Derivative Payoff Versus Profit At Expiration

Payoff and Profit Are Different Quantities

At expiration, a derivative may produce a cash flow determined by the underlying price and the contract terms. That cash flow is the payoff. The position’s profit also reflects what the investor paid or received to enter the contract. For an option, ignoring the premium can change both the calculated profit and the break-even underlying price.

The distinction matters even when the question asks only for a price at expiration. A call with a strike of $100 and an underlying price of $110 has a long payoff of $10. If the call cost money to buy, however, its profit is less than $10 after allowing for the value of that premium at expiration. A forward with no initial payment has no option premium to subtract, but its contractual delivery price still determines its payoff.

Definition: The expiration payoff is the cash flow specified by a derivative contract at expiration, before accounting for the cost or proceeds of entering the position. The expiration profit is the position’s net economic result at expiration after accounting for those initial cash flows on a consistent time basis. A break-even underlying price is a nonnegative expiration price at which the position’s expiration profit is zero.

Model and Notation

Assume one contract on one unit of an underlying asset, with no transaction costs, taxes, or interim payments. The expiration time is $T$ years from now, the underlying price at expiration is $S_T$, and the annual effective accumulation rate is $i$. Assume this rate is fixed and is used to express initial option premiums as equivalent amounts at expiration. This is an economic profit convention: it compares all cash flows at the same date. It is not a claim that a premium will necessarily earn that rate in an actual account.

For a call, let $K$ be its strike price and $C_0$ the premium paid at time 0 by the buyer. For a put, let $K$ be its strike price and $P_0$ its time-0 premium. Their corresponding accumulated premium amounts are

C_T^{\text{prem}}=C_0(1+i)^T, \qquad P_T^{\text{prem}}=P_0(1+i)^T.

The superscript $\text{prem}$ distinguishes an accumulated premium from an option payoff. If a question explicitly defines profit using the undiscounted premium rather than an accumulated premium, use that stated convention. Here, all profit calculations compare amounts at expiration.

For a forward, let $K$ denote the delivery price agreed at time 0. A newly entered forward is assumed to have no initial payment. The following payoff formulas apply at expiration. They do not include daily futures margin settlements; for a futures position with daily settlement, the total realized result can depend on the settlement cash-flow path and its reinvestment.

Forward Payoff and Profit

The long forward agrees to buy the underlying for $K$ at expiration. Its underlying is worth $S_T$ then, so the value of the purchase commitment is the difference between the market price and the delivery price. The short forward has the opposite obligation and payoff.

\begin{aligned} \text{Long forward payoff} &= S_T-K,\\ \text{Short forward payoff} &= K-S_T. \end{aligned}

Under the no-initial-payment assumption, the forward’s expiration profit equals its expiration payoff. The long and short break even at $S_T=K$. A negative payoff is a loss on the position; it does not mean that the underlying itself has a negative price.

Worked Example: Forward Delivery Price and Opposite Positions

A portfolio manager enters a one-year forward to buy one unit of a hypothetical commodity for $84 at expiration. The contract requires no payment at entry. At expiration, the commodity’s price is $79. Find the long and short payoffs, profits, and break-even price.

Method. Apply the forward payoff formulas, then use the zero-initial-payment assumption to identify payoff with profit. The common mistake to avoid is subtracting a premium that the forward buyer did not pay.

Long position. The delivery price is $K=84$ and the expiration price is $S_T=79$. Thus the payoff is $79-84=-5$ dollars. With no initial payment, the long’s profit is also $-\$5$.

Short position. Its payoff is $84-79=5$ dollars, so its profit is $+\$5$. The two position results sum to $-5+5=0$, as expected for opposite positions in the same contract.

Break-even. Set the long payoff equal to zero: $S_T-84=0$, giving $S_T=84$. The short payoff is also zero at that same price. The answer is checked by substitution: $84-84=0$ for the long and $84-84=0$ for the short.

Call Payoff and Profit

A call gives its holder the right, but not the obligation, to buy the underlying for $K$. At expiration, exercising is valuable only when $S_T>K$. Therefore, the long call payoff is the greater of zero and $S_T-K$. The short call payoff is its negative.

\begin{aligned} \text{Long call payoff} &= \max(S_T-K,0),\\ \text{Long call profit} &= \max(S_T-K,0)-C_T^{\text{prem}},\\ \text{Short call payoff} &= -\max(S_T-K,0),\\ \text{Short call profit} &= C_T^{\text{prem}}-\max(S_T-K,0). \end{aligned}

For a positive premium, the long call breaks even above the strike, where its payoff must equal the accumulated premium. Thus its break-even price is $K+C_T^{\text{prem}}$. The short call has the same break-even price: its received premium, accumulated to expiration, exactly offsets its exercise loss there. Below or at the strike, the long call’s payoff is zero and its profit is a loss equal to the accumulated premium; the short call has the opposite profit.

Worked Example: Accumulating a Call Premium Before Finding Profit

A buyer pays $6 for a call with strike price $100. Expiration is two years away, and the assumed annual effective accumulation rate is 5%. At expiration the underlying price is $112. Calculate the long call payoff and profit, and find the break-even price.

Method. First accumulate the premium to expiration, then calculate the payoff and subtract the accumulated cost. This avoids the common error of subtracting the time-0 premium directly from a terminal payoff.

The accumulated premium is

C_T^{\text{prem}} =6(1.05)^2 =6(1.1025) =\$6.615.

Since $S_T=112>K=100$, the call is exercised at expiration. Its payoff is $112-100=\$12$. Its profit is $12-6.615=\$5.385$, or $5.39 to the nearest cent.

The break-even price is

K+C_T^{\text{prem}}=100+6.615=106.615.

At that price the payoff is $106.615-100=\$6.615$, exactly the accumulated premium, so profit is zero. At the actual expiration price, the short call’s payoff is $-\$12$ and its profit is $6.615-12=-\$5.385$. The two profits sum to zero because the positions are opposites and use the same premium amount.

Put Payoff and Profit

A put gives its holder the right, but not the obligation, to sell the underlying for $K$. Exercising is valuable when $S_T<K$, so the long put’s payoff is the greater of zero and $K-S_T$. The short put’s payoff is the negative of that amount.

\begin{aligned} \text{Long put payoff} &= \max(K-S_T,0),\\ \text{Long put profit} &= \max(K-S_T,0)-P_T^{\text{prem}},\\ \text{Short put payoff} &= -\max(K-S_T,0),\\ \text{Short put profit} &= P_T^{\text{prem}}-\max(K-S_T,0). \end{aligned}

For a positive accumulated premium $P_T^{\text{prem}}$, the put break-even equation in the exercise region is $K-S_T=P_T^{\text{prem}}$, giving $S_T=K-P_T^{\text{prem}}$. This is a nonnegative break-even price when $P_T^{\text{prem}}\le K$. The boundary case matters: if $P_T^{\text{prem}}=K$, the long put breaks even at $S_T=0$. If $P_T^{\text{prem}}>K$, there is no nonnegative break-even price; even at $S_T=0$, the maximum payoff $K$ is less than the accumulated premium. The short put has the same break-even condition, with the signs of profit reversed.

Worked Example: Long and Short Put Profit

A trader buys a two-year put with strike price $120 for a premium of $15. The assumed annual effective accumulation rate is 4%. At expiration the underlying price is $95. Find the long and short payoffs and profits, and determine their break-even price.

Method. Accumulate the premium, calculate intrinsic value, and then compute each position’s profit. The main trap is to use the $15 time-0 premium as though it were already measured at expiration.

The accumulated premium is

P_T^{\text{prem}} =15(1.04)^2 =15(1.0816) =\$16.224.

Because $95<120$, the long put is exercised. Its payoff is $120-95=\$25$. The long profit is $25-16.224=\$8.776$, or $8.78 to the nearest cent. The short payoff is $-\$25$, and the short profit is $16.224-25=-\$8.776$, or $-\$8.78$ to the nearest cent.

The break-even price is $120-16.224=\$103.776$, or $103.78 to the nearest cent. It is nonnegative because $16.224<120$. Substitution verifies the result: at $S_T=103.776$, the put payoff is $120-103.776=\$16.224$, exactly the accumulated premium, so both positions have zero profit.

Why the Premium Must Be Compared at Expiration

An option buyer pays the premium before receiving any expiration payoff. To calculate profit at expiration, both amounts must be expressed at the same date. If the time-0 premium is $C_0$, accumulating it for $T$ years at the assumed effective rate gives $C_0(1+i)^T$. Subtracting this amount from the expiration payoff measures the terminal result of the call position under the stated convention.

\begin{aligned} \text{Long call profit at }T &=\text{expiration payoff} -\text{time-0 premium accumulated to }T\\ &=\max(S_T-K,0)-C_0(1+i)^T. \end{aligned}

The same reasoning applies to a put. For a short option, the premium is received at time 0, so its terminal value is added to the short position’s payoff. This also explains why long and short profits are exact negatives when they refer to opposite positions in the same option, with the same premium and accumulation convention.

Identify the contract and position.
Record whether the position is long or short, and identify the strike or forward delivery price.

Calculate expiration payoff.
Use the contract’s payoff rule at the given $S_T$, before including the premium.

Put initial cash flows on the expiration-date basis.
Accumulate an option premium using the rate and time specified by the problem.

Calculate profit and check break-even.
Subtract the accumulated premium for a long option or add it for a short option. Set profit equal to zero and check that the resulting $S_T$ is nonnegative.

Common Mistakes and Exam Checks

A common error is to call an option payoff its profit. For example, a call payoff of $12 is not a $12 profit if the accumulated premium is $6.615; the profit is $5.385. Another error is to subtract the premium paid at time 0 without accumulating it, even though the question asks for an expiration-date profit. Keep the valuation date of every amount visible in the calculation.

The put break-even boundary is another frequent trap. If $P_T^{\text{prem}}=K$, the equation $S_T=K-P_T^{\text{prem}}$ gives zero, which is a valid nonnegative underlying price. Only when $P_T^{\text{prem}}>K$ is the algebraic break-even price negative and therefore unavailable for a nonnegative underlying. At $S_T=0$, the put’s payoff cannot exceed $K$, so this conclusion also follows directly from the maximum possible payoff.

Do not confuse payoff and profit. State which one is requested before substituting numbers.
Do not charge a forward premium by default. A newly entered forward under the assumptions here has no initial payment.
Do not omit the strike-price condition. A call payoff is zero for $S_T\le K$, while a put payoff is zero for $S_T\ge K$.
Check the signs of opposite positions. Long and short payoffs, and their corresponding profits, should be negatives of one another when based on the same contract and cash-flow convention.
Check that a proposed break-even price is feasible. In particular, an underlying price cannot be negative.

A disciplined separation of payoff, premium, and profit is aligned with the SOA Exam FM (Financial Mathematics) syllabus. It is especially useful in multi-step questions, where a correct payoff formula can still lead to an incorrect answer if the premium is handled at the wrong date.

Check Your Understanding

For each problem, distinguish payoff from profit and use the stated accumulation convention.

A one-year long forward has delivery price $73 and no initial payment. The underlying is worth $68 at expiration. Find the long and short payoffs and profits, and state the break-even price. (Answer: Long payoff and profit $-\$5$; short payoff and profit $+\$5$; break-even $S_T=\$73$.)
A buyer pays $4 for a two-year call with strike price $80. The annual effective accumulation rate is 5%, and $S_T=\$90$. Find the long call payoff, profit, and break-even price. (Answer: Accumulated premium $=\$4.41$; payoff $=\$10$; profit $=\$5.59$; break-even $S_T=\$84.41$.)
A one-year put has strike price $150 and premium $12. The annual effective accumulation rate is 4%, and $S_T=\$140$. Find the long and short profits and the break-even price. (Answer: Accumulated premium $=\$12.48$; long profit $=-\$2.48$; short profit $=+\$2.48$; break-even $S_T=\$137.52$.)
A trader writes a one-year call with strike price $60 and receives a premium of $3. The annual effective accumulation rate is 0%, and $S_T=\$70$. Find the short call payoff, profit, and break-even price. (Answer: Payoff $=-\$10$; profit $=-\$7$; break-even $S_T=\$63$.)
A long put has strike price $50 and an accumulated premium at expiration of $50. What is its profit at $S_T=0$, and does it have a nonnegative break-even price? How does the answer change if the accumulated premium is $52? (Answer: With premium $50, profit at zero is $0$ and the break-even price is $S_T=0$. With premium $52, profit at zero is $-\$2$ and there is no nonnegative break-even price.)

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