Risk And Reward Of A Short Put

Introduction

Writing a put can produce a limited maximum reward alongside a substantial downside exposure. The distinction is easiest to see at expiration: the seller keeps the premium if the put finishes out of the money, but may have to buy the underlying for the strike price when its market price is lower. If the underlying price falls to zero, that obligation is at its most costly.

This tutorial focuses on the expiration profit of one short put, stated per share or other single underlying unit. It examines why the premium is the greatest possible profit and how to find the largest possible loss. All examples ignore transaction costs, taxes, and interest earned or paid on the premium.

Definition: A short put is the position of the put seller. At expiration, if the put is exercised, the seller must buy the underlying for the strike price $K$. Let $S_T$ be the underlying price at expiration and $p$ the premium received per underlying unit when the put is sold. The seller's expiration profit per unit is the premium received minus the put's expiration exercise obligation.

Expiration Profit and Its Boundaries

A put is exercised at expiration when $S_T<K$; its exercise obligation per underlying unit is then $K-S_T$. When $S_T\ge K$, the put is not exercised and this obligation is zero. Thus the seller's profit is the premium minus the put's payoff to its holder:

\Pi_{\text{short put}}=p-\max(K-S_T,0)

Equivalently, the profit has two branches:

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

The nonnegative-price condition matters: an ordinary share price cannot fall below zero. When $S_T\ge K$, the seller keeps the full premium, so $p$ is the maximum profit. When the price is below $K$, the exercise obligation reduces profit dollar for dollar as $S_T$ falls. At $S_T=0$, the obligation is $K$, and profit is $p-K$. If $p<K$, the largest possible loss is therefore $K-p$ per underlying unit.

Key result: For a short put with strike $K$ and premium $p$, the maximum profit per underlying unit is $p$, attained whenever $S_T\ge K$. Assuming $S_T\ge0$, the minimum profit is $p-K$, attained at $S_T=0$. The maximum loss, expressed as a nonnegative amount, is $\max(K-p,0)$. In the usual case $p<K$, this is $K-p$.

The formula separates two amounts that are easy to confuse. At a zero underlying price, the put seller's gross exercise obligation is $K$: the seller pays $K$ to acquire an asset worth zero. The premium offsets that obligation, so the net loss is $K-p$, not $K$. The premium is not an additional loss; it was received when the position was opened.

Why the Maximum Profit Is the Premium

The maximum-profit conclusion follows directly from the nonnegative exercise obligation. For every possible expiration price, $\max(K-S_T,0)\ge0$. Subtracting a nonnegative amount from $p$ cannot produce a profit greater than $p$. For any $S_T\ge K$, the obligation equals zero, so the profit reaches $p$. This proves both the upper bound and that it is attainable.

For the lower bound, use the assumption $S_T\ge0$. The put obligation $\max(K-S_T,0)$ cannot exceed $K$: when $S_T<K$, $K-S_T\le K$, and when $S_T\ge K$, the obligation is zero. Therefore profit cannot be below $p-K$. At $S_T=0$, the obligation equals $K$, so profit is exactly $p-K$. This establishes the worst-case result under the stated price assumption.

This is a bounded-loss result for the put seller, not a claim that the loss must be small. If the strike is much larger than the premium, a zero underlying price produces a loss close to the strike. The premium limits the seller's maximum reward, while the strike and premium together determine the maximum loss.

Worked Examples

Worked Example: Finding the Full Profit Range

An investor sells one put with strike $K=\$50$ and receives a premium of $p=\$6$ per share. Find the expiration profit at $S_T=\$35$, $S_T=\$50$, and $S_T=\$64$. Then identify the maximum profit and worst-case loss per share.

Method. Calculate the put obligation $\max(K-S_T,0)$ at each price, then subtract it from the premium. This method keeps the obligation and the seller's net profit distinct.

At $S_T=\$35$, the obligation is $\max(50-35,0)=\$15$. Profit is $6-15=-\$9$ per share.

At $S_T=\$50$, the obligation is $\max(50-50,0)=\$0$. Profit is $6-0=\$6$ per share.

At $S_T=\$64$, the obligation is $\max(50-64,0)=\$0$. Profit is again $6-0=\$6$ per share.

The maximum profit is the premium, $\$6$ per share, attained at any expiration price of at least $\$50$. If the underlying falls to zero, the obligation is $\max(50-0,0)=\$50$, giving profit $6-50=-\$44$. Thus the worst-case loss is $\$44$ per share.

Check. The reported maximum profit does not exceed the $\$6$ premium, and the zero-price loss equals $K-p=50-6=\$44$. A common mistake this method avoids is calling the gross $\$50$ obligation the net loss.

Worked Example: A Loss Below the Strike but Above Zero

A fund sells a put on a hypothetical share for a premium of $\$9.50$ per share, with strike $K=\$82$. At expiration, the share price is $S_T=\$61.25$. Calculate the expiration profit or loss per share and the largest possible loss per share over all nonnegative expiration prices.

Method. First find the exercise obligation at the stated expiration price. Then subtract that obligation from the premium. Separately, apply the zero-price boundary to find the worst case.

Since $\$61.25<\$82$, the put is exercised. Its obligation is $\$82-\$61.25=\$20.75$ per share. The seller's profit is $\$9.50-\$20.75=-\$11.25$, so the position has a loss of $\$11.25$ per share.

For the worst case, set $S_T=0$. The obligation is then $\$82-\$0=\$82$, and profit is $\$9.50-\$82=-\$72.50$. The largest possible loss is $\$72.50$ per share.

Check. The loss at $\$61.25$ is less than the worst-case loss because the share still has positive value. The worst-case result also agrees with $K-p=82-9.50=\$72.50$. The common mistake avoided here is treating the loss at one below-strike price as the maximum possible loss.

Worked Example: Comparing Several Expiration Outcomes

A portfolio manager sells a put on a hypothetical asset with strike $K=\$120$, receiving $p=\$14.50$ per unit. Calculate the profit per unit at expiration prices of $\$0$, $\$100$, $\$120$, and $\$132$. State the maximum profit and the worst-case loss, and explain why the two are not symmetrical.

Method. Apply $\Pi_{\text{short put}}=p-\max(K-S_T,0)$ separately to each price. This avoids applying the exercise obligation when the put is out of the money.

At $S_T=\$0$, the obligation is $\max(120-0,0)=\$120$, so profit is $14.50-120=-\$105.50$.

At $S_T=\$100$, the obligation is $\max(120-100,0)=\$20$, so profit is $14.50-20=-\$5.50$.

At $S_T=\$120$, the obligation is $\max(120-120,0)=\$0$, so profit is $\$14.50$.

At $S_T=\$132$, the obligation is $\max(120-132,0)=\$0$, so profit is also $\$14.50$.

The maximum profit is $\$14.50$ per unit. The worst-case loss is $\$120-\$14.50=\$105.50$ per unit, reached at a zero price. The reward is capped at the premium, but the loss can grow as the underlying price falls below the strike. The amounts are not symmetrical: the premium is not the mirror image of the possible loss.

Check. Each price at or above the strike gives the same maximum profit, while the zero-price result is the minimum profit in the assumed price range. A common mistake avoided is assigning a larger profit at $\$132$ than at $\$120$; once the put expires unexercised, a higher price does not increase the premium already received.

Interpreting the Risk and Reward

The put seller's payoff profile has a flat profit region at $p$ for $S_T\ge K$, followed by a declining profit as $S_T$ falls below $K$. Below the strike, each one-dollar decrease in $S_T$ increases the seller's loss by one dollar per underlying unit. The decline stops at the zero-price boundary in this model. These features describe expiration outcomes; they do not state how likely any outcome is.

The position's maximum loss assumes the underlying cannot have a negative price and can reach zero. It also assumes the put's exercise obligation is the standard amount $K-S_T$ per unit and that the premium is counted without adjustment for interest, fees, or other positions. If those assumptions change, the calculation must be adapted. For example, costs would reduce net profit, while a multi-unit position would require accounting for the number of units.

A useful exam-style discipline is to write the three boundary facts before making a conclusion: the profit is $p$ when $S_T\ge K$; the profit is $p-K+S_T$ when $0\le S_T<K$; and the minimum over nonnegative prices occurs at $S_T=0$. These facts make it harder to confuse a maximum reward with a maximum loss or to forget that the premium offsets the exercise cost.

Common Mistakes

Reporting the strike as the net worst-case loss. At $S_T=0$, the gross obligation is $K$, but the seller has already received $p$. The net loss is $K-p$ when $p<K$.
Calling the premium the profit in every outcome. The seller earns the full premium only if the put obligation is zero, which occurs at expiration when $S_T\ge K$. Below the strike, the obligation reduces profit.
Applying a positive obligation above the strike. If $S_T>K$, then $K-S_T<0$, but the put payoff is zero, not a negative obligation. Use $\max(K-S_T,0)$.
Assuming a larger rise creates a larger short-put profit. Once the put expires unexercised, the seller's profit remains $p$; it does not keep increasing with $S_T$.
Forgetting the lower price boundary. The worst case for this standard underlying-price model is $S_T=0$, not an arbitrarily negative price.

Key takeaway: A short put's maximum expiration profit per underlying unit is the premium received. With a nonnegative underlying price, its worst-case profit occurs at zero and equals $p-K$; when $p<K$, the largest possible loss is $K-p$.

Check Your Understanding

For each problem, work per underlying unit and assume no transaction costs, taxes, or interest adjustments.

A put seller receives a premium of $\$3.20$ for a put with strike $\$44$. What is the profit if the underlying price is zero, and what is the maximum profit? (Answer: zero-price profit $=-\$40.80$, a loss of $\$40.80$; maximum profit $=\$3.20$.)
A short put has strike $\$67$ and premium $\$5.50$. At expiration, the underlying is worth $\$59$. Find the seller's profit or loss. (Answer: $-\$2.50$, a loss of $\$2.50$.)
A seller receives $\$12$ for a put with strike $\$105$. The underlying finishes at $\$118$. Find the expiration profit and state whether this is the maximum profit. (Answer: profit $=\$12$; yes, it is the maximum profit.)
A put has strike $\$36$ and premium $\$2.40$. Find the seller's profit if the underlying reaches zero and identify the largest possible loss. (Answer: profit $=-\$33.60$; largest possible loss $=\$33.60$.)
A short put has strike $\$90$ and premium $\$7.25$. The underlying finishes at $\$78$. Calculate the expiration loss and the worst-case loss over all nonnegative expiration prices. (Answer: expiration loss $=\$4.75$; worst-case loss $=\$82.75$.)

What You'll Learn