "in the money" is whichever side of X helps.

Know it cold European = exercise only at expiration. American = any time up to expiration. A buyer's payoff is never negative.

The four option positions

Optionsthe buyer chooses; the seller must perform

Call (right to buy at X) Memorize
- Long call (buyer) Memorize pays c₀; gains when the underlying rises; loss capped at c₀
- Short call (seller) Memorize receives c₀; loses when the underlying rises; loss unlimited
Put (right to sell at X) Memorize
- Long put (buyer) Memorize pays p₀; gains when the underlying falls; loss capped at p₀
- Short put (seller) Memorize receives p₀; loses when the underlying falls; loss at most X − p₀

Moneyness: exercise now, would it pay?

Moneyness at time tcompare the underlying price S with the exercise price X

In the money Memorize call: S above X. Put: S below X. Positive exercise value
At the money Use it S equals X. Zero exercise value
Out of the money Memorize call: S below X. Put: S above X. Zero exercise value

When an option can be exercised

Exercise stylethe names say nothing about geography

European Memorize only at expiration
American Memorize any time up to expiration

Call payoff at expiration

c_T = \max(0,\, S_T - X)

Quick example

Exercise price 45, underlying at 52 at expiration. max(0, 52 − 45) = 7.00. At 41 the payoff is 0.

On your calculator

52−45= 7

52ENTER45− 7

ENTER enter (on the HP, puts the number on the stack)

Long call profit

\Pi = \max(0,\, S_T - X) - c_0

Quick example

X = 80, premium 3.40, S_T = 85. Payoff 5.00, profit 5.00 − 3.40 = 1.60.

On your calculator

85−80= payoff 5
−3.4= 1.6

85ENTER80− payoff 5
3.4− 1.6

ENTER enter (on the HP, puts the number on the stack)

Short call profit

\Pi = -\max(0,\, S_T - X) + c_0

Quick example

X = 100, premium 5.00, S_T = 108. −8.00 + 5.00 = −3.00.

On your calculator

5−108+100= -3

5ENTER108−100+ -3

ENTER enter (on the HP, puts the number on the stack)

Put payoff at expiration

p_T = \max(0,\, X - S_T)

Quick example

Exercise price 120, underlying at 104. max(0, 120 − 104) = 16.00. At 131 the payoff is 0.

On your calculator

120−104= 16

120ENTER104− 16

ENTER enter (on the HP, puts the number on the stack)

Long put profit

\Pi = \max(0,\, X - S_T) - p_0

Quick example

X = 70, premium 2.80, S_T = 62. Payoff 8.00, profit 8.00 − 2.80 = 5.20.

On your calculator

70−62= payoff 8
−2.8= 5.2

70ENTER62− payoff 8
2.8− 5.2

ENTER enter (on the HP, puts the number on the stack)

Short put profit

\Pi = -\max(0,\, X - S_T) + p_0

Quick example

X = 40, premium 1.60, S_T = 35. −5.00 + 1.60 = −3.40.

On your calculator

1.6−40+35= -3.4

1.6ENTER40−35+ -3.4

ENTER enter (on the HP, puts the number on the stack)

Call breakeven price

S_T^{BE} = X + c_0

Quick example

X = 80 and premium 3.40. The call buyer breaks even at 80 + 3.40 = 83.40.

On your calculator

80+3.4= 83.4

80ENTER3.4+ 83.4

ENTER enter (on the HP, puts the number on the stack)

Put breakeven price

S_T^{BE} = X - p_0

Quick example

X = 70 and premium 2.80. The put buyer breaks even at 70 − 2.80 = 67.20.

On your calculator

70−2.8= 67.2

70ENTER2.8− 67.2

ENTER enter (on the HP, puts the number on the stack)

How it is asked

Profit, not payoff

Buy a call, X = 95, premium 5.30. The stock ends at 101. Profit per share?

max(0, 101 − 95) − 5.30 = 0.70. Traps: 6.00 (the payoff) and 5.30 (the premium).
Seller at a loss

Sell a put, X = 75, premium 4.20, underlying ends at 66. Profit?

−9.00 + 4.20 = −4.80. Traps: −9.00 (ignores premium) and +4.20 (assumes it expired worthless).
Breakeven

Breakeven at expiration for a put buyer?

X − p₀. Trap: X + p₀ is the call.
Maximum gain or loss

Maximum loss for the writer of a put with X = 62 and premium 2.40?

The underlying falls to 0: −62 + 2.40 = −59.60. The call writer's loss has no limit.

Exam traps

Reporting the payoff when the question asks for profit (or the reverse). Profit includes the premium.
Getting the seller's sign wrong. The seller's payoff is the buyer's with a minus sign, and the premium comes in as a gain.
Using X + p₀ as the put breakeven. It is X − p₀; X + premium is the call.
Ignoring that the whole premium is lost when the option expires worthless, and kept by the seller.
Saying the put writer's loss is unlimited. It is capped at X − p₀ because the underlying cannot fall below zero.

Checkpoint

6 exam-style questions. About 90 seconds each.

1An investor buys a call with an exercise price of USD 105 for a premium of USD 6.40. At expiration the stock is USD 112. The investor's profit per share is closest to:
2A trader sells a put with an exercise price of CHF 90 for a premium of CHF 5.10. At expiration the underlying is CHF 82. The trader's profit is closest to:
3The breakeven price at expiration for the buyer of a put with exercise price X and premium p₀ is:
4A trader sells a call with an exercise price of USD 45 and receives a premium of USD 2.30. At expiration the underlying is USD 51.50. The trader's profit is closest to:
5A trader sells a put with an exercise price of USD 48 and receives a premium of USD 3.60. The trader's maximum possible loss is closest to:
6Once the premium has been paid at the start of an option contract, which party bears counterparty credit risk?

Key takeaways

Call payoff max(0, S_T − X); put payoff max(0, X − S_T). A buyer's payoff is never negative.
Profit = payoff − premium for buyers, premium − payoff for sellers.
Breakeven: call X + c₀, put X − p₀. The buyer's maximum loss is the premium.
Only the buyer faces counterparty credit risk, because the seller is paid upfront.

Call payoff at expiration

c_T = \max(0,\, S_T - X)

Quick example

Exercise price 45, underlying at 52 at expiration. max(0, 52 − 45) = 7.00. At 41 the payoff is 0.

On your calculator

52−45= 7

52ENTER45− 7

ENTER enter (on the HP, puts the number on the stack)

Long call profit

\Pi = \max(0,\, S_T - X) - c_0

Quick example

X = 80, premium 3.40, S_T = 85. Payoff 5.00, profit 5.00 − 3.40 = 1.60.

On your calculator

85−80= payoff 5
−3.4= 1.6

85ENTER80− payoff 5
3.4− 1.6

ENTER enter (on the HP, puts the number on the stack)

Short call profit

\Pi = -\max(0,\, S_T - X) + c_0

Quick example

X = 100, premium 5.00, S_T = 108. −8.00 + 5.00 = −3.00.

On your calculator

5−108+100= -3

5ENTER108−100+ -3

ENTER enter (on the HP, puts the number on the stack)

Put payoff at expiration

p_T = \max(0,\, X - S_T)

Quick example

Exercise price 120, underlying at 104. max(0, 120 − 104) = 16.00. At 131 the payoff is 0.

On your calculator

120−104= 16

120ENTER104− 16

ENTER enter (on the HP, puts the number on the stack)

Long put profit

\Pi = \max(0,\, X - S_T) - p_0

Quick example

X = 70, premium 2.80, S_T = 62. Payoff 8.00, profit 8.00 − 2.80 = 5.20.

On your calculator

70−62= payoff 8
−2.8= 5.2

70ENTER62− payoff 8
2.8− 5.2

ENTER enter (on the HP, puts the number on the stack)

Short put profit

\Pi = -\max(0,\, X - S_T) + p_0

Quick example

X = 40, premium 1.60, S_T = 35. −5.00 + 1.60 = −3.40.

On your calculator

1.6−40+35= -3.4

1.6ENTER40−35+ -3.4

ENTER enter (on the HP, puts the number on the stack)

Call breakeven price

S_T^{BE} = X + c_0

Quick example

X = 80 and premium 3.40. The call buyer breaks even at 80 + 3.40 = 83.40.

On your calculator

80+3.4= 83.4

80ENTER3.4+ 83.4

ENTER enter (on the HP, puts the number on the stack)

Put breakeven price

S_T^{BE} = X - p_0

Quick example

X = 70 and premium 2.80. The put buyer breaks even at 70 − 2.80 = 67.20.

On your calculator

70−2.8= 67.2

70ENTER2.8− 67.2

ENTER enter (on the HP, puts the number on the stack)

All Derivatives formulas, every lesson

Practice exam-style questions on Forwards, Futures, Swaps, Options and Credit Derivatives

Next lesson Credit default swaps

Questions about this lesson?

Where this comes back

Forward commitments vs contingent claims Forwards, Futures, Swaps, Options and Credit Derivatives
Pricing and Valuing Options Derivatives
Put–Call Parity and Option Replication Derivatives

Questions candidates ask

What is the difference between option payoff and profit?

Payoff is what the option is worth at expiration, ignoring the price paid for it: max(0, S_T − X) for a call. Profit is the payoff minus the premium for the buyer, or the premium minus the payoff for the seller. Profit is what the trader actually makes.

How do you calculate the breakeven of a put option?

Subtract the premium from the exercise price: X − p₀. A put bought with X = 55 and a premium of 3.20 breaks even at 51.80. Below that price the buyer makes a profit; above it, a loss.

What is the maximum loss for an option seller?

A call seller's loss is unlimited, because the underlying can rise without limit. A put seller's loss is capped at X − p₀, reached if the underlying falls to zero. In both cases the seller's maximum gain is the premium.

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