Why Compare Payoff and Profit?
An option’s payoff at expiration describes the cash value generated by the option at that time. It does not, by itself, show whether the position earned or lost money overall: the buyer paid a premium to acquire the option, while the seller received one. A profit diagram includes that initial premium, so it answers a different question.
The distinction is easiest to see by drawing both diagrams on the same axes. The horizontal axis is the underlying asset price at expiration, \(S_T\), and the vertical axis is dollars per unit of the underlying. For a fixed option position, the premium moves the profit diagram vertically relative to the payoff diagram. It does not alter the option’s expiration-payoff shape.
Throughout this tutorial, consider one option unit and express all dollar amounts per unit. Assume a standard option held to expiration, with no transaction costs or other cash flows. Use the simplified profit convention: do not accrue interest on the premium between purchase and expiration. If a problem instead asks for profit including financing, the premium’s time value must also be included.
From Payoff to Profit
Let \(K\) be the strike price. Let \(c_0\) denote the premium paid for a long call and \(p_0\) the premium paid for a long put. A long option has a negative initial cash flow equal to its premium. Therefore, at expiration, simplified profit is payoff less premium. For a short option, the initial cash flow is a premium received, so simplified profit is payoff plus that premium.
The payoff of a short option is the negative of the corresponding long option’s payoff. Profit follows the same signed-payoff rule, with the premium received by the short position added. In each case, the difference between profit and payoff is a constant for that position: the premium, with the appropriate sign. Thus, at every value of \(S_T\), the profit coordinate is the payoff coordinate shifted vertically by the same amount.
Why the Premium Shifts the Diagram Vertically
The result follows directly from the timing of the two cash flows. At expiration, the option generates its payoff, which depends on \(S_T\). Separately, the holder paid a fixed premium at the start, or the writer received one. Under the simplified convention, adding or subtracting that fixed amount changes the vertical coordinate at every \(S_T\) by the same amount.
Write the option payoff as a function of \(S_T\), keeping the direction of the position clear.
A buyer pays the premium; a seller receives it. This fixes the sign of the adjustment.
Subtract a premium paid or add a premium received to obtain simplified profit.
Check that each profit point is the corresponding payoff point shifted vertically, not horizontally.
For example, a long call’s payoff is zero below the strike and rises above the strike. Subtracting \(c_0\) puts the profit diagram \(c_0\) dollars below the payoff diagram at every underlying price. The point where the profit reaches zero can change, but the point where the option payoff changes formula remains at the strike.
Worked Example: Long Call Payoff and Profit
A hypothetical investor buys one call with strike \(K=\$50\) for a premium of \(c_0=\$6\) per unit. Find payoff and simplified profit at expiration prices of \(\$35\), \(\$50\), \(\$56\), and \(\$70\). Describe how to compare the two diagrams.
Method. For a long call, calculate \(\max(S_T-K,0)\) first, then subtract the premium. This order keeps the option payoff distinct from the purchase cost.
At \(S_T=\$35\), the payoff is \(\max(35-50,0)=\$0\), and profit is \(0-6=-\$6\).
At \(S_T=\$50\), the payoff is \(\max(50-50,0)=\$0\), and profit is \(0-6=-\$6\).
At \(S_T=\$56\), the payoff is \(\max(56-50,0)=\$6\), and profit is \(6-6=\$0\).
At \(S_T=\$70\), the payoff is \(\max(70-50,0)=\$20\), and profit is \(20-6=\$14\).
| \(S_T\) | Payoff | Simplified profit |
|---|---|---|
| \(\$35\) | \(\$0\) | \(-\$6\) |
| \(\$50\) | \(\$0\) | \(-\$6\) |
| \(\$56\) | \(\$6\) | \(\$0\) |
| \(\$70\) | \(\$20\) | \(\$14\) |
To draw the diagrams, plot the listed points with \(S_T\) on the horizontal axis. At every listed price, the profit point is \(\$6\) below the payoff point. The payoff reaches zero at \(S_T=\$50\); the profit reaches zero at \(S_T=\$56\). The latter is the price at which the call’s \(\$6\) payoff has just recovered the \(\$6\) premium. Check: each profit entry equals its payoff entry minus \(\$6\). The common mistake avoided is treating the premium as part of the option’s payoff rather than as a separate adjustment to profit.
Position Direction Determines the Shift
The direction of the vertical shift depends on whether the position paid or received the premium. A long position begins with a cash outflow, so its profit is below its payoff. A short position begins with a cash inflow, so its profit is above its payoff. The option payoff itself does not change when the premium changes: it is determined by the contract terms and the expiration price.
| Position | Premium at inception | Profit adjustment to payoff | Diagram shift |
|---|---|---|---|
| Long call or long put | Paid | Subtract premium | Down |
| Short call or short put | Received | Add premium | Up |
Worked Example: Short Put Diagrams
A hypothetical trader writes one put with strike \(K=\$40\) and receives a premium of \(p_0=\$3.50\) per unit. Find the short put’s payoff and simplified profit when \(S_T\) is \(\$30\), \(\$36.50\), \(\$40\), and \(\$48\). State the vertical shift and the zero-profit price.
Method. The short put payoff is the negative of the long put payoff, \(-\max(K-S_T,0)\). Add the premium received to obtain the writer’s simplified profit.
At \(S_T=\$30\), the short payoff is \(-\max(40-30,0)=-\$10\). Profit is \(-10+3.50=-\$6.50\).
At \(S_T=\$36.50\), the short payoff is \(-\max(40-36.50,0)=-\$3.50\). Profit is \(-3.50+3.50=\$0\).
At \(S_T=\$40\), the short payoff is \(-\max(40-40,0)=\$0\). Profit is \(0+3.50=\$3.50\).
At \(S_T=\$48\), the short payoff is \(-\max(40-48,0)=\$0\). Profit is \(0+3.50=\$3.50\).
| \(S_T\) | Short put payoff | Simplified profit |
|---|---|---|
| \(\$30\) | \(-\$10.00\) | \(-\$6.50\) |
| \(\$36.50\) | \(-\$3.50\) | \(\$0.00\) |
| \(\$40\) | \(\$0.00\) | \(\$3.50\) |
| \(\$48\) | \(\$0.00\) | \(\$3.50\) |
The profit diagram is the payoff diagram shifted upward by \(\$3.50\). Its zero-profit price occurs where the short put’s loss of \(\$3.50\) exactly offsets the received premium. Check: adding \(\$3.50\) to each payoff gives every profit shown. A frequent error is to subtract the premium from the writer’s payoff; that would incorrectly treat the received premium as a payment.
Preserved Shape, Changed Zero Level
A vertical shift preserves the option payoff’s slope and kink: every segment keeps the same direction and rate of change with respect to \(S_T\), and the price at which the payoff changes formula remains the same. What can change is where the profit graph crosses the horizontal axis, because zero profit includes the premium adjustment. This comparison concerns the two diagrams for the same position and contract, not a comparison of options with different strikes or positions.
Worked Example: Long Put Diagrams
An investor buys one put with strike \(K=\$75\) for \(p_0=\$4\) per unit. Compute payoff and simplified profit at \(S_T=\$60\), \(\$71\), \(\$75\), and \(\$90\). Then verify that the premium moves the diagram without changing the payoff’s formula boundary.
Method. For a long put, calculate \(\max(K-S_T,0)\), then subtract the paid premium of \(\$4\).
At \(S_T=\$60\), payoff is \(\max(75-60,0)=\$15\); profit is \(15-4=\$11\).
At \(S_T=\$71\), payoff is \(\max(75-71,0)=\$4\); profit is \(4-4=\$0\).
At \(S_T=\$75\), payoff is \(\max(75-75,0)=\$0\); profit is \(0-4=-\$4\).
At \(S_T=\$90\), payoff is \(\max(75-90,0)=\$0\); profit is \(0-4=-\$4\).
| \(S_T\) | Long put payoff | Simplified profit |
|---|---|---|
| \(\$60\) | \(\$15\) | \(\$11\) |
| \(\$71\) | \(\$4\) | \(\$0\) |
| \(\$75\) | \(\$0\) | \(-\$4\) |
| \(\$90\) | \(\$0\) | \(-\$4\) |
The put payoff changes formula at \(S_T=\$75\), and that boundary remains at \(\$75\) in the profit diagram. For every price shown, profit equals payoff minus \(\$4\). Check: the zero-profit point at \(\$71\) has a \(\$4\) put payoff, exactly covering the \(\$4\) premium. The common mistake avoided is to move the diagram’s boundary when the premium changes; the premium shifts the vertical values, not the underlying price at which the payoff formula changes.
Common Diagram Errors
A diagram comparison is reliable only when its axes and cash-flow signs are consistent. Keep \(S_T\) on the horizontal axis for both diagrams and use the same vertical dollar scale. Then check the premium adjustment point by point.
- Reversing the premium sign. A buyer pays a premium, so subtract it. A writer receives a premium, so add it.
- Shifting horizontally. Premium is an amount of money, not an underlying price. It changes the vertical coordinate, not the horizontal coordinate.
- Confusing payoff with profit. A zero payoff does not imply zero profit. A long option with zero payoff has lost the premium under the simplified convention.
- Changing the contract’s payoff to account for the premium. The premium affects the position’s profit, not the option’s contractual expiration payoff.
- Changing the profit convention without stating it. If financing or transaction costs are included, specify them and adjust the profit calculation consistently.
Check Your Understanding
For each problem, calculate payoff first and then apply the correct premium adjustment. Unless stated otherwise, use one option unit and the simplified expiration-profit convention with no interest accrual or transaction costs.
- A buyer pays \(\$5\) for a call with \(K=\$32\). At \(S_T=\$29\) and \(S_T=\$41\), what are the payoff and simplified profit at each price? (Answer: At \(\$29\), payoff \(\$0\), profit \(-\$5\); at \(\$41\), payoff \(\$9\), profit \(\$4\).)
- A writer receives \(\$2.25\) for a put with \(K=\$28\). At \(S_T=\$24\), what are the short payoff and simplified profit, and how far above the payoff diagram is the profit diagram? (Answer: Payoff \(-\$4\), profit \(-\$1.75\); the profit diagram is shifted up \(\$2.25\).)
- A long put has strike \(\$90\) and premium \(\$7\). At what expiration price does simplified profit equal zero? Give the payoff and profit at \(S_T=\$90\) as a check. (Answer: \(S_T=\$83\); at \(\$90\), payoff \(\$0\) and profit \(-\$7\).)
- A trader writes a call with strike \(\$45\) and receives \(\$3\). At \(S_T=\$40\) and \(S_T=\$52\), find the short call payoff and profit. State whether the profit diagram lies above or below the payoff diagram. (Answer: At \(\$40\), payoff \(\$0\), profit \(\$3\); at \(\$52\), payoff \(-\$7\), profit \(-\$4\); the profit diagram lies \(\$3\) above the payoff diagram.)
- A long call with strike \(\$60\) costs \(\$8\). A student says that adding the premium moves the payoff kink from \(\$60\) to \(\$68\). Give the correct location of the kink and calculate payoff and simplified profit at \(S_T=\$68\). (Answer: The kink remains at \(\$60\); at \(\$68\), payoff is \(\$8\) and profit is \(\$0\).)