Long Call Payoff at Expiration

Introduction

A call option gives its owner the right, but not the obligation, to buy an underlying asset for a specified price. At expiration, the owner compares that contractual price with the asset’s market price. The difference determines whether exercising the call has value. This tutorial focuses on that expiration payoff: the value generated by the option at expiration, before accounting for the premium paid to acquire it.

The distinction matters because a call can have a positive expiration payoff even when the total position has not recovered its initial cost. The premium is a separate cash flow. It is not subtracted when calculating the call’s expiration payoff.

Definition: For a long call with strike price $K$, let $S_T$ be the underlying asset’s market price at expiration $T$. The call’s expiration payoff per unit of the underlying is $\max(S_T-K,0)$. If a position contains $n$ contracts, each covering $M$ units, its total expiration payoff is $nM\max(S_T-K,0)$, assuming all contracts have the same strike and expiration.

Three Terminal-Price Cases

The payoff depends on the relationship between $S_T$ and $K$. If the market price is below the strike, buying the asset at the higher strike price would not be beneficial, so the call is left unexercised and its payoff is zero. If the market price equals the strike, exercising provides no price advantage and the payoff is also zero. If the market price is above the strike, exercising allows the holder to obtain an asset worth more than the exercise price; the difference is the payoff.

\text{Long call payoff per unit} = \begin{cases} 0, & S_T \leq K,\\ S_T-K, & S_T>K. \end{cases}

Equivalently, using the positive-part notation for calls, the payoff is $(S_T-K)^+$. The payoff is measured in the same monetary units as $S_T$ and $K$. It is never negative: a long call holder can decline to exercise, so the option itself does not require a loss at expiration.

This formula describes the value at expiration, not the profit on the full transaction. If a call premium was paid earlier, that payment is not part of the expiration payoff calculation. Determining profit or a break-even price requires a separate calculation.

Why the Positive-Part Formula Holds

Assume a standard call on one unit of an asset, with strike $K$, held until expiration. Ignore transaction costs and any settlement complications. At expiration, exercising means paying $K$ to receive an asset that can be sold for $S_T$. The net value generated by exercise is therefore $S_T-K$. If this amount is negative or zero, the holder can instead choose not to exercise, producing a payoff of zero.

The holder’s available choices are thus to exercise, with value $S_T-K$, or not to exercise, with value zero. The payoff is the larger of those two values:

\max(S_T-K,0)=(S_T-K)^+.

This is an expiration payoff relationship. It does not require a probability distribution for $S_T$, nor does it require an interest-rate assumption: once the terminal price is specified, the payoff follows directly. If a contract covers multiple units, multiply the per-unit payoff by the number of units covered. For cash settlement, the contractual cash amount is designed to reflect the same intrinsic value, subject to the contract’s stated terms.

Worked Example: One Call Below, At, and Above Its Strike

A hypothetical call has strike $K=\$48$ and covers 100 shares. Its premium was $\$2.40$ per share, paid earlier. Find the expiration payoff for terminal share prices of $\$43$, $\$48$, and $\$61.25$. Assume the stated terminal prices are known scenario values; no probability distribution or interest rate is assumed.

Method. Calculate the positive difference $S_T-K$ for each terminal price, replace any nonpositive difference by zero, and multiply by the 100-share contract size.

At $S_T=\$43$. The difference is $\$43-\$48=-\$5$. Since this is below zero, the per-share payoff is $\max(-\$5,0)=\$0$. The contract payoff is $100(\$0)=\$0$.

At $S_T=\$48$. The difference is $\$48-\$48=\$0$. The per-share payoff is $\max(\$0,0)=\$0$, and the contract payoff is $100(\$0)=\$0$.

At $S_T=\$61.25$. The difference is $\$61.25-\$48=\$13.25$. The per-share payoff is $\$13.25$, and the contract payoff is $100(\$13.25)=\$1,325$.

Check. The payoffs are $\$0$, $\$0$, and $\$1,325$, respectively. They are nonnegative and rise above zero only when the terminal price is greater than the strike. The earlier premium was $100(\$2.40)=\$240$, but it is not subtracted from any of these payoff amounts.

Common mistake avoided: Subtracting the premium and reporting $\$1,085$ as the payoff in the last case confuses payoff with a separate profit calculation. The expiration payoff is $\$1,325$.

Scaling from One Unit to a Position

Option quotations commonly state a payoff per unit of the underlying, while a position may contain several contracts, each covering a specified number of units. Keep those units explicit. First find the payoff per unit; then multiply by the units covered across the position. If contracts have different strikes, calculate each contract group separately before adding their payoffs.

The multiplier changes the dollar amount, not the exercise condition. For a position of $n$ identical long calls with contract multiplier $M$, the position has zero payoff when $S_T\leq K$, and a payoff of $nM(S_T-K)$ when $S_T>K$.

Worked Example: Several Contracts at One Strike

An investor holds four hypothetical long call contracts, each covering 100 shares, with strike $K=\$110$. The premium was $\$4.80$ per share. Find the total expiration payoff if the terminal share price is $\$108$, $\$110$, or $\$126$. Treat each terminal price as a specified scenario, with no probability distribution or interest rate assumed.

Method. There are $4(100)=400$ shares covered in total. Apply the positive-part payoff per share, then multiply by 400.

At $S_T=\$108$. The per-share difference is $\$108-\$110=-\$2$, so the per-share payoff is $\max(-\$2,0)=\$0$. Total payoff: $400(\$0)=\$0$.

At $S_T=\$110$. The per-share difference is $\$110-\$110=\$0$, so total payoff is $400(\$0)=\$0$.

At $S_T=\$126$. The per-share difference is $\$126-\$110=\$16$. Total payoff: $400(\$16)=\$6,400$.

Check. The total premium paid was $4(100)(\$4.80)=\$1,920$. That amount is distinct from the expiration payoffs: the three payoff results are $\$0$, $\$0$, and $\$6,400$. In particular, the $\$6,400$ result is not reduced by the premium when the requested quantity is payoff.

Common mistake avoided: Multiplying by four contracts but forgetting that each contract covers 100 shares would give an answer that is too small by a factor of 100.

Adding Payoffs Across Different Strikes

A position can contain long calls on the same underlying asset with different strikes. At a given expiration price, each call has its own positive-part payoff. Calculate each contribution using its own strike and contract quantity, then add the contributions. A call that is below its strike contributes zero, even if another call in the position has a positive payoff.

Worked Example: Two Long Calls with Different Strikes

A hypothetical position contains three long calls with strike $K_1=\$40$ and one long call with strike $K_2=\$55$. Every contract covers 100 shares, and all four calls expire at the same time on the same underlying. The premiums are not needed for this calculation. Find the total expiration payoff if the terminal share price is $\$50$ and if it is $\$62$. No probability distribution or interest rate is assumed.

At $S_T=\$50$. Each of the three calls with strike $\$40$ has a per-share payoff of $\max(\$50-\$40,0)=\$10$. Their combined payoff is $3(100)(\$10)=\$3,000$. The call with strike $\$55$ has payoff per share $\max(\$50-\$55,0)=\$0$, so its contract payoff is $100(\$0)=\$0$. The position total is $\$3,000+\$0=\$3,000$.

At $S_T=\$62$. Each call with strike $\$40$ has a per-share payoff of $\$62-\$40=\$22$. Their combined payoff is $3(100)(\$22)=\$6,600$. The call with strike $\$55$ has a per-share payoff of $\$62-\$55=\$7$, giving $100(\$7)=\$700$. The position total is $\$6,600+\$700=\$7,300$.

Check. At $\$50$, the higher-strike call contributes zero, while the lower-strike calls contribute $\$3,000$. At $\$62$, both groups contribute, and their separate values sum to $\$7,300$.

Common mistake avoided: Applying one strike to every contract would ignore the different exercise prices. Evaluate each group with its own strike before summing.

Common Errors to Avoid

Subtracting the premium from payoff. The premium is a separate earlier cash flow. The expiration payoff is only $\max(S_T-K,0)$ per unit.
Using the wrong sign. For a call, compare terminal price minus strike, $S_T-K$. If the result is negative, the payoff is zero; do not report a negative payoff for a long call.
Giving a positive payoff at the strike. When $S_T=K$, the difference is exactly zero, so the expiration payoff is zero.
Forgetting the contract multiplier. A per-share amount is not automatically the total contract amount. Multiply by the number of covered shares or other specified units.
Combining different strikes too early. Compute each call’s payoff with its own strike, then add the results.

Key takeaway: At expiration, a long call pays the positive part of terminal price minus strike. Calculate that amount per unit first, scale it to the contracts held, and keep the premium separate from the payoff.

Check Your Understanding

For each problem, calculate expiration payoff only. Unless stated otherwise, treat terminal prices as specified scenario values; no probability distribution or interest rate is assumed.

A long call has strike $\$32$ and covers one unit. What is its payoff if the terminal price is $\$29.50$? (Answer: $\$0$)
A position contains three long calls with strike $\$75$, each covering 100 shares. What is the total payoff if the terminal share price is $\$75$? (Answer: $\$0$)
An investor holds two long call contracts with strike $\$41$, each covering 100 shares. Find the total payoff if the terminal price is $\$49.60$. (Answer: $\$1,720$)
A position contains five long calls with strike $\$128$, each covering 10 units. Find the total payoff at terminal prices of $\$119$, $\$128$, and $\$143$. (Answer: $\$0$, $\$0$, and $\$750$, respectively)
A position contains three long calls with strike $\$20$ and two long calls with strike $\$25$, all covering 100 shares each and expiring on the same date. At a terminal share price of $\$31$, what is the total expiration payoff? (Answer: $\$4,500$)

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