Expiration Payoff and Profit Exam Workshop

Expiration Payoff Is Not the Same as Profit

An option’s expiration payoff depends on the underlying price at expiration and the option’s strike. The option premium is a separate cash flow: a purchaser pays it at initiation, while a writer receives it. A reliable solution therefore calculates payoff and profit separately before combining them.

Throughout this tutorial, assume European-style options held to expiration, no transaction costs, and deterministic underlying prices for the stated scenarios. Premiums are quoted per unit of the underlying. We use the standard expiration profit-diagram convention: profit per unit equals expiration payoff per unit less the premium paid, or equals the premium received less the writer’s expiration liability. This convention does not accumulate the initial premium with interest. If a problem instead asks for a time-value-adjusted result, the premium cash flow must be accumulated or discounted using the stated interest assumptions. No mortality model or loss distribution is relevant here.

Definition: For expiration price $S_T$ and strike $K$, a call purchaser’s payoff per underlying unit is $\max(S_T-K,0)$, and a put purchaser’s payoff per unit is $\max(K-S_T,0)$. A writer’s payoff is the negative of the corresponding purchaser’s payoff. Profit also includes the premium: a purchaser subtracts the premium paid, and a writer adds the premium received.

Let $c$ be the call premium and $p$ the put premium, each quoted per underlying unit. The expiration profit per unit is therefore:

\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 formulas keep three ideas distinct: who owns the option, whether it finishes in the money, and which way the premium enters the profit calculation. At the strike, $S_T=K$, both call and put payoffs are zero. That does not mean the option purchaser breaks even: the premium paid remains a cost.

Why the Payoff Formulas Have Two Regions

A call gives its purchaser the right, not the obligation, to buy the underlying for $K$. At expiration, buying for $K$ is beneficial only if the underlying is worth more than $K$. If $S_T>K$, the value of that right per unit is $S_T-K$; if $S_T\leq K$, the purchaser can let it expire and receive zero. This gives the call payoff $\max(S_T-K,0)$. A put gives the right to sell for $K$, so its payoff is $K-S_T$ only when $S_T<K$, and is zero otherwise. A writer has the opposite obligation and thus the opposite payoff.

The break-even prices follow by setting profit equal to zero. For a purchased call, the in-the-money equation is $S_T-K-c=0$, so its break-even price is $K+c$. For a purchased put, the in-the-money equation is $K-S_T-p=0$, so its break-even price is $K-p$. These are break-even points only when they lie in the region where the option has positive payoff. The same prices mark the writer’s break-even points, but the writer’s profit signs are reversed.

Boundary check: At $S_T=K$, a call or put payoff is zero. The purchaser’s profit is then $-c$ for a call or $-p$ for a put; the writer’s profit is $+c$ or $+p$, respectively. Always substitute the strike into the payoff formula rather than assuming the strike is a break-even price.

Worked Expiration Scenarios

Worked Example: Call Payoff, Break-Even, and Contract Scaling

An investor buys two call contracts on a hypothetical asset. Each contract covers 100 units, the strike is $K=\$50$, and the premium is $c=\$4$ per unit. Find the payoff and profit at expiration when $S_T$ is $\$42$, $\$50$, $\$54$, and $\$63$. Assume no interest adjustment to the premium.

Method. First use the long-call payoff $\max(S_T-50,0)$. Subtract $\$4$ per unit to obtain profit, then multiply each per-unit result by $2\times100=200$ units.

At $S_T=\$42$, the payoff is $\max(42-50,0)=\$0$ per unit. Profit is $0-4=-\$4$ per unit, or $-4\times200=-\$800$ for the position.

At the strike, $S_T=\$50$, payoff is $\max(50-50,0)=\$0$. Profit is $0-4=-\$4$ per unit, or $-\$800$ for the position.

The break-even price is $K+c=50+4=\$54$. At $S_T=\$54$, payoff is $\max(54-50,0)=\$4$, and profit is $4-4=\$0$ per unit, or $\$0$ for the position.

At $S_T=\$63$, payoff is $\max(63-50,0)=\$13$. Profit is $13-4=\$9$ per unit, or $9\times200=\$1{,}800$.

Check. At the break-even price, payoff equals the premium. Below the strike, the option expires without payoff and the position loses the total premium, $4\times200=\$800$. The calculation avoids treating the strike itself as break-even.

Worked Example: Written Put at and Around Its Break-Even

A trader writes three put contracts on a hypothetical asset. Each contract covers 50 units, $K=\$80$, and the premium received is $p=\$6$ per unit. Calculate the writer’s expiration payoff and profit if $S_T$ is $\$68$, $\$74$, $\$80$, or $\$95$. Ignore interest on the premium.

Method. The put purchaser’s payoff is $\max(80-S_T,0)$; the writer’s payoff is its negative. The writer’s profit per unit is $6-\max(80-S_T,0)$. There are $3\times50=150$ units in the position.

At $S_T=\$68$, the put purchaser’s payoff is $80-68=\$12$ per unit, so the writer’s payoff is $-\$12$. The writer’s profit is $6-12=-\$6$ per unit, or $-6\times150=-\$900$.

At $S_T=\$74$, the purchaser’s payoff is $80-74=\$6$, and the writer’s payoff is $-\$6$. The writer’s profit is $6-6=\$0$ per unit, or $\$0$ overall. This is the put break-even price, $K-p=80-6=\$74$.

At $S_T=\$80$, the purchaser’s payoff is $\max(80-80,0)=\$0$, and the writer’s payoff is also zero. The writer keeps the premium, so profit is $+\$6$ per unit, or $6\times150=\$900$.

At $S_T=\$95$, the put is out of the money and both payoff amounts are zero. The writer’s profit remains $+\$6$ per unit, or $\$900$.

Check. At $S_T=\$74$, the writer’s liability equals the premium received, giving zero profit. At and above the strike, the put payoff is zero, so the writer retains the premium. The contract multiplier scales the per-unit results; it does not change the break-even price.

Worked Example: Combining Legs Without Losing the Signs

A hypothetical investor holds two long call contracts with $K=\$40$, premium $c=\$5$ per unit, and multiplier 100. The investor also writes one put contract with $K=\$50$, premium $p=\$7$ per unit, and multiplier 100. Find the total expiration profit when $S_T=\$35$, $\$45$, and $\$55$. Ignore interest on both premiums.

Method. Calculate each leg’s payoff and profit separately, including its contract count. Then add the two position profits. The call position has 200 units; the written put has 100 units.

When $S_T=\$35$, each call has zero payoff and profit $0-5=-\$5$ per unit. The two-call position profit is $-5\times200=-\$1{,}000$. The put purchaser’s payoff is $50-35=\$15$ per unit, so the writer’s profit is $7-15=-\$8$ per unit, or $-8\times100=-\$800$. Total profit is $-1{,}000-800=-\$1{,}800$.

When $S_T=\$45$, each call payoff is $45-40=\$5$, so call profit is $5-5=\$0$ per unit and the call position profit is $\$0$. The put payoff to its purchaser is $50-45=\$5$, so the written put earns $7-5=\$2$ per unit, or $2\times100=\$200$. Total profit is $\$200$.

When $S_T=\$55$, each call payoff is $55-40=\$15$, giving call profit $15-5=\$10$ per unit, or $10\times200=\$2{,}000$. The put payoff is zero, so the written put earns its $\$7$ premium per unit, or $7\times100=\$700$. Total profit is $2{,}000+700=\$2{,}700$.

Check. At $\$45$, the call payoff exactly offsets its premium per unit, while the written put has a positive profit. The combined position is therefore profitable by $\$200$, not at its call’s break-even profit of zero. This verifies that break-even checks apply to individual legs unless the question asks for the whole position.

A Reliable Exam Calculation

For a position with more than one option, organize the work by leg. Record whether the position is long or short, calculate the option payoff, add the premium with the appropriate sign, and only then apply the number of contracts and multiplier. This order makes it easier to catch a sign error before it affects the total.

Identify the option and direction.
A long position pays the option purchaser’s payoff; a short position has the opposite payoff. Write down the premium paid or received.

Evaluate the expiration region.
Compare $S_T$ with $K$. At equality, the payoff is zero; do not assign a positive payoff on the basis of the strike alone.

Calculate per-unit profit.
Subtract the premium for a purchased option or subtract the writer’s liability from the premium received.

Scale and combine.
Multiply by contracts and units per contract, then add the signed profits of all legs.

Common Sign Errors and Why They Fail

A frequent mistake is to treat the payoff as if it already includes the premium. Consider the written put in the second example when $S_T=\$68$. A mistaken calculation might use $S_T-K-p=68-80-6=-\$18$ per unit. That expression is not the writer’s profit: it subtracts the premium even though the writer received it, and it uses a negative put expression without first identifying the purchaser’s liability.

The correct calculation is the writer’s received premium minus the put purchaser’s payoff: $6-\max(80-68,0)=6-12=-\$6$ per unit. A useful sign check is to ask what happens when the option expires worthless. The purchaser then loses the premium, while the writer earns it. If a calculation gives the opposite signs in that situation, the premium sign has been reversed.

A second trap is scaling too early or applying the multiplier to the strike or break-even price. Contract scaling changes total dollars, not the price of the underlying at which one unit breaks even. For example, two call contracts with a 100-unit multiplier still have the same per-unit break-even $K+c$ as one such contract; they simply have twice the total dollar profit or loss at every expiration price.

Key takeaway: Calculate payoff from the option right, calculate profit by including the premium with its correct sign, and scale only after obtaining per-unit results. At the strike, payoff is zero; at a break-even price, profit—not payoff—is zero.

Check Your Understanding

For each problem, calculate the requested expiration profit using the stated per-unit premium and contract multiplier. Ignore interest on premiums.

An investor buys one call contract covering 100 units, with $K=\$72$ and $c=\$5$. Find payoff and profit per unit at $S_T=\$68$, $\$72$, and $\$81$. (Answer: Payoffs $\$0,\$0,\$9$; profits $-\$5,-\$5,+\$4$ per unit.)
A trader writes one put contract covering 60 units, with $K=\$36$ and $p=\$4.50$. Find the writer’s total profit at $S_T=\$30$, $\$36$, and $\$42$. (Answer: $-\$90,+\$270,+\$270$, respectively.)
An investor buys two put contracts, each covering 40 units, with $K=\$125$ and $p=\$9$. Find total profit at $S_T=\$110$. (Answer: $+\$480$.)
An investor buys one call contract covering 100 units with $K=\$25$, $c=\$3$, and writes one put contract covering 100 units with $K=\$20$, $p=\$2$. Find total profit at $S_T=\$18$, calculating the legs separately. (Answer: $-\$300$.)
An investor buys three call contracts, each covering 50 units, with $K=\$60$ and $c=\$7$. Find total profit at $S_T=\$70$, and verify the per-unit break-even. (Answer: Total profit $+\$450$; break-even $S_T=\$67$.)

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