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Economic Modelling · Principles of option pricing

Options Basics and Payoff Diagrams for Calls and Puts

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 K. At expiry, a long call pays max(S − K, 0) and a long put pays max(K − S, 0). A short position has the opposite sign. Profit equals payoff minus the premium paid, ignoring interest.

Understand Options Basics and Payoff Diagrams

An option is a contract. The buyer (the long position) pays a price, called the premium, today. In return the buyer gets a right, but no duty, to trade an underlying asset at a fixed strike price K on or before an expiry date. The seller (the writer, or short position) receives the premium and must honour the trade if the buyer chooses to exercise.

A call gives the right to buy. A put gives the right to sell. You exercise a call only when the share price S is above K, because you buy cheaply and could sell at S. You exercise a put only when S is below K. If exercise would lose money, you walk away. This is why payoffs are never negative for the buyer.

The style of the option matters for timing. A European option can be exercised only at expiry. An American option can be exercised at any time up to and including expiry. Payoff diagrams at expiry look the same for both. The difference shows up when you value the option before expiry.

Moneyness describes where S sits against K today. A call is in the money if S > K, at the money if S = K, and out of the money if S < K. For a put, reverse it: in the money if S < K, out of the money if S > K. Moneyness is about whether immediate exercise would give a positive payoff, not about whether you made a profit overall.

A payoff diagram plots the value at expiry against the share price S at expiry. A profit diagram shifts the payoff line by the premium. Long and short positions are mirror images about the horizontal axis. The buyer's loss is capped at the premium. The writer's gain is capped at the premium. The writer of a call has unlimited potential loss; the writer of a put has a loss capped at K minus the premium, because S cannot fall below zero.

Key rules to remember

Long call payoff at expiry
max(S_T − K, 0)
Zero when S_T ≤ K. Rises one-for-one with S_T above K.
Long put payoff at expiry
max(K − S_T, 0)
Zero when S_T ≥ K. Maximum value is K, when S_T = 0.
Short call payoff
−max(S_T − K, 0) = min(K − S_T, 0)
The negative of the long call payoff.
Short put payoff
−max(K − S_T, 0) = min(S_T − K, 0)
The negative of the long put payoff.
Profit of a long option (ignoring interest)
Profit = Payoff − Premium
For the writer, Profit = Premium − Payoff owed. If asked, include interest: premium × (1 + i) or premium × e^(δT).
Break-even share price
Call: S_T = K + c. Put: S_T = K − p
c and p are the premiums paid. The break-even price is where profit equals zero.
Maximum profit and loss, long call
Max loss = c. Max profit = unlimited
Short call is the reverse: max profit = c, max loss unlimited.
Maximum profit and loss, long put
Max loss = p. Max profit = K − p
Short put: max profit = p, max loss = K − p.

How to solve Options Basics and Payoff Diagrams questions

Use this routine for any question asking you to find a payoff, profit, break-even or sketch for a single option or a simple combination.

  1. 1Identify each position: call or put, long or short, strike K and premium paid or received.
  2. 2Write the payoff at expiry for each position using max(·, 0), with the correct sign for long or short.
  3. 3Split the share price range at each strike. Write each payoff as a simple linear expression in each range.
  4. 4Add the payoffs of all positions in each range to get the total payoff.
  5. 5Subtract premiums paid (add premiums received) to get profit. Include interest on premiums only if the question says so.
  6. 6Find the break-even points by setting profit to zero in each range, and check the answer lies in that range.
  7. 7Sketch: mark K on the horizontal axis, draw straight line segments with kinks at strikes, and label intercepts, slopes and maximum profit and loss.

Quickest way: Kink-and-slope sketching

When to use it: Use when a question asks you to sketch a payoff or profit diagram or to read off maximum gain or loss quickly.

  1. Remember the shape: a call is flat then slopes up after K. A put slopes down until K then is flat.
  2. For a short position, flip the picture upside down.
  3. Compute the payoff at S = 0 and at a large S. These two ends fix the line.
  4. Mark the kink at each strike. Slopes are +1 for a long underlying-like part and −1 for a short one.
  5. Shift the whole line down by the premium for a long option, up for a short option, to get profit.
  6. Read the break-even from the point where the profit line crosses zero.

Common mistakes in Options Basics and Payoff Diagrams

  • Treating profit as the same as payoff.

    The payoff diagram is the first picture you learn and the premium is easy to forget.

    Fix: Always ask: is the question about payoff or profit? Profit = payoff − premium paid. Write it down as a final step.

  • Giving a long put an unlimited maximum profit.

    Students copy the long call shape, which is unlimited.

    Fix: A share price cannot go below zero. The maximum put payoff is K, so the maximum profit is K minus the premium.

  • Confusing the in the money condition for puts.

    Students remember S > K for calls and apply it to puts.

    Fix: A put is in the money when S < K. Ask whether exercising now would give a positive payoff.

  • Saying American and European payoff diagrams differ at expiry.

    Students mix up the exercise rights with the payoff formula.

    Fix: At expiry both pay the same. The difference is when exercise is allowed, which affects value before expiry. An American option is worth at least as much as an otherwise identical European option.

  • Drawing the short position with the wrong sign or the wrong premium effect.

    Students flip the payoff but forget that the writer receives the premium.

    Fix: Short payoff = negative of long payoff. Short profit = premium received minus payoff owed. The line sits above zero for small losses and falls below as the payoff grows.

  • Putting the break-even at K.

    The kink of the payoff diagram is at K, so it feels like the key point.

    Fix: The break-even is at K + c for a long call and K − p for a long put. The profit line is shifted by the premium.

Worked examples

Example 1

An investor buys a European call on a share with strike ₹500 for a premium of ₹30. Ignore interest. Find the payoff and profit if the share price at expiry is (a) ₹470, (b) ₹520, (c) ₹580. State the break-even price.

Show the solution
  1. Payoff = max(S_T − 500, 0). Profit = payoff − 30.
  2. (a) S_T = 470: payoff = max(−30, 0) = 0. Profit = 0 − 30 = −₹30.
  3. (b) S_T = 520: payoff = 20. Profit = 20 − 30 = −₹10.
  4. (c) S_T = 580: payoff = 80. Profit = 80 − 30 = ₹50.
  5. Break-even: S_T − 500 = 30, so S_T = ₹530.

Answer: Payoffs are ₹0, ₹20 and ₹80. Profits are −₹30, −₹10 and ₹50. Break-even share price is ₹530. The maximum loss is the premium, ₹30.

Example 2

A writer sells a European put with strike ₹200 and receives a premium of ₹12. Ignore interest. (a) Write the writer's payoff and profit at expiry. (b) Find the break-even share price. (c) Find the maximum profit and the maximum loss.

Show the solution
  1. (a) The long put pays max(200 − S_T, 0). The writer's payoff = −max(200 − S_T, 0).
  2. Writer's profit = 12 − max(200 − S_T, 0).
  3. (b) For S_T < 200 the profit is 12 − (200 − S_T) = S_T − 188. Set this to zero: S_T = ₹188.
  4. (c) For S_T ≥ 200 the profit is 12, which is the maximum profit.
  5. The loss is largest when S_T = 0: profit = 0 − 188 = −₹188.

Answer: Profit = 12 − max(200 − S_T, 0). Break-even is ₹188. Maximum profit is ₹12, reached when S_T ≥ 200. Maximum loss is ₹188, reached when S_T = 0.

Exam tips

  • In MCQs, check the sign and whether the question asks for payoff or profit before you compute. These are the usual traps.
  • In written answers, state your assumptions: European exercise, no dividends, and whether interest on the premium is ignored.
  • When asked to sketch, label axes, the strike, the premium, break-even and the maximum gain and loss. Examiners award marks for these labels.
  • Practise short and long versions of both options until you can draw all four in under a minute. Combinations in later topics build on them.
  • In Paper B style work, you may be asked to compute payoffs in Excel or R. Use a max function on a vector of share prices and subtract the premium for profit.

Practice questions from Principles of option pricing

Options Basics and Payoff Diagrams in other exams

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

Options Basics and Payoff Diagrams: frequently asked questions

What is the difference between European and American options?

A European option can be exercised only at expiry. An American option can be exercised at any time up to expiry. The payoff at expiry is the same formula for both, but an American option is worth at least as much as the equivalent European option.

How do I draw the payoff diagram for a long put?

Plot share price at expiry on the horizontal axis. Draw a line sloping down from K at S = 0 to zero at S = K, then flat at zero for S > K. For profit, shift the whole line down by the premium.

Why can the buyer of an option never lose more than the premium?

The buyer has a right, not an obligation. If exercise would lose money, the buyer lets the option lapse. So the payoff is never negative, and the only cost is the premium paid at the start.

How is the short put different from the long put?

The short put is the mirror image. The writer receives the premium but pays out if the share price ends below K. The maximum loss is K minus the premium, and the maximum profit is the premium.