Long Put Profit and Break-Even

From Put Payoff to Buyer’s Profit

A purchased put gives its holder the right to sell the underlying asset for the strike price at expiration. Its expiration payoff depends on the terminal asset price, but the buyer’s profit also reflects the premium paid to acquire the option. Keeping those two amounts separate is essential: a put can have a positive payoff and still produce a loss if the payoff is smaller than the premium.

This tutorial uses the standard expiration-profit convention: subtract the premium paid at initiation from the payoff received at expiration, without accumulating or discounting the premium. The underlying asset is assumed to have a nonnegative terminal price. The formulas omit transaction costs, taxes, and other fees. This treatment of long put profit and break-even is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Let $K\ge 0$ be the put’s strike price, let $p\ge 0$ be the premium paid per unit of the underlying, and let $S_T\ge 0$ be the underlying’s price at expiration. The long put’s payoff per unit is $\max(K-S_T,0)$. Its expiration profit per unit is the payoff minus the initial premium $p$.

\Pi_T=\max(K-S_T,0)-p

Here, $\Pi_T$ denotes profit per unit at expiration. A contract covering more than one unit would require scaling both the premium and the payoff by the contract’s unit amount. The examples below use one unit so that the option’s payoff, premium, and profit can be compared directly.

Deriving the Break-Even Price

A break-even terminal price is a value of $S_T$ at which the buyer’s profit equals zero. First consider the region $S_T<K$. In this region the holder can sell at the strike, which exceeds the market price, so the payoff is $K-S_T$. Subtracting the premium gives the profit. Setting that expression equal to zero identifies the candidate break-even price.

\begin{aligned} \Pi_T&=K-S_T-p &&\text{when }S_T<K,\\ 0&=K-S_T-p,\\ S_T&=K-p. \end{aligned}

This algebra produces a candidate, not an automatic guarantee that a break-even price exists. The candidate must satisfy the assumptions used to obtain it: the terminal price must be nonnegative, and it must lie in the region $S_T<K$. If $p>0$, then $K-p<K$, so the second condition is satisfied whenever the candidate is nonnegative. The nonnegative-price condition requires $K-p\ge 0$, or $p\le K$.

The other region is $S_T\ge K$. There the put payoff is zero, so the profit is $-p$. If $p>0$, this region cannot contain a break-even price. Thus, for a positive premium, the put breaks even at a feasible terminal price precisely when $p\le K$, and the break-even price is $K-p$. If $p>K$, the algebraic candidate is negative and has no interpretation as a possible asset price.

\begin{array}{ll} 0\le p<K: & \text{break-even at }S_T=K-p>0,\\ p=K: & \text{break-even only at }S_T=0,\\ p>K: & \text{no nonnegative break-even terminal price.} \end{array}

If $p=0$, the same profit equation gives a break-even at $S_T=K$: the buyer pays nothing and receives zero payoff there. For positive premiums, the break-even lies below the strike. The boundary cases are useful checks, especially when a problem gives a premium close to or greater than the strike.

Worked Calculations

Worked Example: Profit Below the Break-Even Price

A hypothetical one-unit put has strike $K=\$120$ and premium $p=\$9$. At expiration the underlying price is $\$102$. Calculate the payoff and profit, find the break-even terminal price, and check the result at break-even. Assume no transaction costs or premium accumulation.

Method. Calculate the expiration payoff first, then subtract the premium. Separately set the profit to zero to find break-even. This avoids the common mistake of calling the payoff itself the profit.

Payoff at $\$102$. Since $102<120$, the payoff is the difference between the strike and terminal price:

\max(120-102,0)=\max(18,0)=\$18.

Profit at $\$102$. Subtract the $\$9$ premium:

\$18-\$9=\$9.

Break-even. The premium is less than the strike, so the break-even is feasible:

S_T=K-p=120-9=\$111.

Check. At $S_T=\$111$, the payoff is $\$120-\$111=\$9$, and profit is $\$9-\$9=\$0$. At $\$102$, the profit is $\$9$, which is consistent with the terminal price being $\$9$ below break-even in the positive-payoff region.

Worked Example: Finding Prices That Meet a Profit Target

A hypothetical investor pays $\$7$ for a one-unit put with strike $K=\$64$. Find the highest nonnegative expiration price at which the investor earns at least $\$5$ per unit. Also find the break-even price and verify the profit at the highest qualifying price.

Method. A positive profit target requires a positive payoff, so first work in the region $S_T<64$. Apply the target to profit, not payoff. This avoids forgetting that the $\$7$ premium must be recovered before the trade earns a profit.

In the positive-payoff region, profit is $64-S_T-7$. The requirement is:

64-S_T-7\ge 5.

Combining the constants gives $57-S_T\ge 5$, so $-S_T\ge-52$, or $S_T\le52$. The underlying price must also be nonnegative. Therefore the full range meeting the target is $0\le S_T\le \$52$. The highest qualifying price is $\$52$.

Break-even. The premium is below the strike, so:

S_T=64-7=\$57.

Check at the highest qualifying price. At $S_T=\$52$, the payoff is $64-52=\$12$. Subtracting the $\$7$ premium gives profit $12-7=\$5$, exactly the target. At the break-even price of $\$57$, the payoff is $64-57=\$7$, so profit is zero. Prices above $\$57$ cannot meet the $\$5$ target; in particular, at or above the strike the payoff is zero and profit is $-\$7$.

Worked Example: When the Algebraic Threshold Is Impossible

A hypothetical put has strike $K=\$48$ and premium $p=\$53$ per unit. Determine whether any nonnegative terminal price breaks even, and calculate the profit if the terminal price is $\$0$ and if it is $\$60$. Assume the stated premium is paid at initiation and is not accumulated.

Method. Check the feasible domain before treating the algebraic solution as an attainable price. This avoids reporting a negative break-even price as though an underlying asset could reach it.

The zero-profit equation in the positive-payoff region gives the candidate:

S_T=K-p=48-53=-\$5.

A terminal asset price cannot be negative under the stated model. Hence there is no nonnegative break-even price. This conclusion can also be checked across the entire domain. If $0\le S_T<48$, the payoff is at most $\$48$, so profit is at most $48-53=-\$5$. If $S_T\ge48$, the payoff is zero and profit is $-\$53$. Neither region contains zero profit.

At $S_T=\$0$: payoff is $\max(48-0,0)=\$48$, and profit is $48-53=-\$5$.

At $S_T=\$60$: payoff is $\max(48-60,0)=\$0$, and profit is $0-53=-\$53$.

Check. The most favorable possible terminal price for this put is zero, where its payoff is the full $\$48$ strike. Because that payoff is still $\$5$ less than the premium, the buyer cannot break even at any nonnegative terminal price.

Reading the Threshold Correctly

For a positive premium not exceeding the strike, $K-p$ has a direct interpretation: it is the terminal price at which the put’s exercise value exactly repays the premium. A terminal price below that level produces a positive profit; a price between that level and the strike produces a positive payoff that is insufficient to recover the premium; and a price at or above the strike produces no payoff. This description follows directly from comparing the payoff with $p$, rather than from the payoff alone.

A frequent exam trap is to solve $K-S_T=p$, obtain $S_T=K-p$, and stop without checking the domain. For example, with $K=\$35$ and $p=\$39$, the equation gives $S_T=-\$4$. The algebra is correct, but the conclusion “break-even is $-\$4$” is not: the assumed asset price cannot be negative. The correct conclusion is that no nonnegative terminal price breaks even. At $S_T=0$, the payoff is only $\$35$, still less than the $\$39$ premium.

Another mistake is to report the payoff $K-S_T$ as the profit whenever the put finishes in the money. The payoff is the amount received from exercising; the profit must subtract the premium. For instance, a payoff of $\$12$ with a $\$7$ premium gives a $\$5$ profit, not a $\$12$ profit. Conversely, if a question explicitly changes the accounting convention by asking for a terminal value that includes financing the premium, the premium must be carried forward at the specified rate. That is a different calculation from the expiration-profit convention used here.

Identify the inputs.
Record the strike $K$, premium $p$, and terminal price $S_T$, using consistent per-unit amounts.

Calculate the payoff.
Use $\max(K-S_T,0)$; a put’s payoff cannot be negative.

Calculate profit or solve for zero.
Subtract $p$ from the payoff. For break-even, solve the zero-profit equation in the positive-payoff region.

Check feasibility and verify.
Require $S_T\ge0$, then substitute the candidate price into the payoff and profit formulas.

The central result is compact: a long put with strike $K$ and premium $p$ has break-even terminal price $K-p$ only when that value is nonnegative. Always report the domain check as part of the answer; it distinguishes a valid threshold from a merely algebraic one.

Check Your Understanding

For each problem, use the expiration-profit convention stated in the question and check that any proposed break-even price is feasible.

A hypothetical one-unit put has strike $\$92$ and premium $\$11$. Find its break-even price and its profit if the terminal price is $\$75$. (Answer: Break-even $\$81$; profit $\$6$.)
A put has strike $\$70$ and premium $\$8$. Find the highest terminal price at which its buyer earns at least $\$4$ profit, and verify the result. (Answer: $\$58$; payoff $\$12$, profit $\$4$.)
A put has strike $\$27$ and premium $\$31$. Determine whether it has a nonnegative break-even price and find its profit when the terminal price is zero. (Answer: No nonnegative break-even price; profit at zero is $-\$4$.)
A put’s strike is $\$84$, and its break-even terminal price is $\$69$. Find the premium and calculate profit if the terminal price is $\$60$. (Answer: Premium $\$15$; profit $\$9$.)
A buyer pays $\$18$ for a put with strike $\$50$. Determine the terminal-price range for strictly positive profit, and calculate profit at $S_T=\$32$ and $S_T=\$40$. (Answer: $0\le S_T<\$32$; profit at $\$32$ is $\$0$, and at $\$40$ is $-\$8$.)

What You'll Learn