From Option Payoff To Profit

Payoff Is Not Profit

An option payoff describes the value of the option position at expiration, before accounting for the premium exchanged when the contract was initiated. Profit includes that earlier premium cash flow. The distinction matters because a position can have a positive payoff at expiration and still have a loss overall if the premium paid was larger.

This tutorial uses one option on one share of an underlying asset. Let $S_T$ be the share price at expiration, $K$ the strike price, $c$ the call premium, and $p$ the put premium. Premiums and payoffs are stated in dollars per share. We calculate total profit over the option’s life without interest, transaction costs, taxes, or other cash flows. If the premium were invested or borrowed and its time value were included, the premium cash flow would need to be accumulated to expiration; that is not the convention used here.

Definition: The expiration payoff is the value of an option position at expiration, excluding its initial premium. The expiration profit is the payoff plus the premium cash flow, with a premium paid by a long position counted as a cost and a premium received by a short position counted as income.

Payoff and Profit Formulas

For a call, the holder can buy the underlying share for $K$. The call’s long payoff is therefore the positive amount, if any, by which $S_T$ exceeds $K$. For a put, the holder can sell for $K$, so the long payoff is the positive amount, if any, by which $K$ exceeds $S_T$.

\begin{aligned} \text{Long call payoff} &= \max(S_T-K,0),\\ \text{Long put payoff} &= \max(K-S_T,0). \end{aligned}

The holder pays the premium at initiation, so that cash flow is $-c$ for a call or $-p$ for a put. A short position has the opposite expiration payoff and receives the premium at initiation. Thus, for a short position, the premium cash flow is positive.

\begin{aligned} \text{Long call profit} &= \max(S_T-K,0)-c,\\ \text{Short call profit} &= c-\max(S_T-K,0),\\ \text{Long put profit} &= \max(K-S_T,0)-p,\\ \text{Short put profit} &= p-\max(K-S_T,0). \end{aligned}

These expressions treat payoff and profit from the perspective of the named position. The short holder’s payoff is the negative of the long holder’s payoff, while the premium cash flow reverses sign as well. A short call, for example, receives the call premium but owes the call holder’s expiration payoff if the call finishes with value.

Why the Premium Is Added or Subtracted

The profit formulas follow by adding the position’s cash flows over the life of the contract. For a long call, there are two relevant cash flows under the stated assumptions: $-c$ at initiation and $\max(S_T-K,0)$ at expiration. Their sum is the total profit. For a short call, the initial cash flow is $+c$, and its expiration cash flow is $-\max(S_T-K,0)$. The same reasoning applies to puts.

\begin{aligned} \text{Long call profit} &= \underbrace{(-c)}_{\text{premium paid}} + \underbrace{\max(S_T-K,0)}_{\text{expiration payoff}}\\ &= \max(S_T-K,0)-c,\\[4pt] \text{Short call profit} &= \underbrace{c}_{\text{premium received}} - \underbrace{\max(S_T-K,0)}_{\text{amount owed at expiration}}\\ &= c-\max(S_T-K,0). \end{aligned}

This cash-flow argument also shows why the short position’s profit is the negative of the long position’s profit when both sides refer to the same option and the same premium: one side’s premium payment is the other side’s receipt, and one side’s expiration payoff is the other side’s obligation.

\text{Short profit}=-\text{Long profit}.

Worked Examples

Worked Example: Long Call With Two Expiration Prices

A portfolio analyst buys one call on a share with strike $K=\$50.00$ and pays a premium of $c=\$4.20$ per share. Find the expiration payoff and profit if the share price is (a) $\$61.50$ and (b) $\$47.50$. Ignore interest and all costs other than the premium.

Method. First calculate the call payoff, $\max(S_T-K,0)$, for each expiration price. Then subtract the premium paid, $\$4.20$, to obtain profit.

(a) Expiration price $\$61.50$. The amount by which the share price exceeds the strike is $\$61.50-\$50.00=\$11.50$. The call payoff is therefore $\max(\$11.50,0)=\$11.50$. Subtract the premium: $\$11.50-\$4.20=\$7.30$ profit.

(b) Expiration price $\$47.50$. The share price does not exceed the strike, so the call payoff is $\max(\$47.50-\$50.00,0)=\max(-\$2.50,0)=\$0.00$. The option expires without value, but the premium was still paid. Profit is $\$0.00-\$4.20=-\$4.20$.

Check. In each case, profit equals payoff minus the premium. The first profit is $\$7.30$, which is $\$11.50-\$4.20$; the second is a loss equal to the $\$4.20$ premium.

Common mistake avoided: Do not call the $\$11.50$ payoff the total profit. It excludes the $\$4.20$ paid to acquire the call.

Worked Example: Long Put With a Premium Cost

A small investment account buys one put on a share with strike $K=\$80.00$ for a premium of $p=\$3.60$ per share. Calculate the payoff and profit if $S_T=\$71.00$, and then if $S_T=\$84.00$. Use the same assumptions of no interest or other costs.

Method. A long put’s payoff is $\max(K-S_T,0)$. Since the buyer pays the premium, subtract $\$3.60$ from the payoff in each case.

When $S_T=\$71.00$. The difference $K-S_T$ is $\$80.00-\$71.00=\$9.00$, so the put payoff is $\max(\$9.00,0)=\$9.00$. Profit is $\$9.00-\$3.60=\$5.40$.

When $S_T=\$84.00$. Now $K-S_T=\$80.00-\$84.00=-\$4.00$, so the put payoff is $\max(-\$4.00,0)=\$0.00$. Profit is $\$0.00-\$3.60=-\$3.60$.

Check. The put has positive payoff only in the first case because the share price is below the strike. The profits reconcile: $\$9.00-\$3.60=\$5.40$ and $\$0.00-\$3.60=-\$3.60$.

Common mistake avoided: A put payoff uses $K-S_T$, not $S_T-K$. Reversing the subtraction would incorrectly give a zero payoff in the first case.

Worked Example: Short Call and Premium Received

A hypothetical fund writes one call on a share with strike $K=\$120.00$ and receives $c=\$6.25$ per share. Calculate its profit if the share price at expiration is (a) $\$113.00$ and (b) $\$132.50$. Exclude interest, fees, and other cash flows.

Method. The short call’s profit is the premium received minus the long call’s payoff: $c-\max(S_T-K,0)$. The amount owed at expiration is zero when the call has no value and positive when $S_T>K$.

(a) $S_T=\$113.00$. The call payoff to its holder is $\max(\$113.00-\$120.00,0)=\max(-\$7.00,0)=\$0.00$. The writer retains the premium, giving profit $\$6.25-\$0.00=\$6.25$.

(b) $S_T=\$132.50$. The holder’s payoff is $\max(\$132.50-\$120.00,0)=\$12.50$. The writer’s profit is $\$6.25-\$12.50=-\$6.25$.

Check. For each expiration price, the writer’s profit is the negative of the call holder’s profit: at $\$113.00$, the holder’s profit is $\$0.00-\$6.25=-\$6.25$; at $\$132.50$, it is $\$12.50-\$6.25=\$6.25$.

Common mistake avoided: The writer does not subtract the premium received as though it were a cost. The initial premium is a positive cash flow to the short position.

Worked Example: Short Put and the Expiration Obligation

A hypothetical dealer writes one put with strike $K=\$40.00$, receiving $p=\$2.80$ per share. Find the dealer’s profit if $S_T=\$36.50$ and if $S_T=\$43.00$. Assume there are no other cash flows or costs.

Method. Use the short put formula $p-\max(K-S_T,0)$. The premium received is positive, and the put payoff to its holder is the amount the writer owes at expiration.

When $S_T=\$36.50$. The holder’s payoff is $\max(\$40.00-\$36.50,0)=\$3.50$. The dealer’s profit is $\$2.80-\$3.50=-\$0.70$.

When $S_T=\$43.00$. The holder’s payoff is $\max(\$40.00-\$43.00,0)=\max(-\$3.00,0)=\$0.00$. The dealer’s profit is $\$2.80-\$0.00=\$2.80$.

Check. The corresponding long put profits are $\$3.50-\$2.80=\$0.70$ and $\$0.00-\$2.80=-\$2.80$. Each is the negative of the dealer’s profit.

Common mistake avoided: In the first case, do not treat the $\$3.50$ put payoff as the dealer’s profit. It is the amount owed to the holder before crediting the $\$2.80$ premium received.

A Reliable Calculation Process

For any single option position, keep the expiration value and the initial premium in separate steps. This makes the sign of each cash flow visible and reduces errors when changing from a long position to a short position.

Identify the option and position.
Record whether the contract is a call or put and whether the position is long or short. Read the strike, expiration share price, and premium from the problem.

Calculate the long holder’s payoff.
For a call use $\max(S_T-K,0)$; for a put use $\max(K-S_T,0)$. This payoff is never negative for the long option holder.

Apply the position and premium signs.
For a long position, subtract the premium paid. For a short position, subtract the holder’s payoff from the premium received.

Check the result.
Verify that profit equals payoff plus the signed premium cash flow. For opposite positions in the same contract, confirm that their profits sum to zero under the stated assumptions.

Common Sign and Interpretation Errors

The most frequent error is to report payoff when the question asks for profit. Payoff is determined by the contract terms and $S_T$; profit also depends on the premium. A call can have a positive expiration payoff while its buyer has a negative total profit, because the payoff may be smaller than the premium. Conversely, a short position can have positive profit even when the option has a positive payoff, if the premium received exceeds the amount owed.

Forgetting the initial cash flow: Long positions pay a premium, so subtract it. Short positions receive a premium, so add it before subtracting any expiration obligation.
Using a negative payoff for a long option: The maximum operator ensures that an unexercised call or put has a payoff of zero, not a negative amount.
Mixing up the holder’s payoff and the writer’s profit: The option payoff formula describes the holder’s expiration value. The writer’s profit is the premium received minus that payoff.
Changing only one sign when reversing positions: Both the premium cash flow and the expiration payoff reverse between the long and short sides.

In an accounting or finance analysis, naming the cash flows makes the result easier to interpret: a payoff is an expiration amount associated with the option, while profit is the net result after the premium. This distinction is also useful when comparing strategies, because the same expiration payoff can correspond to different profits when initial premiums differ.

Key takeaway: Calculate the option payoff from the expiration price and strike first. Then include the premium with its cash-flow sign: subtract a premium paid by a long position, or subtract the option payoff from a premium received by a short position.

Check Your Understanding

For each question, calculate the requested expiration profit under the stated no-interest, no-fee convention. Premiums and results are per share.

A long call has strike $\$72.00$ and premium $\$5.10$. If $S_T=\$80.00$, what are its payoff and profit? (Answer: payoff $\$8.00$; profit $\$2.90$)
A long put has strike $\$55.00$ and premium $\$2.40$. If $S_T=\$57.00$, what is its profit? (Answer: $-\$2.40$)
A short call has strike $\$95.00$ and receives a premium of $\$4.75$. If $S_T=\$102.00$, what is the short position’s profit? (Answer: $-\$2.25$)
A short put has strike $\$30.00$ and receives a premium of $\$1.60$. If $S_T=\$27.50$, calculate the holder’s payoff and the writer’s profit. (Answer: holder payoff $\$2.50$; writer profit $-\$0.90$)
A call with strike $\$44.00$ has a premium of $\$3.25$. At $S_T=\$48.00$, calculate the long holder’s profit and the short writer’s profit, then verify their relationship. (Answer: long profit $\$0.75$; short profit $-\$0.75$; the profits sum to $\$0.00$)

What You'll Learn