Long Put Break-Even Price

What Long Put Break-Even Means

A long put gives its holder the right, but not the obligation, to sell an underlying asset for the strike price. At expiration, its payoff depends on the relationship between the underlying price and the strike. To find the break-even underlying price, however, payoff alone is not enough: the holder must also recover the premium paid for the option.

This tutorial focuses on the expiration break-even price: the underlying price at expiration for which the long put’s profit, after subtracting its premium, is zero. The calculation uses the option payoff at expiration and the premium paid. It does not include transaction costs, taxes, or interest earned or paid on the premium; if a problem specifies any of those, incorporate them separately.

Definition: Let $K>0$ be the put’s strike price, $S_T\ge 0$ the underlying price at expiration, and $p\ge 0$ the premium paid per unit of the underlying. The long put’s expiration payoff per unit is $\max(K-S_T,0)$. Its expiration profit per unit, under the convention used here, is $\max(K-S_T,0)-p$. A break-even price is a feasible value of $S_T$ for which this profit equals zero.

The premium and payoff must use the same unit. If the quoted premium is per share, use a per-share payoff. If a contract covers $M$ units and $p$ is the premium per unit, total expiration profit is $M[\max(K-S_T,0)-p]$. For $M>0$, multiplying by $M$ changes the dollar profit but not the underlying price at which profit is zero.

Deriving the Break-Even Price

The payoff is piecewise: when $S_T<K$, the put has payoff $K-S_T$; when $S_T\ge K$, it has payoff zero. A proposed break-even price must lie in the region whose payoff was used to solve the equation. This condition matters, particularly when the premium is zero or large relative to the strike.

\text{Profit}= \begin{cases} K-S_T-p, & 0\le S_T<K,\\ -p, & S_T\ge K. \end{cases}

In the first region, setting profit equal to zero gives $K-S_T-p=0$, hence $S_T=K-p$. In the second region, profit is $-p$, which equals zero only when $p=0$. Therefore, the equation $S_T=K-p$ is a valid solution in the positive-payoff region only when it actually falls below the strike and is feasible.

Write expiration profit.
Subtract the premium from the long put payoff: $\max(K-S_T,0)-p$.

Use the positive-payoff region.
For $0\le S_T<K$, set $K-S_T-p=0$, which gives $S_T=K-p$.

Check feasibility and the other region.
Confirm that the solution is a feasible expiration price below $K$, then check whether any $S_T\ge K$ also gives zero profit.

Theorem: For a long put with $K>0$ and $p\ge 0$, the expiration break-even prices are as follows:

If $0<p<K$, there is one break-even price, $S_T=K-p$, which lies strictly between zero and $K$.
If $p=0$, every $S_T\ge K$ is a break-even price. For $S_T<K$, the put has positive profit.
If $p=K$, the only break-even price is $S_T=0$.
If $p>K$, there is no feasible break-even price.

These cases follow directly from the payoff regions. When $0<p<K$, $0<K-p<K$, so the candidate price is feasible and lies in the region where the equation was derived. At $S_T\ge K$, profit is $-p<0$, so there is no second solution. When $p=0$, profit is zero throughout the entire region $S_T\ge K$, not just at $S_T=K$. When $p=K$, the equation gives $S_T=0$, and that is the only feasible zero-profit price: for $0<S_T<K$, profit is $-S_T<0$, while for $S_T\ge K$, it is $-K<0$. Finally, if $p>K$, the payoff can never exceed $K$, so even at $S_T=0$, profit is at most $K-p<0$.

Worked Calculations

Worked Example: A Premium Below the Strike

A hypothetical investor buys a put with strike $K=\$84.00$ for a premium of $p=\$7.25$ per share. Find the expiration break-even price, then verify the result at the break-even price and at two other expiration prices.

Method. Since $0<\$7.25<\$84.00$, the unique break-even price is in the positive-payoff region. Set the put payoff equal to the premium:

$$ K-S_T=p $$

Solving gives $S_T=K-p=\$84.00-\$7.25=\$76.75$. This is below the strike and above zero, as required.

Check at the proposed break-even. At $S_T=\$76.75$, payoff is $\max(\$84.00-\$76.75,0)=\$7.25$. Profit is $\$7.25-\$7.25=\$0.00$.

Check below it. At $S_T=\$70.00$, payoff is $\$84.00-\$70.00=\$14.00$, so profit is $\$14.00-\$7.25=\$6.75$.

Check above it but below the strike. At $S_T=\$79.00$, payoff is $\$84.00-\$79.00=\$5.00$, so profit is $\$5.00-\$7.25=-\$2.25$. The sign changes around $\$76.75$, as expected. The common mistake avoided here is adding the premium to the strike; for a long put, the premium is recovered by a price decrease below the strike.

Worked Example: Converting a Contract Premium to a Per-Unit Break-Even

A hypothetical put contract covers 100 shares, has strike $K=\$52.00$ per share, and costs $\$235.00$ in total. Find the expiration break-even share price. Verify the total contract profit at $S_T=\$49.00$ and at $S_T=\$52.00$.

Method. First convert the premium to a per-share amount: $\$235.00/100=\$2.35$ per share. Because $0<\$2.35<\$52.00$, the break-even price is $K-p=\$52.00-\$2.35=\$49.65$.

The total-contract formula gives the same result. At break-even, the payoff on 100 shares must equal the $\$235.00$ premium:

100(\$52.00-S_T)=\$235.00

Dividing by 100 gives $\$52.00-S_T=\$2.35$, so $S_T=\$49.65$. This price is below the strike, making the positive-payoff equation applicable.

Verification at $S_T=\$49.00$. Per-share payoff is $\$52.00-\$49.00=\$3.00$, and total payoff is $100(\$3.00)=\$300.00$. Subtracting the total premium gives profit $\$300.00-\$235.00=\$65.00$.

Verification at $S_T=\$52.00$. The payoff is zero, so total profit is $\$0.00-\$235.00=-\$235.00$. The break-even price is a share price, not a total contract value. Applying the total premium directly to the per-share strike would mix units and give an invalid result.

Worked Example: Zero, Equal-to-Strike, and Above-Strike Premiums

Consider three hypothetical long puts, each with strike $K=\$30.00$, but with premiums of (a) $\$0.00$, (b) $\$30.00$, and (c) $\$33.50$. Determine the break-even price or prices in each case.

(a) Premium $p=\$0.00$. For $S_T<\$30.00$, profit is $\$30.00-S_T$, which is positive. For every $S_T\ge \$30.00$, payoff is zero and profit is also zero. Thus the break-even prices are all feasible prices $S_T\ge \$30.00$, not a single price. The expression $K-p=\$30.00$ identifies the boundary, but does not describe the full set of break-even prices.

(b) Premium $p=\$30.00=K$. At $S_T=\$0.00$, payoff is $\$30.00$, so profit is $\$30.00-\$30.00=\$0.00$. If $0<S_T<\$30.00$, profit is $\$30.00-S_T-\$30.00=-S_T<0$; if $S_T\ge \$30.00$, profit is $-\$30.00$. The only break-even price is $\$0.00$.

(c) Premium $p=\$33.50>K$. The greatest possible payoff is $\$30.00$, attained at $S_T=\$0.00$. Even there, profit is $\$30.00-\$33.50=-\$3.50$. At any larger underlying price, payoff is no greater, so profit remains negative. There is no feasible break-even price. The common mistake avoided is treating $K-p$ as a valid solution without checking whether it is a nonnegative underlying price.

Common Errors and Practical Checks

A reliable answer includes both an equation and a feasibility check. The most frequent sign error is to use the long call expression $K+p$ for a long put. For example, with $K=\$84.00$ and $p=\$7.25$, that incorrect calculation gives $\$91.25$. But at $S_T=\$91.25$, the put payoff is zero and its profit is $-\$7.25$, not zero. The correct put calculation is $K-p=\$76.75$, where payoff exactly equals the premium.

Other checks help catch different errors:

Check the region. A solution obtained from $K-S_T-p=0$ must satisfy $0\le S_T<K$. If it does not, do not report it without analyzing the boundary cases.
Check the premium range. For $p=0$, the break-even set is not unique. For $p=K$, the only break-even is zero. For $p>K$, there is no break-even.
Check the units. If the premium is stated per contract, divide by the number of underlying units before using the per-unit formula, or use total payoff and total premium consistently.
Check profit, not just payoff. A put can have a positive payoff while still producing a loss if that payoff is less than the premium.

This result concerns profit at expiration under the stated convention. Before expiration, an option may have a market value that differs from its expiration payoff; the expiration break-even calculation alone does not determine interim profit. For exam calculations, identify whether the question asks about payoff or profit, use the specified premium units, and then test the candidate price in the original profit expression.

Key takeaway: For $K>0$ and $0<p<K$, a long put’s unique expiration break-even price is $K-p$. Always verify the price is feasible and check the zero-premium, equal-to-strike, and above-strike cases separately.

Check Your Understanding

For each problem, use expiration profit and check any candidate price in the appropriate payoff region.

A hypothetical put has strike $K=\$96.00$ and premium $p=\$5.60$ per share. Find its unique expiration break-even price. (Answer: $\$90.40$ per share)
A put has strike $K=\$41.00$ and premium $p=\$0.75$ per share. Find the break-even price and the per-share profit if $S_T=\$39.50$. (Answer: break-even $\$40.25$; profit $\$0.75$ per share)
A put has strike $K=\$68.00$ and premium $p=\$68.00$. State all feasible expiration break-even prices. (Answer: $S_T=\$0.00$ only)
A put has strike $K=\$25.00$ and premium $p=\$27.00$. Determine whether a feasible break-even exists and find the greatest possible profit. (Answer: no break-even; greatest profit is $-\$2.00$ per share at $S_T=\$0.00$)
A hypothetical contract covers 100 shares, has strike $K=\$63.00$, and costs $\$185.00$ in total. Find the break-even share price and the total contract profit if $S_T=\$60.90$. (Answer: break-even $\$61.15$ per share; profit $\$25.00$)

What You'll Learn