Written Call Payoff at Expiration

Introduction

A call gives its holder the right to buy an underlying asset at a specified strike price. The writer takes the other side of that contract: if the call is exercised, the writer must meet the contractual obligation. At expiration, the writer’s option payoff is therefore the opposite of the holder’s payoff. The sign is central: a call that finishes above its strike has a positive payoff to the holder and a negative payoff to the writer.

This tutorial focuses on that expiration payoff, not on the premium received when the call was written and not on the writer’s total profit. Keeping those quantities separate prevents a common sign error and makes the payoff calculation work consistently for terminal prices below, at, or above the strike.

Definition: For a call written on one unit of an underlying asset, let $K$ be the strike price and $S_T$ the asset’s price at expiration $T$. The writer’s expiration payoff is the cash flow attributable to the option contract at expiration, excluding any premium paid or received earlier. Under standard call settlement, it is $$ -\max(S_T-K,0). $$

The Written Call Payoff

The expression $\max(S_T-K,0)$ is the call holder’s expiration payoff per underlying unit. The writer’s payoff is its negative. In piecewise form, that means the writer has no option cash flow when the asset price is below or equal to the strike, and a negative payoff when the asset price is above the strike.

\text{Writer payoff per unit} = -\max(S_T-K,0) = \begin{cases} 0, & S_T\leq K,\\ K-S_T, & S_T>K. \end{cases}

If the call is written on $q$ units, multiply the per-unit payoff by $q$. If a contract covers $M$ units and the writer has written $N$ contracts, the total payoff is

-NM\max(S_T-K,0).

Here $N$, $M$, and $S_T$ are taken as nonnegative quantities. The minus sign records the writer’s position; it does not mean that the underlying price or the call’s holder payoff is negative. The formula assumes the contract is held to expiration and uses the standard terminal payoff, without transaction costs or other contractual adjustments. The payoff is measured at expiration, so it is not discounted to an earlier date.

Worked Example: A Written Call Below and Above Its Strike

A trader has written three call contracts on a hypothetical share. Each contract covers 100 shares, the strike is $K=\$52$ per share, and the contracts are held to expiration. Calculate the trader’s option payoff if the expiration share price is (a) $\$47$ and (b) $\$61$. Ignore premiums and other costs.

Method. Calculate the payoff per share using $-\max(S_T-K,0)$, then multiply by the total number of shares covered. The total quantity is $3\times100=300$ shares.

(a) Expiration price $\$47$. The price is below the strike, so

S_T-K=47-52=-5,\qquad \max(-5,0)=0.

The writer’s payoff is $0$ per share and hence $300\times \$0=\$0$ in total.

(b) Expiration price $\$61$. The price is above the strike, so

S_T-K=61-52=9,\qquad \max(9,0)=9.

The writer’s payoff per share is $-\$9$. Across 300 shares, the total is $300\times(-\$9)=-\$2{,}700$. The negative sign means a $\$2{,}700$ loss in option payoff at expiration, before accounting for any premium received earlier.

Check. At $\$47$, the call has no positive intrinsic amount; at $\$61$, the holder’s payoff is $300\times\$9=\$2{,}700$. The writer’s payoff is exactly its negative. A common mistake is to report $+\$2{,}700$ for the writer because the price is $\$9$ above the strike. That is the holder’s payoff, not the writer’s.

Why the Writer’s Payoff Is the Negative

The negative sign follows from the two sides of one option contract. For every dollar of expiration payoff the contract provides to its holder, the writer has the corresponding dollar obligation. Thus the holder’s payoff and writer’s payoff sum to zero when considered as option payoffs between the two parties. This does not include fees or other external cash flows.

Find the holder’s terminal amount.
For a call on one unit, the holder’s payoff is $\max(S_T-K,0)$.

Reverse the position sign.
The writer owes the amount the holder receives, so the writer’s payoff is $-\max(S_T-K,0)$.

Scale to the contract quantity.
For $NM$ underlying units, multiply the per-unit writer payoff by $NM$.

For $S_T\leq K$, the positive-part expression is zero, so its negative is also zero. For $S_T>K$, the positive part equals $S_T-K$, and its negative is $-(S_T-K)=K-S_T$. These two cases produce the piecewise formula. At $S_T=K$, both expressions give zero; there is no discontinuity in the payoff at the strike.

\max(S_T-K,0)+\bigl[-\max(S_T-K,0)\bigr]=0.

This identity is a useful sign check, but it is about the option payoff only. The writer may have received a premium at an earlier date. That earlier cash flow is not part of the expiration payoff calculated here.

Worked Example: Applying a Contract Multiplier

A hypothetical exchange-traded call has a contract multiplier of 25 underlying units. An investor has written four contracts with strike $K=\$80$. At expiration, the underlying price is $\$83.20$. Calculate the writer’s total option payoff. Ignore premiums, fees, and taxes.

Method. First find the call payoff per underlying unit, then apply the contract multiplier and number of written contracts. There are $4\times25=100$ underlying units in total.

The amount by which the terminal price exceeds the strike is

$$ S_T-K=83.20-80=3.20. $$

The holder’s payoff is $\$3.20$ per underlying unit, so the writer’s payoff per unit is $-\$3.20$. Scaling to 100 units gives

100\times(-\$3.20)=-\$320.

The writer’s total option payoff is therefore $-\$320$. Check: the holder’s payoff across the same 100 units is $100\times\$3.20=\$320$, the exact opposite. A common mistake is to multiply the $\$3.20$ price difference by four contracts only, omitting the multiplier of 25; that would understate the magnitude of the payoff.

Several Written Calls and Boundary Cases

When a position includes calls with different strikes, calculate each call’s payoff separately and then add the results. Calls that are not above their strikes contribute zero; calls that are above their strikes contribute negative amounts for the writer. Do not apply one strike to all contracts unless the contracts actually share that strike.

Worked Example: Combining Two Written Call Positions

An investor has written two calls with strike $\$105$ and one call with strike $\$98$, all on the same hypothetical asset. Each call covers 10 units. Find the combined expiration payoff when the asset price is (a) $\$102$ and (b) $\$110$. Ignore premiums and other costs.

Method. Evaluate each strike separately using the writer’s per-unit payoff, multiply by the number of contracts and 10 units per contract, and add.

(a) $S_T=\$102$. For the two calls with strike $\$105$, the terminal price is below the strike. Their combined payoff is $2\times10\times\$0=\$0$. For the one call with strike $\$98$,

-\max(102-98,0)=-\$4

per unit, so its total payoff is $10\times(-\$4)=-\$40$. The combined payoff is $\$0-\$40=-\$40$.

(b) $S_T=\$110$. For each call with strike $\$105$, the writer’s payoff is $-\max(110-105,0)=-\$5$ per unit. For two contracts covering 10 units each, the total is $2\times10\times(-\$5)=-\$100$. For the call with strike $\$98$, the payoff is $-\max(110-98,0)=-\$12$ per unit, or $10\times(-\$12)=-\$120$. The combined payoff is $-\$100-\$120=-\$220$.

Check. At $\$102$, only the $\$98$ strike is exceeded; at $\$110$, both strikes are exceeded. The combined payoffs, $-\$40$ and $-\$220$, are the negatives of the corresponding combined holder payoffs. A common mistake is to calculate every call using the $\$105$ strike, which would incorrectly give no payoff at $\$102$ for the call written at $\$98$.

Common Mistakes and a Reliable Check

The written call payoff is simple to calculate, but errors often arise from confusing the position, the unit of measurement, or the date of the cash flow. Before finalizing a result, check the following:

Reversing the sign. Above the strike, the writer’s payoff is negative. The positive amount belongs to the holder.
Using a premium in the payoff. The payoff formula uses $S_T$ and $K$; it does not include an earlier premium. A total profit calculation is a different quantity.
Ignoring the contract multiplier. Calculate the payoff per underlying unit first, then scale by all units covered.
Applying the wrong strike. For a position with multiple strikes, evaluate each written call separately.
Treating the strike boundary incorrectly. At $S_T=K$, the payoff is zero, not a positive or negative amount.

A quick consistency test is to calculate the holder’s payoff for the same quantity. The writer’s expiration payoff must have the opposite sign and the same magnitude. If the terminal price is no greater than the strike, the call writer’s payoff cannot be negative under this standard payoff model; if the price is above the strike, it cannot be positive.

Key takeaway: For a written call on $NM$ underlying units, the expiration payoff is $-NM\max(S_T-K,0)$. First calculate the amount above the strike, replace any negative amount with zero, reverse the sign for the writer, and then apply the full contract quantity.

Check Your Understanding

For each problem, calculate the written call’s expiration payoff only; exclude premiums and other costs.

An investor has written one call on 40 units with strike $\$35$. The terminal price is $\$32$. What is the payoff? (Answer: $\$0$)
A trader has written two contracts, each covering 50 units, with strike $\$24$. The terminal price is $\$27$. Find the total payoff. (Answer: $-\$300$)
A writer has sold three calls with strike $\$140$, each covering 10 units. If the terminal price is exactly $\$140$, what is the total payoff? (Answer: $\$0$)
An investor has written one 20-unit call at strike $\$76$ and one 20-unit call at strike $\$83$. The terminal price is $\$80$. Find the combined payoff. (Answer: $-\$80$)
A trader has written five call contracts, each covering 15 units, at strike $\$48.50$. The terminal price is $\$51.10$. What is the total payoff? (Answer: $-\$195$)

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