Piecewise Profit Equations for Puts

From Put Payoff to Piecewise Profit

A put gives its holder the right to sell the underlying asset for the strike price at expiration. The holder’s expiration payoff therefore depends on whether the terminal asset price is below the strike. Profit adds the premium to that exercise decision: the purchaser subtracts the premium, while the writer receives it.

This tutorial considers one put position on one unit of an underlying asset, with strike price $K>0$, terminal asset price $S_T\geq 0$, and premium $\pi$. Assume the put is evaluated at expiration, with no transaction costs or taxes. To compare profit amounts at expiration, treat $\pi$ as the premium amount on that same date. If a premium is paid earlier and the question requires interest to be recognized, accumulate it to expiration before using the equations.

Definition: The payoff of a put at expiration is the amount delivered by the option before accounting for the premium. The profit is the payoff less the purchaser’s premium, or the writer’s premium received less the amount owed under the option. A long put is a purchased put; a written put is a sold put.

If $S_T<K$, exercising the put produces $K-S_T$ per unit: the holder sells for $K$ an asset worth $S_T$. If $S_T\geq K$, exercising does not improve on selling at the market price, so the holder does not exercise and the payoff is zero. Thus the put payoff is $\max(K-S_T,0)$. The profit equations split at $K$, because that is where the exercise decision changes.

The Two Piecewise Equations

For a purchased put, subtract the premium from the payoff in either region. For a written put, reverse the cash-flow signs: the writer receives the premium and owes the payoff when the holder exercises.

\text{Long put profit}= \begin{cases} K-S_T-\pi, & S_T<K,\\ -\pi, & S_T\geq K. \end{cases}

\text{Written put profit}= \begin{cases} \pi-K+S_T, & S_T<K,\\ \pi, & S_T\geq K. \end{cases}

The two pieces agree at the strike. At $S_T=K$, the long-put expression below the strike would give $K-K-\pi=-\pi$, matching the expression at or above the strike. The written-put expressions likewise both give $\pi$. Writing one branch with a strict inequality and the other with an inclusive inequality avoids assigning the strike to two separate cases; either branch gives the same profit at $K$.

Key result: The strike $K$ determines which payoff branch applies; it is not, in general, the break-even price. For either position, setting profit to zero in the exercise region gives the candidate break-even price $S_T=K-\pi$. This candidate is meaningful only if it is nonnegative and lies in the region $S_T\leq K$.

Why the Equations Hold

The piecewise form follows directly from the exercise decision and the direction of the premium cash flow. This short derivation is also a useful check on the signs in an exam calculation.

Find the put payoff.
If $S_T<K$, the holder exercises and receives $K-S_T$. If $S_T\geq K$, the holder does not exercise and receives zero. The payoff is therefore $K-S_T$ below the strike and $0$ at or above it.

Account for the purchaser’s premium.
The purchaser pays $\pi$, so subtract it from each payoff branch. This gives $K-S_T-\pi$ below $K$ and $0-\pi=-\pi$ at or above $K$.

Reverse the cash flows for the writer.
The writer receives $\pi$ and owes the holder’s payoff. Subtract the payoff from $\pi$, giving $\pi-(K-S_T)=\pi-K+S_T$ below $K$ and $\pi-0=\pi$ at or above $K$.

This derivation uses the payoff at expiration, not a forecast of the asset price and not the option’s value before expiration. A profit calculated at expiration also does not automatically include interest on an earlier premium payment. If interest is required, first express the premium in expiration-date dollars.

Worked Equations and Break-Even Checks

Worked Example: Purchased Put Across the Strike

A hypothetical investor buys one put with strike $K=\$64$. The premium is $\pi=\$7$, expressed in expiration-date dollars. Write the piecewise profit equation, calculate profit when $S_T$ is $\$52$, $\$64$, and $\$70$, and find the break-even price.

Method. First split at the strike. Below $\$64$, the put is exercised and its payoff is $64-S_T$; at or above $\$64$, its payoff is zero. Subtract the $\$7$ premium in both regions:

\text{Profit}= \begin{cases} 64-S_T-7=57-S_T, & S_T<64,\\ -7, & S_T\geq64. \end{cases}

At $S_T=\$52$, use the below-strike branch. The payoff is $64-52=\$12$, so profit is $12-7=\$5$. At $S_T=\$64$, the put is not exercised; payoff is $\$0$, so profit is $0-7=-\$7$. At $S_T=\$70$, the payoff is also $\$0$, so profit remains $-\$7$.

For break-even, set the below-strike profit to zero: $57-S_T=0$, hence $S_T=\$57$. This is a valid break-even because $57<64$. Check: at $\$57$, the payoff is $64-57=\$7$, exactly equal to the premium, leaving zero profit. A common mistake is to report the strike, $\$64$, as break-even; at that price the option expires unused and the purchaser loses the $\$7$ premium.

Worked Example: Written Put and Reversed Premium Sign

A hypothetical investor writes one put with strike $K=\$150$ and receives a premium of $\pi=\$11$ in expiration-date dollars. Write the profit equation, calculate profit at terminal prices of $\$132$, $\$139$, $\$150$, and $\$166$, and identify the break-even price.

Method. Below $\$150$, the holder exercises and the writer owes $150-S_T$. Subtract that liability from the premium received. At or above $\$150$, there is no exercise liability:

\text{Profit}= \begin{cases} 11-(150-S_T)=S_T-139, & S_T<150,\\ 11, & S_T\geq150. \end{cases}

At $S_T=\$132$, the writer owes $150-132=\$18$, so profit is $11-18=-\$7$. At $S_T=\$139$, the liability is $150-139=\$11$, so profit is $11-11=\$0$. At $S_T=\$150$, the put is not exercised and profit is the premium, $\$11$. At $S_T=\$166$, it is also not exercised, so profit is $\$11$.

The below-strike equation gives break-even from $S_T-139=0$, or $S_T=\$139$; this is below the strike, as required. Check: at that price the writer’s liability is $\$11$, exactly offset by the premium received. The common sign error is to subtract the premium from the writer’s liability as though the writer purchased the put. The writer instead calculates premium received minus payoff owed.

Worked Example: Accumulating an Upfront Premium

A hypothetical investor buys a put with strike $K=\$100$ and pays $\$10$ today. The expiration date is two years away, and the effective annual interest rate is $i=10\%$. Assume annual compounding and no other cash flows. Write the expiration-profit equation and calculate profit for $S_T=\$85$, $\$100$, and $\$105$. Find the break-even price.

Method. Because profit is measured at expiration, first accumulate the premium for two years:

10(1.10)^2=10(1.21)=\$12.10.

Thus the terminal-equivalent premium is $\pi=\$12.10$, not $\$10$. Below the strike, profit is $100-S_T-12.10=87.90-S_T$; at or above the strike, it is $-\$12.10$:

\text{Profit}= \begin{cases} 87.90-S_T, & S_T<100,\\ -12.10, & S_T\geq100. \end{cases}

At $S_T=\$85$, the payoff is $100-85=\$15$, so profit is $15-12.10=\$2.90$. At $S_T=\$100$, the payoff is zero, giving profit $-\$12.10$. At $S_T=\$105$, the payoff is again zero, giving profit $-\$12.10$.

Set the below-strike branch equal to zero: $87.90-S_T=0$, so the break-even price is $\$87.90$, which is below the strike. Check: at $\$87.90$, the payoff is $100-87.90=\$12.10$, equal to the accumulated premium. Using $\$10$ directly would instead report a break-even of $\$90$, which compares amounts from different dates and ignores the stated interest rate.

Interpreting Break-Even and Avoiding Errors

For a long put, the profit below the strike is $K-S_T-\pi$, so it is positive when $S_T<K-\pi$, provided that threshold is a feasible nonnegative price. For a written put, profit below the strike is $\pi-K+S_T$, so it is positive when $S_T>K-\pi$ within that region. At or above the strike, the long position’s profit is $-\pi$, and the writer’s profit is $\pi$.

If $\pi=K$, the candidate break-even $K-\pi$ is zero. If $\pi>K$, the candidate is negative and there is no nonnegative terminal price at which profit is zero. For the long put, profit is then negative at every feasible price. For the written put, profit is positive at every feasible price. These conclusions follow from the piecewise equations and the assumption $S_T\geq0$; they are not a reason to use a break-even outside the permitted price range.

Common exam trap: The put’s exercise region is below the strike, $S_T<K$, not above it. For example, with $K=\$100$, premium $\pi=\$8$, and $S_T=\$92$, a long put has payoff $100-92=\$8$ and profit $8-8=\$0$. Incorrectly using the at-or-above-strike branch would give $-\$8$. Also check whose profit is requested: the writer receives the premium, so its profit is the negative of the long position’s profit for the same option and premium.

A reliable solution therefore has three checks: use the terminal price to select the correct side of the strike; subtract the premium for a purchaser or add it before subtracting the liability for a writer; and verify any break-even by substituting it into the applicable branch. If the premium is paid before expiration and interest matters, convert it to the same valuation date as the payoff first.

Check Your Understanding

For each problem, write or use the appropriate piecewise profit equation and check that any break-even belongs to the stated price region.

A purchased put has strike $\$72$ and expiration-date premium $\$5$. Write its profit equation and calculate profit at $S_T=\$60$ and $S_T=\$75$. (Answer: $72-S_T-5$ for $S_T<72$, and $-5$ for $S_T\geq72$; profits are $\$7$ and $-\$5$.)
A written put has strike $\$48$ and expiration-date premium $\$6$. Find its break-even price and profit at $S_T=\$43$ and $S_T=\$55$. (Answer: break-even $\$42$; profits $\$1$ and $\$6$.)
A purchased put has strike $\$125$ and premium $\$9$, stated in expiration-date dollars. At what terminal prices is its profit positive? State the break-even and verify it is in the exercise region. (Answer: positive when $0\leq S_T<\$116$; break-even $\$116$, which is below $\$125$.)
A written put has strike $\$90$ and premium $\$14$, stated in expiration-date dollars. Write both branches and calculate profit at $S_T=\$70$ and $S_T=\$95$. (Answer: $S_T-76$ for $S_T<90$, and $14$ for $S_T\geq90$; profits are $-\$6$ and $\$14$.)
A purchased put has strike $\$80$, costs $\$12$ today, and expires in three years. The effective annual interest rate is $5\%$, compounded annually. Find its terminal-equivalent premium, piecewise profit equation, and break-even price. (Answer: premium $12(1.05)^3=\$13.8915$; profit $80-S_T-13.8915$ for $S_T<80$, and $-13.8915$ for $S_T\geq80$; break-even $\$66.1085$.)

What You'll Learn