Short Put Profit And Break-Even

From the Short Put to Expiration Profit

A short put position receives a premium when the option is sold, but the writer may have to buy the underlying asset for the strike price at expiration. The premium is income to the writer; the possible purchase obligation is a cost. To find the position’s profit, compare those two amounts for the underlying price at expiration.

The calculations in this tutorial are per underlying share. Let $S_T$ be the underlying price at expiration, $K$ the put’s strike price, and $p$ the premium received per share. Assume the option is held to expiration, the premium is not invested, and there are no transaction fees or other cash flows. No interest-rate, mortality, or probability-distribution assumption is needed: these are deterministic calculations conditional on the stated expiration price.

Definition: The short put’s expiration profit per share is the premium received less the put writer’s expiration obligation. For a standard put held to expiration, that obligation is $\max(K-S_T,0)$. Thus, $\text{profit}=p-\max(K-S_T,0)$.

If the underlying finishes below the strike, the put is exercised under the standard expiration convention: the writer buys the underlying for $K$ when it is worth $S_T$. The economic cost of that obligation, measured against the expiration price, is $K-S_T$ per share. If the underlying finishes at or above the strike, the put has no exercise value to its holder, so the writer has no purchase obligation. In either case, the writer has received $p$.

The Piecewise Profit Formula

The maximum expression can be written in two cases. Below the strike, subtract the obligation from the premium. At or above the strike, there is no obligation and the profit is the premium alone.

\Pi_{\text{short put}}(S_T) = p-\max(K-S_T,0) = \begin{cases} p-K+S_T, & S_T<K,\\ p, & S_T\ge K. \end{cases}

At $S_T=K$, both descriptions give the same result: the obligation is zero and profit is $p$. This boundary check matters because a put’s value does not become negative when the underlying price rises above the strike; the writer simply keeps the premium.

Break-even result: When $p<K$, the short put breaks even at $S_T=K-p$. This price is below the strike, so the put has an expiration obligation of $p$, exactly offsetting the premium received. When $p\ge K$, the algebraic solution $K-p$ is nonpositive. Since an underlying price is ordinarily nonnegative, there is no positive break-even price; if $p=K$, profit is zero only at $S_T=0$, and if $p>K$, profit is positive for every $S_T\ge0$.

Why the Break-Even Formula Holds

A break-even price is a price at which profit equals zero. For the usual case $0<p<K$, the candidate price must be below the strike, so the relevant branch of the piecewise formula is $p-K+S_T$. Set that branch equal to zero and solve for $S_T$:

0=p-K+S_T \quad\Longrightarrow\quad S_T=K-p.

The condition $p<K$ ensures that $K-p>0$, and because $p>0$, it also ensures $K-p<K$. The solution therefore lies in the below-strike case used to derive it. Substitution confirms the result: the obligation is $K-(K-p)=p$, so profit is $p-p=0$. This check is important; solving an equation is not enough unless the solution satisfies the case assumption.

Worked Example: Profit at Several Expiration Prices

A hypothetical put has strike $K=\$52.00$, and its writer receives a premium of $p=\$4.50$ per share. Find the profit per share if the underlying finishes at $\$60.00$, $\$52.00$, $\$49.00$, $\$47.50$, and $\$0.00$. Also find the break-even price. Use the expiration-only assumptions stated above; no interest rate or probability distribution is assumed.

Method. For each price, calculate the writer’s obligation $\max(52-S_T,0)$, then subtract it from the $\$4.50$ premium.

At $S_T=\$60.00$, the underlying is above the strike, so the obligation is $\max(52-60,0)=\$0.00$. Profit is $4.50-0.00=\$4.50$.

At $S_T=\$52.00$, the obligation is $\max(52-52,0)=\$0.00$. Profit is $4.50-0.00=\$4.50$.

At $S_T=\$49.00$, the obligation is $\max(52-49,0)=\$3.00$. Profit is $4.50-3.00=\$1.50$.

At $S_T=\$47.50$, the obligation is $\max(52-47.50,0)=\$4.50$. Profit is $4.50-4.50=\$0.00$.

At $S_T=\$0.00$, the obligation is $\max(52-0,0)=\$52.00$. Profit is $4.50-52.00=-\$47.50$.

For break-even, $K-p=52.00-4.50=\$47.50$, which agrees with the direct calculation. Check: at break-even, the $\$4.50$ obligation exactly uses the $\$4.50$ premium. The common mistake avoided here is treating the premium as the total profit even when the put finishes in the money.

Applying the Formula and Checking the Case

A reliable calculation starts by identifying whether the expiration price is below or at/above the strike. Then calculate the obligation, subtract it from the premium, and check the result against the break-even price. For a premium less than the strike, prices above break-even produce positive profit, the break-even price produces zero, and prices below break-even produce a loss.

Record the contract inputs.
Identify $K$, $p$, and the specified expiration price $S_T$, using consistent per-share units.

Calculate the writer’s obligation.
Use $\max(K-S_T,0)$. The writer’s obligation cannot be negative.

Subtract the obligation from the premium.
Profit is premium received minus expiration obligation, not obligation minus premium.

Verify the result.
For $0<p<K$, compare $S_T$ with $K-p$, and check that at $S_T=K-p$ the obligation equals the premium.

Worked Example: A Below-Strike Price Near Break-Even

A hypothetical short put has strike $K=\$83.00$ and premium $p=\$6.25$ per share. Calculate profit if the underlying finishes at $\$76.40$, $\$76.75$, and $\$70.50$. Find the break-even price and use it to check the results. Assume the option is held to expiration, with no fees or investment return on the premium; no interest rate or distribution assumption is required.

Method. First solve for break-even: $K-p=83.00-6.25=\$76.75$. Since $p<K$, this is a feasible price below the strike.

At $S_T=\$76.40$, the put obligation is $83.00-76.40=\$6.60$. Profit is $6.25-6.60=-\$0.35$. The price is $\$0.35$ below break-even, consistent with a loss of $\$0.35$.

At $S_T=\$76.75$, the obligation is $83.00-76.75=\$6.25$. Profit is $6.25-6.25=\$0.00$.

At $S_T=\$70.50$, the obligation is $83.00-70.50=\$12.50$. Profit is $6.25-12.50=-\$6.25$. This price is $\$6.25$ below break-even, and the profit is a loss of $\$6.25$.

Check: each dollar decrease in $S_T$ below the strike increases the obligation by one dollar per share. Relative to break-even, the first price is $\$0.35$ lower and the last is $\$6.25$ lower, matching the calculated losses. The method avoids rounding the break-even price prematurely or confusing a small loss with a positive profit.

Interpreting the Premium and the Boundary

The premium is received whether or not the put is exercised. If $S_T\ge K$, the writer’s profit is $p$, because there is no expiration obligation under the contract. If $S_T<K$, the premium offsets part of the obligation. The break-even price is therefore not the strike: it is the strike reduced by the premium, provided the resulting price is feasible.

The zero-price boundary is a useful check. If $S_T=0$, the put’s obligation per share is $K$, and profit is $p-K$. For the common case $p<K$, this is negative. The writer’s loss at that boundary is $K-p$, which is the magnitude of the break-even price, though the two values describe different things: one is a profit at zero underlying price, and the other is the underlying price that makes profit zero.

Worked Example: Checking the Zero-Price Boundary

A hypothetical put writer receives $p=\$9.75$ per share for a put with $K=\$120.00$. Calculate profit when $S_T=\$114.50$, $S_T=\$105.25$, and $S_T=\$0.00$, and identify break-even. Assume expiration-only profit, with no fees and no interest credited on the premium; no interest rate or probability distribution is needed.

Method. Calculate $K-p=120.00-9.75=\$110.25$. This is below the strike and is the candidate break-even price.

At $S_T=\$114.50$, the obligation is $120.00-114.50=\$5.50$. Profit is $9.75-5.50=\$4.25$, positive because $\$114.50$ is above break-even.

At $S_T=\$105.25$, the obligation is $120.00-105.25=\$14.75$. Profit is $9.75-14.75=-\$5.00$, negative because the price is $\$5.00$ below break-even.

At $S_T=\$0.00$, the obligation is $120.00-0.00=\$120.00$. Profit is $9.75-120.00=-\$110.25$.

At break-even, $S_T=\$110.25$, the obligation is $120.00-110.25=\$9.75$, leaving profit $9.75-9.75=\$0.00$. Check: the zero-price profit is $-\$110.25$, while the break-even underlying price is $+\$110.25$; their equal magnitudes follow from $p-K=-(K-p)$.

Common Exam Trap: Reversing the Cash-Flow Sign

A frequent error is to calculate $\max(K-S_T,0)-p$. That expression is the writer’s obligation less the premium, which reverses the writer’s profit sign. For example, with $K=\$52.00$, $p=\$4.50$, and $S_T=\$49.00$, the wrong calculation is $3.00-4.50=-\$1.50$. The correct calculation is premium received minus obligation: $4.50-3.00=+\$1.50$.

A second error is to use $K+p$ for the short put’s break-even price. That adds the premium to the strike, even though the premium reduces the amount of expiration decline the writer can absorb before reaching zero profit. The correct equation is $p-(K-S_T)=0$, which gives $S_T=K-p$.

Expiration condition	Writer’s obligation per share	Profit per share
$S_T\ge K$	$\$0$	$p$
$K-p<S_T<K$, when $p<K$	Positive, but less than $p$	Positive
$S_T=K-p$, when $p<K$	$p$	$\$0$
$0\le S_T<K-p$, when $p<K$	Greater than $p$	Negative

Key takeaway: For a short put, expiration profit per share equals premium received minus $\max(K-S_T,0)$. In the usual case $0<p<K$, break-even is $K-p$; verify it by checking that the put obligation at that price equals the premium.

Check Your Understanding

For each hypothetical position, calculate expiration profit per share or the requested break-even result. Assume the option is held to expiration, the premium is not invested, and there are no fees. No interest rate or probability distribution is assumed. Round money to the nearest cent.

A short put has $K=\$40.00$, $p=\$3.20$, and $S_T=\$38.00$. Find the obligation, profit, and break-even price. (Answer: obligation $\$2.00$; profit $+\$1.20$ per share; break-even $\$36.80$.)
A short put has $K=\$67.50$ and $p=\$4.50$. Find its break-even price and profit if $S_T=\$59.00$. (Answer: break-even $\$63.00$; obligation $\$8.50$; profit $-\$4.00$ per share.)
A short put has $K=\$92.00$, $p=\$7.25$, and $S_T=\$100.00$. Find the obligation and profit, then explain whether the position is at break-even. (Answer: obligation $\$0.00$; profit $+\$7.25$ per share; not break-even.)
A short put has $K=\$150.00$, $p=\$8.60$, and $S_T=\$140.25$. Calculate profit and compare $S_T$ with break-even. (Answer: break-even $\$141.40$; obligation $\$9.75$; profit $-\$1.15$ per share; the price is $\$1.15$ below break-even.)
A hypothetical short put has $K=\$24.00$ and $p=\$25.50$. For nonnegative expiration prices, determine whether a break-even price exists and find the minimum profit. (Answer: the algebraic break-even price is $-\$1.50$, so no nonnegative break-even price exists; minimum profit is $+\$1.50$ per share at $S_T=\$0.00$.)

Expiration condition	Writer’s obligation per share	Profit per share
\(S_T\ge K\)	\(\$0\)	\(p\)
\(K-p<S_T<K\), when \(p<K\)	Positive, but less than \(p\)	Positive
\(S_T=K-p\), when \(p<K\)	\(p\)	\(\$0\)
\(0\le S_T<K-p\), when \(p<K\)	Greater than \(p\)	Negative

What You'll Learn