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Economic Modelling · Binomial option-pricing model

Option Payoffs and Payoff Diagrams Explained

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 K. At expiry, a call pays max(S − K, 0) and a put pays max(K − S, 0). Profit is payoff minus the premium paid. Draw the kinked line, then subtract the premium.

Understand Options Basics and Payoff Diagrams

An option is a contract. The holder (long position) pays a premium today and gets a right. The writer (short position) receives the premium and must honour the contract if the holder exercises.

A call option gives the right to buy the underlying asset at the strike price K on or before the expiry date. A put option gives the right to sell at K. You exercise only when it helps you. That is why the payoff can never be negative for the holder.

A European option can be exercised only at expiry. An American option can be exercised at any time up to expiry. Because an American option has every right a European one has, plus more, its price is at least as high as the European option with the same terms.

Moneyness describes where the share price S sits against K right now. 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 it is the reverse: in the money if S < K. Moneyness uses the current price; the payoff diagram uses the price at expiry.

Six factors drive an option's value: the current share price, the strike price, the time to expiry, the volatility of the share, the risk-free interest rate and the dividends expected during the option's life. Calls gain when S rises. Puts gain when S falls. Higher volatility raises the value of both, because the holder keeps the upside and the loss is capped at the premium.

Key rules to remember

Long call payoff at expiry
max(S_T − K, 0)
S_T is the share price at expiry. Never negative for the holder.
Long put payoff at expiry
max(K − S_T, 0)
Pays when the share price ends below the strike.
Short positions
Short call payoff = −max(S_T − K, 0); short put payoff = −max(K − S_T, 0)
The writer's payoff is the negative of the holder's. The writer's payoff is never positive; the writer's maximum profit is the premium received.
Profit at expiry (ignoring interest)
Profit = payoff − premium (long); profit = premium − payoff (short)
If asked to allow for interest, accumulate the premium to expiry at the risk-free rate first.
Breakeven share price
Call: K + premium; Put: K − premium
Holder's breakeven, ignoring interest on the premium.
Moneyness
Call in the money if S > K; put in the money if S < K
Intrinsic value now = max(S − K, 0) for a call, max(K − S, 0) for a put.
Effect of factors on a European option (other factors fixed)
S up: call up, put down. K up: call down, put up. Volatility up: both up. Rate up: call up, put down. Dividends up: call down, put up.
Time to expiry: more time raises American options; for European options the effect is not always upward (e.g. deep in-the-money European puts).

How to solve Options Basics and Payoff Diagrams questions

Use this order for any question on option payoffs, profit or the factors affecting price.

  1. 1Identify the contract: call or put, long or short, European or American, and note K and the premium.
  2. 2Write the payoff formula for the long position, then change the sign if the position is short.
  3. 3Compare S_T with K to decide whether the option is exercised. Compute the payoff for the given prices.
  4. 4Subtract the premium (long) or add it (short) to get profit. Accumulate the premium at the risk-free rate if the question says to allow for interest.
  5. 5Find key points for the diagram: the kink at K, the value at S_T = 0, the breakeven and the maximum gain and loss.
  6. 6Draw or describe the shape: flat part, then a 45-degree line. Label axes (S_T horizontal, payoff or profit vertical) and the kink.
  7. 7For price-factor questions, take one factor at a time, hold the others fixed, and state the direction with a one-line reason.

Quickest way: Kink, slope, shift

When to use it: Use for MCQs and quick sketches of payoff or profit at expiry.

  1. Kink: mark K on the horizontal axis. The payoff line bends there.
  2. Slope: long call rises with slope +1 to the right of K, flat to the left. Long put has slope −1 to the left of K, flat to the right. Short positions are mirror images in the horizontal axis.
  3. Shift: for profit, move the whole payoff line down by the premium (long) or up (short).
  4. Check extremes: long call and long put lose at most the premium. A long put's maximum profit is K − premium (when S_T = 0); its maximum payoff is K. A short call has unlimited loss.

Common mistakes in Options Basics and Payoff Diagrams

  • Drawing a call payoff that goes negative below K.

    You treat the option like a forward contract, which does have an obligation.

    Fix: The holder can walk away, so the payoff is max(S_T − K, 0). It is flat at zero below K.

  • Confusing payoff with profit.

    The premium is paid at the start and is easy to forget at expiry.

    Fix: Read the question for the word profit. Then always subtract the premium (long) or add it (short).

  • Getting moneyness of a put the wrong way round.

    You copy the call rule S > K.

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

  • Saying a European option can be worth more than an equivalent American option (reversing the inequality).

    The names are mixed up, and the extra right is forgotten.

    Fix: American can be exercised early, so American ≥ European for the same terms. They are equal for a call on a non-dividend-paying share.

  • Stating that higher volatility lowers option value because risk is bad.

    You carry over intuition from risk-averse investors.

    Fix: The holder's loss is capped at the premium while the gain is open. More volatility raises the chance of a large payoff, so both calls and puts gain value.

  • Treating the writer's maximum loss as the premium.

    You mix up the holder's and the writer's positions.

    Fix: The holder's maximum loss is the premium. A call writer's loss is unlimited; a put writer's maximum loss is K minus the premium received.

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 at expiry if the share price is (a) ₹470, (b) ₹520, (c) ₹560. State the breakeven 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 = 560: payoff = 60. Profit = 60 − 30 = ₹30.
  5. Breakeven: payoff must equal premium, so S_T = 500 + 30 = ₹530.

Answer: Payoffs: ₹0, ₹20, ₹60. Profits: −₹30, −₹10, ₹30. Breakeven share price is ₹530. Maximum loss is ₹30.

Example 2

A writer sells a European put with strike ₹200 and receives a premium of ₹12. Ignore interest. Find the writer's profit if the share price at expiry is (a) ₹230 and (b) ₹170. State the writer's maximum possible loss.

Show the solution
  1. Writer's payoff = −max(200 − S_T, 0). Profit = premium + payoff = 12 − max(200 − S_T, 0).
  2. (a) S_T = 230: put is out of the money, payoff to holder = 0. Writer's profit = 12.
  3. (b) S_T = 170: holder's payoff = 200 − 170 = 30. Writer's profit = 12 − 30 = −18.
  4. Maximum loss occurs when S_T = 0: holder's payoff = 200. Writer's profit = 12 − 200 = −188.

Answer: (a) profit ₹12; (b) loss of ₹18; maximum loss ₹188 (when the share price falls to zero).

Exam tips

  • Sketch first. A quick labelled diagram with K, the premium and the breakeven often earns marks even if arithmetic slips.
  • Read whether the question asks for payoff or profit, and whether to allow for interest on the premium. State your assumption.
  • For factor questions, change one variable at a time and give direction plus a reason. Mention European or American if it matters.
  • In MCQs, check the sign: for a short position, flip the diagram about the horizontal axis.
  • In Paper B, a payoff function is a one-line formula. In R use pmax(S − K, 0) for a call to handle many prices at once.

Practice questions from Binomial option-pricing model

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. So an American option is worth at least as much as the equivalent European option.

What is the difference between payoff and profit for an option?

Payoff is the amount received at expiry, such as max(S_T − K, 0) for a call. Profit also deducts the premium paid (and interest on it if required). A payoff is never negative for the holder, but profit can be.

What determines the price of an option?

The main factors are the current share price, the strike, the time to expiry, the share's volatility, the risk-free rate and expected dividends. Higher volatility raises the value of both calls and puts. Rising share price helps calls and hurts puts.

What does in the money mean?

A call is in the money when the share price is above the strike. A put is in the money when the share price is below the strike. It means exercising now would give a positive payoff.