Long Put Payoff at Expiration

What a Long Put Pays at Expiration

A purchased put gives its holder a payoff tied to the difference between the strike price and the underlying asset’s price at expiration. The key comparison is between the strike $K$ and the terminal price $S_T$. If the terminal price is below the strike, the payoff is positive; if it is equal to or above the strike, the payoff is zero.

This tutorial considers the payoff at expiration only. It does not include the amount originally paid for the option, interest, transaction costs, or any other cash flows. Accordingly, the payoff is not the same as the position’s profit. The calculations assume a standard put contract with a stated strike and contract multiplier, with no special contract adjustment. No probability distribution for $S_T$ is assumed: each calculation is conditional on the terminal price given in the problem. The method is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: For a purchased put with strike $K$, the expiration payoff per underlying unit is the greater of $K-S_T$ and zero. Here, $S_T$ is the underlying asset’s price at expiration. If one contract covers $M$ underlying units and the position contains $n$ contracts, the total expiration payoff is $nM$ times the per-unit payoff.

The positive-part notation makes the same rule compact. For any real number $x$, $x^+=\max(x,0)$. The long put payoff per underlying unit is therefore $(K-S_T)^+$. If $S_T<K$, the difference $K-S_T$ is positive and is the payoff. If $S_T\ge K$, the difference is zero or negative, so its positive part is zero. This gives the equivalent piecewise form:

\text{Long put payoff per unit} =(K-S_T)^+ = \begin{cases} K-S_T, & S_T<K,\\ 0, & S_T\ge K. \end{cases}

Reading the Three Terminal-Price Cases

The formula has three useful cases for exam calculations: below the strike, exactly at the strike, and above the strike. At the strike, subtracting the two prices gives zero. Above the strike, $K-S_T$ is negative, but a payoff cannot be negative, so the positive-part rule assigns a payoff of zero. Do not report the negative difference as the option payoff.

The price difference is measured per underlying unit. A contract multiplier converts it to a per-contract amount; the number of contracts then gives the total. Keep these steps separate, particularly when the quoted price is per share but each contract covers many shares. A zero terminal price is a useful boundary case: the per-unit payoff equals the full strike $K$, not zero.

Terminal-price condition	Per-unit payoff	Interpretation
$S_T<K$	$K-S_T$	Positive difference between strike and terminal price
$S_T=K$	0	The two prices are equal
$S_T>K$	0	The price difference is negative, so the positive part is zero
$S_T=0$	$K$	The full strike remains as the per-unit difference

Worked Example: One Put Below, At, and Above Its Strike

A hypothetical investor holds one put on an asset with strike $K=\$50$. The option covers one underlying unit. Calculate the payoff per unit if the terminal price is (a) $\$32$, (b) $\$50$, and (c) $\$67$. Also calculate it if the terminal price is zero. The prices are given as deterministic scenarios; no interest rate or probability model is needed.

Method. For each scenario, calculate $K-S_T$ and take the greater of that difference and zero. Because the contract covers one unit, the per-unit and total contract payoffs are the same.

(a) With $S_T=\$32$, the terminal price is below the strike:

K-S_T=\$50-\$32=\$18,\qquad (\$18)^+=\$18.

The payoff is $\$18$ per unit and $\$18$ for the one-unit contract.

(b) With $S_T=\$50$, the terminal price equals the strike:

K-S_T=\$50-\$50=\$0,\qquad (\$0)^+=\$0.

The payoff is $\$0$.

K-S_T=\$50-\$67=-\$17,\qquad (-\$17)^+=\$0.

The negative difference is not a negative option payoff; its positive part is zero. The payoff is $\$0$.

At $S_T=0$, the calculation is:

K-S_T=\$50-\$0=\$50,\qquad (\$50)^+=\$50.

The payoff is $\$50$. As a check, all four results follow the same rule: a positive strike-minus-terminal-price difference is paid, and a zero or negative difference produces no payoff. The common mistake avoided here is keeping the negative $-\$17$ in case (c) as though it were the payoff.

Scaling from One Unit to Multiple Contracts

Many problems quote an option price per underlying share or unit while asking for the payoff on a position. If each contract covers $M$ units and there are $n$ contracts, the position covers $nM$ units altogether. Multiply the per-unit payoff by this total number of units. This scaling assumes the contracts have the same strike, underlying, and terminal price and that the stated multiplier applies to each contract.

\text{Total long put payoff} =nM(K-S_T)^+.

The order matters conceptually: first apply the positive-part rule to the price difference, then scale the nonnegative payoff. Multiplying a negative price difference by the contract count before taking the positive part would produce a negative number of dollars, which is not the payoff of the purchased put.

Worked Example: Several Contracts at and Away from the Strike

A hypothetical position consists of four purchased put contracts. Each contract covers 100 shares and has strike $K=\$72$. Calculate the total payoff if the underlying closes at (a) $\$68.50$, (b) $\$72$, and (c) $\$79$. Treat the quoted terminal prices as exact; no interest rate or distribution assumption is required.

Step 1: Find the total number of underlying shares covered.

nM=4(100)=400\text{ shares}.

(a) At $S_T=\$68.50$, the price is below the strike. The per-share payoff is:

(K-S_T)^+=\max(\$72-\$68.50,0) =\max(\$3.50,0)=\$3.50.

Multiplying by 400 shares gives the total:

400(\$3.50)=\$1{,}400.

(b) At $S_T=\$72$, the price difference and payoff are both zero:

(K-S_T)^+=\max(\$72-\$72,0)=\$0, \qquad 400(\$0)=\$0.

(K-S_T)^+=\max(\$72-\$79,0)=\max(-\$7,0)=\$0, \qquad 400(\$0)=\$0.

The total payoffs are $\$1{,}400$, $\$0$, and $\$0$, respectively. As a check, the only positive result occurs below the strike, and the positive per-share amount is multiplied by exactly 400 shares. The common mistake avoided is multiplying by four contracts but forgetting the 100-share multiplier.

Combining Put Series at One Terminal Price

A position can contain more than one put series, with different strikes or contract multipliers. Calculate the payoff for each series using its own strike and multiplier, then add the resulting nonnegative amounts. Do not use one series’ strike for the entire position. The terminal price is shared when the puts refer to the same underlying and expiration, but the strike and number of covered units can differ.

Worked Example: Two Put Series on the Same Asset

A hypothetical investor holds three put contracts with strike $K_1=\$90$, each covering 50 units, and two put contracts with strike $K_2=\$75$, each covering 100 units. Both series have the same underlying and expiration. If the terminal price is $S_T=\$82$, find the combined payoff. Assume the quoted prices and multipliers are exact; no probability distribution or interest rate is used.

Method. Apply the positive-part formula separately to each series, scale by that series’ total covered units, and add.

For the first series, the total covered units are:

$$ 3(50)=150. $$

Its per-unit payoff is:

(K_1-S_T)^+=\max(\$90-\$82,0)=\$8.

So the first series pays:

150(\$8)=\$1{,}200.

For the second series, the total covered units are:

$$ 2(100)=200. $$

Its per-unit payoff is:

(K_2-S_T)^+=\max(\$75-\$82,0)=\max(-\$7,0)=\$0.

So the second series pays $200(\$0)=\$0$. Adding the two series gives:

\$1{,}200+\$0=\boxed{\$1{,}200}.

As a check, the terminal price is below the first strike but above the second, so only the first series has a positive payoff. The common mistake avoided is treating the two strikes as interchangeable or applying the first strike to all 350 covered units.

Common Exam Trap: Reversing the Price Difference

A long put uses strike minus terminal price, $K-S_T$, not terminal price minus strike, $S_T-K$. Reversing the subtraction can make the payoff appear negative below the strike and positive above it, exactly opposite to the correct pattern.

For example, with $K=\$60$ and $S_T=\$41$, the wrong calculation $S_T-K=\$41-\$60=-\$19$ might be reported as a $-\$19$ payoff. The correction is:

(K-S_T)^+=\max(\$60-\$41,0)=\$19.

A separate but related error is subtracting the initial option premium while calculating expiration payoff. The payoff formula above contains only the strike and terminal price. Any premium is a separate cash flow and is not included in this tutorial’s payoff calculation.

Identify the contract terms.
Record the strike, terminal price, number of contracts, and contract multiplier.

Calculate the per-unit amount.
Compute $K-S_T$, then take its positive part.

Scale and check.
Multiply by the total units covered. Confirm that the result is nonnegative and that it is zero when $S_T\ge K$.

The central check is simple: a purchased put’s expiration payoff cannot be negative. Below the strike, the payoff per unit decreases dollar for dollar as the terminal price rises toward the strike; at and above the strike, it is zero. This check is useful for catching a reversed subtraction, a missed contract multiplier, or an incorrect nonzero payoff above the strike. Payoff remains distinct from profit, which involves additional cash flows not included here.

Check Your Understanding

For each problem, calculate expiration payoff only; do not subtract a premium.

A purchased put has strike $\$43$, covers one unit, and the terminal price is $\$38.60$. What is its payoff? (Answer: $\$4.40$ per unit)
A purchased put has strike $\$125$ and the terminal price is also $\$125$. What is the payoff per unit? (Answer: $\$0$)
An investor holds seven put contracts, each covering 10 units, with strike $\$58$. If the terminal price is zero, what is the total payoff? (Answer: $\$4{,}060$)
A position has two put contracts with strike $\$96$ and multiplier 100, plus four put contracts with strike $\$90$ and multiplier 50. The two series share a terminal price of $\$91.25$. Find the combined payoff. (Answer: $\$950$)
A position contains three put contracts with strike $\$140$, each covering 25 units, and two put contracts with strike $\$130$, each covering 50 units. At a common terminal price of $\$128$, calculate the payoff from each series and the combined payoff. (Answer: $\$900$ from the first series, $\$200$ from the second, and $\$1{,}100$ combined)

Terminal-price condition	Per-unit payoff	Interpretation
\(S_T<K\)	\(K-S_T\)	Positive difference between strike and terminal price
\(S_T=K\)	0	The two prices are equal
\(S_T>K\)	0	The price difference is negative, so the positive part is zero
\(S_T=0\)	\(K\)	The full strike remains as the per-unit difference

What You'll Learn