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FRM Exam Part I · Introduction to Derivatives

Options: Calls, Puts and Payoffs Explained for FRM Part 1

Updated 11 October 2026 · Fact-checked

An option gives the holder the right, not the obligation, to buy (call) or sell (put) an asset at a strike price. Payoff at expiry: long call = max(S − K, 0), long put = max(K − S, 0). Profit equals payoff minus the premium paid. Short positions reverse the signs.

Understand Options: Calls, Puts and Payoffs

An option is a contract. The buyer (long) pays a premium up front and gets a right. The seller (short, or writer) receives the premium and takes an obligation. This is the key difference from forwards and futures, where both sides are obligated.

A call gives the right to buy the underlying at the strike price K. You exercise only if the asset price S is above K. A put gives the right to sell at K. You exercise only if S is below K. Because the holder can walk away, the long payoff can never be negative. The loss is limited to the premium.

The seller's position is the mirror image. The short call and short put payoffs are the negatives of the long payoffs. The writer's gain is capped at the premium, while the loss can be large. A short call on a stock has unlimited loss. A short put has a maximum loss of K − premium, because the price cannot fall below zero.

A payoff diagram plots value at expiry against the asset price S. A profit diagram subtracts the premium paid (or adds the premium received). For a long call, the break-even price is K + premium. For a long put, it is K − premium. In the simple exam setting, ignore interest on the premium unless the question says to include it.

Moneyness describes the position of S relative to K. A call is in the money when S > K, at the money when S = K and out of the money when S < K. A put is the reverse. European options can be exercised only at expiry. American options can be exercised at any time up to expiry. Most payoff questions assume exercise at expiry.

Key formulas to remember

Long call payoff
max(S − K, 0)
S is the asset price at expiry, K the strike.
Long put payoff
max(K − S, 0)
Rises as the asset price falls below K.
Short call payoff
−max(S − K, 0) = min(K − S, 0)
Never positive. Loss is unlimited as S rises.
Short put payoff
−max(K − S, 0) = min(S − K, 0)
Maximum loss is K before the premium.
Profit, long option
Payoff − premium paid
For a short option: premium received − payoff owed.
Break-even price
Call: K + premium; Put: K − premium
Same for the long and short side of the same option.
Moneyness
Call ITM if S > K; put ITM if S < K
At the money when S = K. Otherwise out of the money.
Intrinsic value
Call: max(S − K, 0); Put: max(K − S, 0)
Option price = intrinsic value + time value. American options have non-negative time value because they can be exercised early, so they never trade below intrinsic value. European options cannot be exercised early, so they can trade below intrinsic value. Examples are a deep in-the-money put, or a deep in-the-money call on an asset that pays large dividends.

How to solve Options: Calls, Puts and Payoffs questions

Use the same sequence for any payoff, profit or moneyness question.

  1. 1Identify the option type (call or put) and the position (long or short).
  2. 2Write down S, K and the premium. Check if the question asks for payoff or profit.
  3. 3Compute the long payoff using max(S − K, 0) for a call or max(K − S, 0) for a put.
  4. 4If you are short, flip the sign of the payoff.
  5. 5For profit, subtract the premium paid (long) or add the premium received (short).
  6. 6Check the sign logic: a long position cannot have a payoff below zero, and a short position cannot have one above zero.
  7. 7For break-even or moneyness, compare S with K or with K ± premium and state the result.

Quickest way: Sign and direction shortcut

When to use it: Use it when options are tested on a combined position or a diagram, with little time.

  1. Ask: does the holder gain when S goes up or down? Long call and short put gain when S rises. Long put and short call gain when S falls.
  2. Long means a limited loss (the premium) and short means a limited gain (the premium).
  3. Find the kink at K. Profit is zero at K ± premium.
  4. Compute only at the S value the question gives. Do not draw the full diagram.
  5. Match the answer to the options by checking the sign and the size.

Common mistakes in Options: Calls, Puts and Payoffs

  • Reporting payoff when profit is asked, or the reverse.

    The premium is given in the question but not used, or used when it should not be.

    Fix: Underline the word payoff or profit first. Profit always includes the premium.

  • Using S − K for a put.

    Students memorize the call formula only.

    Fix: Put payoff is K − S. The put holder sells at K and profits when S is below K.

  • Giving a long option a negative payoff.

    Students forget the holder can choose not to exercise.

    Fix: The payoff is max(…, 0). The loss for the long side is only the premium in the profit calculation.

  • Mixing up in the money for puts.

    Students carry over the call rule, S > K.

    Fix: A put is in the money when S < K. Check the direction of the right.

  • Confusing the break-even of a put with a call.

    Adding the premium to the strike by habit.

    Fix: Call break-even: K + premium. Put break-even: K − premium.

  • Thinking American options always cost the same as European ones.

    Students overlook the early exercise right.

    Fix: An American option is worth at least as much as an otherwise identical European option, because it has extra rights.

Worked examples

Example 1

An investor buys a European call on a stock with a strike of $50 for a premium of $3. At expiry, the stock price is $58. Find the payoff, the profit and the break-even price.

Show the solution
  1. Payoff = max(58 − 50, 0) = $8.
  2. Profit = payoff − premium = 8 − 3 = $5.
  3. Break-even = K + premium = 50 + 3 = $53.

Answer: Payoff $8, profit $5, break-even $53.

Example 2

A trader writes a European put with a strike of $40 and receives a premium of $2.50. At expiry, the stock price is $34. What is the trader's profit, and what is the maximum possible loss?

Show the solution
  1. Put payoff to the holder = max(40 − 34, 0) = $6.
  2. The writer pays $6, so the payoff to the writer is −$6.
  3. Profit = premium received − payoff owed = 2.50 − 6 = −$3.50.
  4. Maximum loss occurs if S falls to 0: payoff owed = $40, so loss = 40 − 2.50 = $37.50.

Answer: Loss of $3.50. The maximum loss is $37.50.

Exam tips

  • Read whether the question asks for payoff or profit and whether the position is long or short before you calculate.
  • Sketch the shape in your head: call (flat then rising), put (falling then flat), then flip for short.
  • Moneyness questions test the put direction. Check S against K for the right type.
  • Expect call and put positions to be combined later in spread and straddle questions, so be sure about the single-option payoffs first.
  • Unless the question gives interest, ignore the time value of the premium.

Practice questions from Introduction to Derivatives

Options: Calls, Puts 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.

Options: Calls, Puts and Payoffs: frequently asked questions

What is the difference between an American and a European option?

A European option can be exercised only at expiry. An American option can be exercised at any time up to and including expiry. The label does not refer to geography.

How do I calculate option profit?

Work out the payoff at expiry, then subtract the premium paid if you are long. If you are short, add the premium received to the payoff you owe, which is negative. Profit shows the net result of the position.

What does in the money, at the money and out of the money mean?

A call is in the money when the asset price is above the strike, and a put is in the money when it is below. At the money means the price equals the strike. Out of the money is the opposite of in the money.

Can the loss on a long option exceed the premium?

No. The holder can choose not to exercise, so the most a long option holder can lose is the premium paid. The writer faces the large risks.