Maximum Gain and Loss for Each Option Position

Introduction

An option’s profit can change substantially as the underlying price at expiration changes. For risk analysis, however, it is often useful to ask a different question from “What is the profit at this particular price?”: What is the best possible outcome, and what is the worst? The answer depends both on whether the position is purchased or written and on whether it is a call or a put.

This tutorial compares those four positions using expiration profit, not just expiration payoff. The key domain assumption is that the underlying price cannot be negative. That assumption makes put outcomes bounded, while a written call can still have an unbounded loss. This topic is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Let $S_T$ be the underlying price at expiration and $K$ the option’s strike price, with $S_T\geq 0$ and $K\geq 0$. Let $c\geq 0$ be the call premium and $p\geq 0$ the put premium, each paid or received when the position is initiated. For one option, ignoring transaction costs and financing, profit at expiration equals the expiration payoff less the premium for a purchased option, or the premium less the payoff for a written option. A maximum loss is stated as a nonnegative dollar amount; a negative profit means a loss.

Payoff and Profit at Expiration

The payoff of a call is the amount received by its holder from exercise; the payoff of a put is defined in the same way. A holder chooses whether to exercise, so the payoff cannot be negative. A written position has the opposite payoff, but its profit also includes the premium received at initiation.

\text{Call payoff}=\max(S_T-K,0), \qquad \text{Put payoff}=\max(K-S_T,0).

Subtracting the premium for a purchased option, or subtracting the payoff from the premium for a written option, gives the four expiration-profit formulas:

\begin{aligned} \text{Long call: }&\max(S_T-K,0)-c,\\ \text{Short call: }&c-\max(S_T-K,0),\\ \text{Long put: }&\max(K-S_T,0)-p,\\ \text{Short put: }&p-\max(K-S_T,0). \end{aligned}

These formulas describe profit at expiration under the stated convention. If a question instead asks for an accumulated profit that includes interest earned on the premium, the premium must be accumulated to expiration before comparing it with the payoff. Do not silently change the profit convention.

Maximum Outcomes for the Four Positions

The following results follow by looking at the range of each payoff as $S_T$ varies over all nonnegative values. The short-put loss is written using $\max(K-p,0)$, so the statement remains accurate even if the premium is unusually large relative to the strike.

Position	Maximum gain or highest profit	Maximum loss	Why
Long call	Unbounded	$c$	The call payoff has no upper bound as $S_T$ increases; it is zero when $S_T\leq K$.
Short call	$c$	Unbounded	The payoff can be zero, but increases without bound as $S_T$ increases.
Long put	$K-p$ as the highest profit, at $S_T=0$	$p$	The put payoff ranges from zero to $K$.
Short put	$p$	$\max(K-p,0)$	The put payoff ranges from zero to $K$, so the written position’s profit ranges from $p-K$ to $p$.

For a long put, $K-p$ is the maximum value of the profit function. If $p\leq K$, it is a nonnegative maximum profit. If $p>K$, the largest profit is still $K-p$, but it is negative: the position loses money for every possible expiration price. In the usual case $p\leq K$, a short put’s maximum loss is $K-p$, attained at $S_T=0$.

Key takeaway: A long call has limited loss and unbounded gain; a short call has limited gain and unbounded loss. A long put and a short put both have bounded outcomes because a nonnegative underlying price limits the put payoff to at most $K$.

Why the Results Hold

The result can be justified directly from the domain and the payoff formulas. For a call, when $S_T\leq K$, its payoff is zero. When $S_T>K$, its payoff is $S_T-K$, which grows without bound as $S_T$ grows. For a put, the restriction $S_T\geq0$ implies $K-S_T\leq K$; since the put payoff is also nonnegative, it must lie between zero and $K$.

Find the payoff range.
The call payoff ranges from zero upward with no finite upper limit. The put payoff is between zero and $K$, inclusive.

Convert payoff to profit.
For a long position, subtract the premium. For a short position, subtract the payoff from the premium received.

Identify the extreme.
Use the lowest and highest possible payoffs to find the lowest and highest profits. A profit with no finite lower bound represents unbounded loss.

For example, the long-put profit is the put payoff minus $p$. Its payoff is at most $K$, so its highest profit is $K-p$; its payoff can also be zero, so its lowest profit is $-p$. Thus its maximum loss is $p$. The short put reverses those payoff effects: its profit ranges from $p-K$ to $p$. This gives a maximum gain of $p$, and a maximum loss of $K-p$ when that value is positive.

Worked Examples

Worked Example: Maximum Outcomes for a Purchased Call

A hypothetical investor buys one call with strike $K=\$72$ and pays premium $c=\$5$. Find the maximum gain and maximum loss, then calculate the profit if the underlying finishes at $\$0$, $\$72$, or $\$100$.

Method. Write the long-call profit as $\max(S_T-72,0)-5$, then examine the payoff’s lowest and highest possible values.

At $S_T=0$, the payoff is $\max(0-72,0)=0$, so profit is $0-5=-\$5$. At $S_T=72$, the payoff is $\max(72-72,0)=0$, so profit is again $-\$5$. These outcomes establish the maximum loss: the investor can lose the $\$5$ premium, but no more under this expiration-profit convention.

At $S_T=100$, the payoff is $\max(100-72,0)=\$28$, and profit is $28-5=\$23$. In general, for $S_T>72$, profit is $S_T-72-5=S_T-77$. As $S_T$ increases without an upper bound, the profit also increases without an upper bound. Therefore, the maximum gain is unbounded, and the maximum loss is $\$5$.

Check. The break-even price is $\$77$, since $77-72-5=0$. This is consistent with the profit formula. The common mistake avoided here is treating the strike as an upper limit on a call’s value; a nonnegative underlying price has no such upper limit.

Worked Example: Maximum Outcomes for a Written Call

A hypothetical trader writes one call with strike $K=\$45$ and receives premium $c=\$3$. Determine the maximum gain and maximum loss, and find the profit when the underlying finishes at $\$40$, $\$45$, and $\$61$.

Method. The short-call profit is $3-\max(S_T-45,0)$. Its greatest value occurs when the call payoff is smallest, and its loss grows as the call payoff grows.

At $S_T=40$, the payoff is $\max(40-45,0)=0$, so profit is $3-0=\$3$. At $S_T=45$, the payoff is also $\max(45-45,0)=0$, so profit remains $\$3$. The maximum gain is therefore the premium received, $\$3$.

At $S_T=61$, the payoff is $\max(61-45,0)=\$16$, so profit is $3-16=-\$13$, a loss of $\$13$. For any $S_T>45$, profit is $3-(S_T-45)=48-S_T$. As $S_T$ grows without bound, $48-S_T$ decreases without bound. Thus the maximum loss is unbounded.

Check. The break-even price is $\$48$, because $3-(48-45)=0$. Above $\$48$, the written call loses money. The common mistake avoided is calling the premium the maximum loss: it is the maximum gain for the writer, not the maximum loss.

Worked Example: Comparing Purchased and Written Puts

A hypothetical investor considers either buying or writing one put with strike $K=\$30$ and premium $p=\$4$. Find the maximum gain and loss for each position, then calculate both profits at expiration prices of $\$0$, $\$24$, $\$30$, and $\$38$.

Method. The put payoff is $\max(30-S_T,0)$. It is at most $\$30$, attained at $S_T=0$, and is zero whenever $S_T\geq30$.

For the long put, profit is payoff minus $\$4$. At $S_T=0$, the payoff is $\max(30-0,0)=\$30$, so profit is $30-4=\$26$. This is the maximum profit. At $S_T=24$, payoff is $30-24=\$6$, and profit is $6-4=\$2$. At $S_T=30$ and $S_T=38$, payoff is zero and profit is $-\$4$. Thus the maximum loss is the $\$4$ premium.

For the short put, profit is premium minus payoff. At $S_T=0$, profit is $4-30=-\$26$, the maximum loss. At $S_T=24$, profit is $4-6=-\$2$. At $S_T=30$ and $S_T=38$, profit is $4-0=\$4$, the maximum gain.

Check. The long put’s break-even price is $\$26$, since $30-26-4=0$. The short put has the opposite profit at each listed price: for example, at $S_T=0$, $+\$26$ for the long put and $-\$26$ for the short put. The common mistake avoided is assuming a put can have an arbitrarily large payoff: with $S_T\geq0$, its payoff cannot exceed its strike.

Common Mistakes and Exam Traps

A maximum gain or loss is an extreme over all allowed expiration prices, not simply the profit at the strike or at the break-even price. Check the whole range $S_T\geq0$, including its boundary at zero and the possibility that $S_T$ increases without bound.

Wrong: “A long call’s maximum gain is $K-c$, because the underlying cannot fall below zero.” This confuses a lower bound with an upper bound. Correction: $S_T$ can increase without limit, so a long call’s gain is unbounded.
Wrong: “A written put has unbounded loss because the underlying might fall indefinitely.” This incorrectly allows a negative underlying price. Correction: With $S_T\geq0$, the put payoff is at most $K$, so the short-put loss is at most $K-p$ when $K>p$.
Wrong: “The premium is the maximum loss for every position.” Correction: That is true for purchased options under the stated convention, but a written call can lose without bound and a written put can lose up to $K-p$.
Wrong: “The maximum profit for a long put is always positive.” Correction: Its highest profit is $K-p$. If $p>K$, even that highest value is negative, so the position loses at every expiration price.

Key takeaway: First determine the range of the option payoff, then account for the premium. Calls have no finite upper payoff when the underlying price is unbounded above; puts have payoff no greater than the strike when the underlying price is nonnegative.

Check Your Understanding

For each position, use expiration profit and assume the underlying price cannot be negative. Answers identify maximum gain and maximum loss unless a question asks for a specific expiration profit.

A purchased call has strike $\$64$ and premium $\$7$. State its maximum gain and maximum loss. (Answer: Maximum gain is unbounded; maximum loss is $\$7$.)
A written put has strike $\$28$ and premium $\$3$. State its maximum gain and maximum loss. (Answer: Maximum gain is $\$3$; maximum loss is $\$25$.)
A purchased put has strike $\$90$ and premium $\$8$. Find its highest profit, the expiration price at which it occurs, its maximum loss, and its break-even price. (Answer: Highest profit is $\$82$ at $S_T=0$; maximum loss is $\$8$; break-even is $\$82$.)
A written call has strike $\$115$ and premium $\$9$. Find its profit if $S_T=\$140$, then state its maximum gain and maximum loss. (Answer: Profit is $9-(140-115)=-\$16$; maximum gain is $\$9$; maximum loss is unbounded.)
A written put has strike $\$42$ and premium $\$6$. Find its profit at $S_T=0$ and $S_T=50$, then state its maximum gain and loss. (Answer: Profit is $-\$36$ at $S_T=0$ and $\$6$ at $S_T=50$; maximum gain is $\$6$ and maximum loss is $\$36$.)

Position	Maximum gain or highest profit	Maximum loss	Why
Long call	Unbounded	\(c\)	The call payoff has no upper bound as \(S_T\) increases; it is zero when \(S_T\leq K\).
Short call	\(c\)	Unbounded	The payoff can be zero, but increases without bound as \(S_T\) increases.
Long put	\(K-p\) as the highest profit, at \(S_T=0\)	\(p\)	The put payoff ranges from zero to \(K\).
Short put	\(p\)	\(\max(K-p,0)\)	The put payoff ranges from zero to \(K\), so the written position’s profit ranges from \(p-K\) to \(p\).

What You'll Learn