Written Option Break-Even Thresholds

Introduction

For a written option, the seller receives a premium but may owe a payment at expiration. The break-even terminal price is the price at which the premium received exactly offsets that expiration payment. Finding it requires keeping the seller’s sign straight: the premium is a gain, while the option payoff is a cost.

This tutorial derives the break-even thresholds for written calls and puts, then uses them to identify the regions of positive profit, zero profit, and loss. The distinction between a threshold and the full set of break-even prices matters in boundary cases, especially when a premium is zero or a put premium is at least as large as the strike.

Model and notation: Consider one option on an asset with nonnegative terminal price $S_T$, strike $K>0$, and expiration at time $T$. Let $c\geq 0$ be the call premium received per unit and $p\geq 0$ the put premium received per unit. Assume the premium is received at the start, and define expiration profit as premium received minus the option payoff, without interest on the premium, transaction costs, or taxes. The call payoff per unit is $(S_T-K)^+$, and the put payoff per unit is $(K-S_T)^+$, where $x^+=\max(x,0)$.

Written Call: Premium Above the Strike

The writer of a call receives $c$ and is responsible for the call payoff at expiration. Thus the writer’s profit per unit is the premium received less that payoff. A break-even price is a value of $S_T$ at which this profit is zero.

\Pi_{\text{written call}}(S_T)=c-(S_T-K)^+.

When $S_T\leq K$, the call payoff is zero, so the writer keeps the premium. When $S_T>K$, the payoff is $S_T-K$. Solving for zero profit in this latter region gives $c-(S_T-K)=0$, or $S_T=K+c$. If $c>0$, this value is indeed greater than $K$, so it lies in the region where the call payoff formula used in the calculation applies.

Theorem: If $c>0$, a written call has one break-even terminal price, $K+c$. Its profit is positive for $0\leq S_T<K+c$, zero at $S_T=K+c$, and negative for $S_T>K+c$. If $c=0$, profit is zero for every $0\leq S_T\leq K$ and negative for $S_T>K$; in that case $K$ is a boundary break-even value, but it is not the only break-even price.

The zero-premium case is worth stating separately. Substituting $c=0$ gives profit $-(S_T-K)^+$: the writer does not make a positive profit at any terminal price, because no premium was received. Below or at the strike the profit is zero, not positive.

Worked Example: Break-Even for a Written Call

A writer sells one call with strike $K=\$52$ and receives a premium of $c=\$4.50$ per unit. Find the break-even terminal price and the writer’s profit if the asset finishes at $\$49$, $\$55$, and $\$60$.

Method. Subtract the call payoff from the premium received. Since $c>0$, set the profit equal to zero in the region above the strike:

4.50-(S_T-52)=0 \quad\Longrightarrow\quad S_T=52+4.50=56.50.

The proposed break-even price, $\$56.50$, is above the strike $\$52$, as required for this calculation. Check it directly: the payoff is $\$56.50-\$52=\$4.50$, so profit is $\$4.50-\$4.50=\$0$.

At $S_T=\$49$, the payoff is $\max(49-52,0)=\$0$, so profit is $\$4.50-\$0=\$4.50$.

At $S_T=\$55$, the payoff is $\max(55-52,0)=\$3$, so profit is $\$4.50-\$3=\$1.50$.

At $S_T=\$60$, the payoff is $\max(60-52,0)=\$8$, so profit is $\$4.50-\$8=-\$3.50$.

The results have the expected signs: the first two terminal prices are below $\$56.50$ and give positive profit, while $\$60$ is above it and gives a loss. The calculation avoids the common mistake of adding the payoff to the premium instead of subtracting it.

Written Put: Premium Below the Strike

The writer of a put also receives a premium, but the put payoff is based on how far the terminal price falls below the strike. The profit per unit is therefore the premium less the put payoff.

\Pi_{\text{written put}}(S_T)=p-(K-S_T)^+.

For $S_T<K$, the payoff is $K-S_T$. Setting profit to zero in that region gives $p-(K-S_T)=0$, so the candidate break-even value is $S_T=K-p$. This value is below $K$ only when $p>0$. Whether it is a possible terminal price, and whether there is a loss region, depends on its sign.

Theorem: If $0<p<K$, the written put has one break-even terminal price, $K-p$. It makes a loss for $0\leq S_T<K-p$, breaks even at $S_T=K-p$, and makes a positive profit for $S_T>K-p$. If $p=0$, profit is negative for $0\leq S_T<K$ and zero for $S_T\geq K$. If $p\geq K$, there is no loss at any positive terminal price: for $p=K$, profit is zero at $S_T=0$ and positive for $S_T>0$; for $p>K$, profit is positive for every $S_T\geq0$.

The condition $p\geq K$ explains when a written put has no positive-price loss region. If $p=K$, then at $S_T=0$ the maximum possible put payoff $K$ exactly uses up the premium. At any positive terminal price, the payoff is less than $K$, so the premium exceeds the payoff. If $p>K$, even the maximum payoff at zero is less than the premium. A stipulated premium in this boundary case is used to examine the algebra; this calculation alone does not establish whether a quoted premium is consistent with a market model.

Worked Example: A Written Put With No Positive-Price Loss

A hypothetical written put has strike $K=\$45$ and premium $p=\$45$ per unit. Determine its break-even terminal price and whether the writer loses money at any positive terminal price.

Method. Compare the premium with the strike, which is the largest possible payoff when $S_T\geq0$. Here $p=K$, so the candidate threshold is $K-p=\$45-\$45=\$0$.

At $S_T=\$0$, the payoff is $\max(45-0,0)=\$45$, and profit is $\$45-\$45=\$0$. This verifies the boundary break-even.

For any $S_T>0$ below $\$45$, the payoff is $45-S_T$, which is less than $\$45$. Thus profit is $45-(45-S_T)=S_T>0$. For $S_T\geq\$45$, the payoff is zero and profit is $\$45$, also positive.

Therefore, the writer has no loss at any positive terminal price. The only break-even price is zero. The trap is to treat $K-p=0$ as a negative-price loss threshold and overlook that terminal prices are nonnegative.

Deriving and Checking the Thresholds

Both thresholds follow from the same accounting principle: at break-even, the premium received equals the payoff owed. The positive-part payoff formulas make the derivation precise and also show when a candidate threshold is outside the relevant payoff region.

Write the seller’s profit.
For a call, profit is $c-(S_T-K)^+$; for a put, it is $p-(K-S_T)^+$. The payoff is subtracted because it is paid by the writer.

Use the nonzero-payoff branch.
For a call above $K$, set $c-(S_T-K)=0$, giving $S_T=K+c$. For a put below $K$, set $p-(K-S_T)=0$, giving $S_T=K-p$.

Check the candidate and the boundary.
The call candidate $K+c$ is above $K$ when $c>0$. The put candidate $K-p$ is below $K$ when $p>0$, but it is a feasible terminal price only when $p\leq K$. If a threshold is at or below zero, use $S_T\geq0$ to describe the actual break-even and loss regions.

Substitute to verify.
At a proposed break-even, calculate the payoff and confirm it equals the premium. This catches sign errors and thresholds that were derived using a payoff branch that does not apply.

For a put with $p<K$, the threshold $K-p$ is positive. Below that price, $K-S_T>p$, so the payoff exceeds the premium and profit is negative. Above the threshold but still below the strike, the payoff is less than the premium. At or above the strike, the payoff is zero and the premium is retained. For $p\geq K$, the minimum possible profit occurs at $S_T=0$, where it is $p-K$; this is nonnegative, confirming there is no loss at positive prices.

Worked Example: A Written Put With a Positive Loss Region

A writer receives $p=\$9.50$ for a put with strike $K=\$72$. Find the break-even terminal price, then calculate profit if the asset finishes at $\$58$, $\$62.50$, or $\$70$.

Method. Since $0<\$9.50<\$72$, the written put has the positive threshold $K-p$:

$$ 72-9.50=62.50. $$

At $S_T=\$62.50$, the payoff is $\max(72-62.50,0)=\$9.50$, equal to the premium. Profit is $\$9.50-\$9.50=\$0$.

At $S_T=\$58$, the payoff is $72-58=\$14$, so profit is $\$9.50-\$14=-\$4.50$.

At $S_T=\$62.50$, as checked above, profit is $\$0$.

At $S_T=\$70$, the payoff is $72-70=\$2$, so profit is $\$9.50-\$2=\$7.50$.

The signs agree with the threshold: $\$58$ is below $\$62.50$ and produces a loss; $\$70$ is above it and produces a gain. The common mistake is to use $K+p=\$81.50$ for a written put. The payoff below the strike is $K-S_T$, so solving for zero profit subtracts the premium from $K$, not adds it.

Common Mistakes and Practical Checks

Reversing the premium sign. A writer receives the premium, so it is positive in the profit formula. The payoff is an amount owed, so it is subtracted.
Using the long-option threshold for the writer. A written call’s threshold is $K+c$; a written put’s candidate threshold is $K-p$. The signs differ because the payoff formulas differ.
Assuming every threshold is a unique break-even price. A zero-premium written call breaks even for all $S_T\leq K$, not only at $K$. A zero-premium written put breaks even for all $S_T\geq K$. State the full set when a premium is zero.
Ignoring the nonnegative-price constraint. If $p>K$, then $K-p<0$, but a negative asset price is outside the model. The correct conclusion is that the written put has positive profit at every feasible terminal price, not that it loses below a negative threshold.
Forgetting the profit convention. These formulas compare the premium received with the expiration payoff without accumulating the premium with interest. If a problem explicitly asks for terminal value including interest on the premium, the premium must first be accumulated to expiration.

A reliable answer states the premium and strike, writes profit as premium minus payoff, solves for zero profit on the appropriate payoff branch, and checks the result against $S_T\geq0$. This sequence is aligned with the SOA Exam FM (Financial Mathematics) syllabus and prevents a mathematically correct equation from being applied outside its valid region.

Check Your Understanding

For each situation, find the break-even value or values and classify the requested terminal-price outcomes. Treat all profits as per unit.

A written call has strike $K=\$31$ and premium $c=\$2.75$. Find its break-even price and its profit at terminal prices of $\$33$, $\$35$, and $\$38$. (Answer: Break-even $\$33.75$; profits are $\$0.75$, $-\$1.25$, and $-\$4.25$, respectively.)
A written call has strike $K=\$84$ and receives no premium. Identify all break-even terminal prices and the loss region. (Answer: Break-even for every $0\leq S_T\leq\$84$; loss for $S_T>\$84$; no positive-profit region.)
A written put has strike $K=\$50$ and premium $p=\$6.40$. Find the break-even price and profit at $S_T=\$40$ and $S_T=\$48$. (Answer: Break-even $\$43.60$; profit at $\$40$ is $-\$3.60$, and at $\$48$ it is $\$4.40$.)
A written put has strike $K=\$27$ and premium $p=\$27$. Determine whether any positive terminal price gives a loss and state the break-even price. (Answer: No positive terminal price gives a loss; the break-even price is $S_T=\$0$.)
A written put has strike $K=\$63$ and premium $p=\$4.25$. State the loss region and calculate the writer’s profit at $S_T=\$55$ and $S_T=\$60$. (Answer: Break-even $\$58.75$, with loss for $0\leq S_T<\$58.75$; profit at $\$55$ is $-\$3.75$, and at $\$60$ it is $\$1.25$.)

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