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FRM Exam Part I · Options Markets

Option Basics and Payoffs at Expiration

Updated 11 October 2026 · Fact-checked

An option gives its holder the right, not the obligation, to buy (call) or sell (put) an asset at a strike price. At expiration, a long call pays max(S − K, 0) and a long put pays max(K − S, 0). Profit is payoff minus the premium paid. Short positions reverse the signs.

Understand Option Basics and Payoffs

An option is a contract. The buyer (long) pays a premium and gets a right. The seller (writer, short) receives the premium and takes on an obligation. The seller must deliver if the buyer exercises.

A call gives the right to buy the underlying at the strike price (K). A put gives the right to sell at K. A European option can be exercised only at expiration. An American option can be exercised at any time up to expiration. Payoff diagrams at expiration are the same shape for both.

The buyer's loss is limited to the premium. The buyer's gain on a call is unlimited as the price rises. A put gains as the price falls, with a maximum gain of K minus the premium, because the price cannot go below zero. The seller has the mirror image: a small capped gain (the premium) and a large, possibly unlimited, loss on a short call.

Moneyness describes the payoff if exercised now. A call is in the money (ITM) when S > K, at the money (ATM) when S = K, and out of the money (OTM) when S < K. A put is ITM when S < K and OTM when S > K. Moneyness uses the strike and the current price, not the premium paid.

The payoff ignores the premium. The profit includes it. Unless a question says to adjust for interest on the premium, profit = payoff − premium for the buyer, and payoff reversed plus premium for the seller. The long and short sides of the same option always sum to zero, so it is a zero-sum contract.

Key formulas to remember

Long call payoff
max(S_T − K, 0)
S_T is the underlying price at expiration. Never negative.
Long put payoff
max(K − S_T, 0)
Never negative.
Short call payoff
−max(S_T − K, 0) = min(K − S_T, 0)
Mirror image of the long call.
Short put payoff
−max(K − S_T, 0) = min(S_T − K, 0)
Mirror image of the long put.
Profit for the buyer
Profit = Payoff − Premium
Ignores interest on the premium unless stated.
Profit for the seller
Profit = Premium − Buyer's payoff
Maximum gain is the premium.
Breakeven price
Call: K + premium. Put: K − premium
For the long position. Short has the same breakeven.
Maximum gain on a long put
K − premium
Occurs when S_T = 0.

How to solve Option Basics and Payoffs questions

Use the same routine for any single-option payoff or profit question.

  1. 1Identify the option type (call or put) and your side (long or short).
  2. 2Write down K, the premium and the expiration price S_T.
  3. 3Compute the buyer's payoff: max(S_T − K, 0) for a call, max(K − S_T, 0) for a put.
  4. 4If you are short, flip the sign of the payoff.
  5. 5Adjust for the premium: subtract it if long, add it if short.
  6. 6For breakeven, solve profit = 0: K + premium for a call, K − premium for a put.
  7. 7Check the answer against the shape: long loss is capped at the premium, short gain is capped at the premium.

Quickest way: Kink-and-slope shortcut

When to use it: For multiple-choice questions on profit, breakeven or position identification.

  1. Find the kink at K. Left or right of it, the payoff is flat on one side.
  2. Call: flat for S < K, slope +1 above. Put: flat for S > K, slope −1 below.
  3. Long: profit flat at −premium. Short: flat at +premium.
  4. Start from the flat level, then add the intrinsic value to get profit.
  5. Eliminate options that have the wrong maximum loss or gain.

Common mistakes in Option Basics and Payoffs

  • Forgetting the premium when asked for profit.

    Payoff formulas look complete, and the premium is given as a side detail.

    Fix: Read whether the question says payoff or profit. For profit, always subtract the premium if long.

  • Confusing a short put with a long call.

    Both gain when the price rises, so they look alike.

    Fix: A long call has a capped loss and unlimited gain. A short put has a capped gain (premium) and a large loss if the price falls.

  • Letting the premium decide moneyness.

    Students think an option is ITM when profit is positive.

    Fix: Moneyness compares S and K only. A call can be ITM and still show a loss after the premium.

  • Giving a long option a negative payoff.

    Applying S − K without the max with zero.

    Fix: The holder will not exercise at a loss, so the payoff is never below zero.

  • Reversing ITM and OTM for puts.

    Memorising the call rule and applying it to both.

    Fix: A put is ITM when S < K. Ask yourself whether you would want to sell above the market price.

  • Treating the maximum loss of a short call as limited.

    Assuming the premium is the only amount at stake.

    Fix: The short call loss grows with S and has no upper limit. Only the long side has a loss capped at the premium.

Worked examples

Example 1

An investor buys a European call on a stock with strike $50 and premium $3. At expiration the stock is at $58. What is the payoff and profit for the buyer, and the profit for the seller?

Show the solution
  1. Payoff = max(58 − 50, 0) = $8.
  2. Buyer's profit = 8 − 3 = $5.
  3. Seller's profit = 3 − 8 = −$5 (zero-sum).

Answer: Payoff $8, buyer profit $5, seller loss $5.

Example 2

An investor writes a European put with strike $40 and receives a premium of $2.50. What is the profit if the stock is at $35 at expiration, and what is the breakeven price?

Show the solution
  1. Buyer's payoff = max(40 − 35, 0) = $5.
  2. Writer's profit = 2.50 − 5 = −$2.50.
  3. Breakeven for the put: K − premium = 40 − 2.50 = $37.50.
  4. Check at S = 37.50: buyer's payoff 2.50, writer's profit 2.50 − 2.50 = 0.

Answer: Loss of $2.50; breakeven at $37.50.

Exam tips

  • Read whether the question asks for payoff or profit, and whether you are long or short. These two reads decide the answer.
  • Use the kink at K and the premium level to match a diagram to a position quickly.
  • Remember the put's maximum gain is K minus the premium, because the price cannot go below zero.
  • Check moneyness against S and K only. Distractor answers often mix in the premium.
  • Premium is usually not discounted in these questions. Only adjust for interest if the question states it.

Practice questions from Options Markets

Option Basics and Payoffs in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Option Basics and Payoffs: frequently asked questions

What is the difference between a long call and a short put?

Both gain when the underlying rises. A long call pays a premium, has a limited loss and an unlimited gain. A short put receives a premium, has a limited gain and a large loss if the price falls toward zero.

How do I calculate option profit at expiration?

Compute the payoff first: max(S − K, 0) for a call, max(K − S, 0) for a put. For a buyer, subtract the premium. For a seller, reverse the payoff sign and add the premium.

When is an option in the money?

A call is in the money when the underlying price is above the strike. A put is in the money when the underlying price is below the strike. At the money means the two are equal.

Does the premium affect moneyness?

No. Moneyness depends only on the underlying price and the strike. The premium affects profit and breakeven, not whether an option is in, at or out of the money.