Tell apart an option's payoff and the profit it earns
The idea
The payoff of a long call at expiry is the greater of zero and the underlying price less the exercise price - it can never be negative, which is the whole point of buying optionality. Profit subtracts the premium paid, so the position loses money for any underlying price below the exercise price plus the premium even where the payoff is positive. For the writer both figures mirror exactly: the writer's maximum profit is the premium and the loss on a naked call is unbounded. Confusing payoff with profit is a common error in reading option diagrams.
Formulas
Profit on a long call
call profit = max(0, Sₜ − X) − premium
- Sₜ
- the price of the underlying at expiry
- X
- the exercise price
- premium
- what was paid for the option at the start
Payoff is what the option pays; profit is what the trade made. The breakeven is X + premium, so an option can finish in the money and still lose money - the commonest confusion in the topic.
A practice question
Can the payoff of a long call option at expiry be negative?
- No, the payoff is floored at zero - correct
- Yes, if the underlying falls far enoughWhy not: A falling underlying makes the option worthless, which is a payoff of zero rather than a negative one.
- Yes, by the amount of the premiumWhy not: The premium makes the profit negative, not the payoff. The two lines differ by exactly that amount.
A call gives you the right, not the duty, to buy at the exercise price (the agreed price). If buying is a bad deal, you simply do not use the right, so you cannot lose on the payoff. Call payoff = maximum of (Underlying price − Exercise price, 0). The premium (the price you paid for the option) affects profit, not payoff. Example: exercise price 50, underlying at 45. Payoff = maximum of (45 − 50, 0) = 0. With a premium of 3, profit = 0 − 3 = −3.
A harder one
A European call with an exercise price of 50 cost 4 per share. The underlying finishes at 52 at expiry. Ignoring the time value of the premium, what are the payoff and the profit per share?
- Payoff 0, profit -4Why not: Treats the option as out of the money. At 52 against a 50 strike it is in the money by 2.
- Payoff 2, profit -2 - correct
- Payoff 2, profit 2Why not: Gets the payoff right and forgets to subtract the 4 paid for it.
Payoff is what the option pays at expiry. Profit is the payoff minus what you paid for the option. Call payoff = maximum of (Underlying − Exercise price, 0) = 52 − 50 = 2. Profit = Payoff − Premium = 2 − 4 = −2. So you make a loss of 2 per share. It is still sensible to use the option, because collecting 2 beats letting it expire worthless and losing the whole 4.
The hardest tier
A student draws the payoff diagram and the profit diagram for a long call option. How do the two lines relate?
- The profit line is the payoff line moved down by the premium - correct
- The profit line is steeper than the payoff lineWhy not: Both lines have the same slope wherever the option is in the money. Only their vertical position differs.
- The two lines coincide once the option is in the moneyWhy not: They never coincide. At every price the profit is the payoff less the premium paid.
Profit = Payoff − Premium at every price. So the profit line is simply the payoff line moved down by the amount of the premium, with the same shape and same slope. Example: call with strike 60 bought for 3. At a price of 70 the payoff is 10 and the profit is 7. The payoff starts to rise at 60, but profit turns positive only at the break-even of 63 (strike plus premium). The two lines never meet. Mixing them up puts break-even in the wrong place.
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