Written Put Payoff at Expiration

What a Written Put Pays at Expiration

A written put position can require its writer to make a payment at expiration. The amount depends on the strike price and the underlying asset’s terminal price. If the terminal price is below the strike, the put holder’s contractual payoff is positive and the writer’s corresponding option cash flow is negative. If the terminal price is at or above the strike, neither side has an expiration payment under the put contract.

This tutorial considers the expiration payoff only. It does not include a premium received when the option was written, interest earned on that premium, transaction costs, taxes, or any separate position in the underlying asset. These exclusions matter: the option’s cash flow at expiration is not the same quantity as the overall profit on the position.

Definition: Consider one put option on one unit of an underlying asset, with strike price $K$ and expiration time $T$. Let $S_T$ be the underlying asset’s price at expiration. The holder’s expiration payoff is $(K-S_T)^+=\max(K-S_T,0)$. The writer’s expiration cash flow from the option is the negative of the holder’s payoff: $-(K-S_T)^+$. A negative amount denotes a payment by the writer; zero denotes no expiration cash flow.

The setup assumes that the option’s terms are stated in currency per unit of the underlying asset and that the quoted terminal price and strike use the same currency and unit. The option payoff is settled at expiration, and the calculation is for the option alone. If a contract represents more than one unit, or if there are several contracts, the one-unit payoff must be scaled by the relevant quantity.

Payoff and Cash-Flow Signs

The positive-part notation $(x)^+=\max(x,0)$ means that a negative value of $x$ is replaced by zero, while a positive value is left unchanged. For a put, the expression $K-S_T$ is positive precisely when the terminal price is below the strike. The holder’s payoff is therefore $K-S_T$ in that case and zero otherwise.

The writer has the opposite cash flow for the option. A transfer paid to the holder is paid by the writer; if the contract produces no payment, neither side has an expiration cash flow from it. Thus the writer’s amount is the negative of the holder’s amount in every terminal-price case. This is a sign relation, not a claim that the writer’s whole investment result is simply the negative of the holder’s whole investment result: any earlier premium is outside the expiration payoff.

\text{Writer's expiration cash flow} = -\text{Holder's expiration payoff} = -(K-S_T)^+ = \begin{cases} S_T-K, & S_T<K,\\ 0, & S_T\ge K. \end{cases}

When $S_T<K$, the writer’s amount $S_T-K$ is negative. The size of the payment is $K-S_T$, so the amount shown as a signed cash flow is its negative. At $S_T=K$, the difference is zero. When $S_T>K$, the positive-part expression for the holder is zero, and its negative remains zero.

Terminal price	Holder’s payoff per unit	Writer’s cash flow per unit
$S_T<K$	$K-S_T$	$S_T-K$, a negative amount
$S_T=K$	0	0
$S_T>K$	0	0

Result: The written put’s expiration cash flow is never positive under this option-only sign convention. It is negative when $S_T<K$, and zero when $S_T\ge K$. This statement does not include the premium received earlier or cash flows from other positions.

Why the Writer’s Formula Is the Negative

The relationship follows from the two sides of the same option contract. For each unit of payoff the contract transfers to the holder at expiration, the writer has the corresponding obligation. The holder’s payoff is $(K-S_T)^+$; the writer’s cash flow for that same contract is the opposite signed amount. Negating the positive-part expression gives the writer’s formula directly.

\text{Writer's cash flow} = -\max(K-S_T,0) = \begin{cases} -(K-S_T)=S_T-K, & K-S_T>0,\\ 0, & K-S_T\le 0. \end{cases}

The condition $K-S_T>0$ is equivalent to $S_T<K$, which gives the first case. When $K-S_T\le 0$, the maximum with zero is zero, giving the second case. At equality, $S_T=K$, both cases agree at zero. This also checks that the piecewise expression and the positive-part expression describe the same cash flow at the boundary.

Scaling for Contracts and Position Size

The formula above is for one option on one unit of the underlying. Let $q$ be the number of written put contracts, and let $M$ be the number of underlying units represented by each contract. If all contracts have the same strike and expiration and the position is written in the same direction, the total cash flow is $qM$ times the per-unit writer cash flow. Here $q$ and $M$ are nonnegative quantities; the fact that the position is written is already represented by the negative sign in the formula.

\text{Total writer cash flow at }T = -qM(K-S_T)^+ = \begin{cases} qM(S_T-K), & S_T<K,\\ 0, & S_T\ge K. \end{cases}

If the problem states the contract size in a different way, use that stated number of underlying units rather than assuming a standard multiplier. If there are written puts with different strikes, calculate each position’s cash flow separately and add the signed amounts. A put with a terminal price above its own strike contributes zero even if another put in the portfolio has a lower strike.

Worked Example: One Written Put at Three Terminal Prices

A hypothetical investor writes one put on one share with strike $K=\$50$. Calculate the writer’s expiration cash flow per share if the terminal share price is (a) $\$42$, (b) $\$50$, and (c) $\$63$. Ignore the premium and all other cash flows.

Method. Apply the piecewise writer formula to each terminal price. Because this is one option on one share, no contract multiplier is needed.

(a) Terminal price $\$42$. Since $42<50$, the writer’s cash flow is $S_T-K$:

$$ 42-50=-8. $$

The writer’s expiration cash flow is $-\$8$ per share. As a check, the holder’s payoff is $50-42=\$8$, whose negative is $-\$8$.

(b) Terminal price $\$50$. Since $S_T=K$, the difference is zero:

$$ 50-50=0. $$

The writer’s expiration cash flow is $\$0$ per share. The holder’s payoff is also $\max(50-50,0)=\$0$.

(c) Terminal price $\$63$. Since $63>50$, the writer’s cash flow is zero:

$$ -(50-63)^+=-0=0. $$

The writer’s expiration cash flow is $\$0$ per share. The holder also has zero payoff because $50-63$ is negative. This example avoids the common mistake of treating $K-S_T$ as a payment in every case without applying the positive-part rule.

Worked Example: Applying a Contract Multiplier

A hypothetical account writes 4 put contracts. Each contract represents 100 shares, the strike is $K=\$72$, and the terminal share price is $S_T=\$65$. Find the account’s total expiration cash flow from these puts, excluding any premium.

Method. First calculate the per-share writer cash flow, then multiply by the number of shares covered by the written contracts.

Step 1: Find the per-share amount. Because $65<72$, use $S_T-K$:

$$ 65-72=-7. $$

The cash flow is $-\$7$ per share. Equivalently, the holder’s payoff is $72-65=\$7$ per share, and the writer’s amount is its negative.

Step 2: Find the total number of shares.

4\text{ contracts}\times100\text{ shares per contract}=400\text{ shares}.

Step 3: Scale the cash flow.

400\times(-\$7)=-\$2{,}800.

The account’s total expiration cash flow from the written puts is $-\$2{,}800$. As a check, the holder-side total payoff is $400\times\$7=\$2{,}800$, so the writer-side amount is its negative. The multiplier scales the option cash flow; it does not change which terminal-price case applies.

Worked Example: Combining Two Written Put Positions

A hypothetical portfolio contains 3 written put contracts with strike $\$40$ and 2 written put contracts with strike $\$55$. Each contract represents 100 shares. Both positions have the same expiration, when the underlying price is $\$47$. Find the total option cash flow at expiration, excluding premiums.

Method. Evaluate each strike separately using the writer’s piecewise formula, apply each position’s contract count and multiplier, and add the signed cash flows.

Position 1: strike $\$40$. The terminal price is above this strike: $47>40$. Therefore, the cash flow per share is zero. The position represents $3\times100=300$ shares, so its total cash flow is:

300\times0=\$0.

Position 2: strike $\$55$. The terminal price is below this strike: $47<55$. The writer’s cash flow per share is:

$$ S_T-K=47-55=-8. $$

This position represents $2\times100=200$ shares. Its total cash flow is:

200\times(-\$8)=-\$1{,}600.

Portfolio total. Add the signed results:

\$0+(-\$1{,}600)=-\$1{,}600.

The total expiration cash flow from the two written put positions is $-\$1{,}600$. The check is that only the put with strike $\$55$ has a positive holder payoff at $S_T=\$47$; the strike-$\$40$ put contributes zero. Applying the same terminal-price comparison to both strikes would produce an incorrect portfolio total.

Common Sign and Case Errors

A frequent mistake is to calculate $K-S_T$ for the writer and report it as the writer’s cash flow. For example, with $K=\$60$ and $S_T=\$52$, that calculation gives $60-52=\$8$. The $\$8$ is the holder’s payoff, not the writer’s cash flow. The writer’s amount is:

-(60-52)^+=-\$8.

A second error is to use $S_T-K$ when $S_T>K$. If $K=\$60$ and $S_T=\$68$, direct subtraction gives $68-60=\$8$, but that is not a payment received by the writer from the put. The correct rule is piecewise: because $S_T\ge K$, the put’s expiration cash flow is zero.

A third error is to multiply by a contract count but overlook the number of underlying units in each contract. In a question with 4 contracts representing 100 shares each, the multiplier is $4\times100=400$, not 4. Always identify whether the requested amount is per unit, per contract, or for the entire position before reporting the answer.

Finally, do not add an earlier premium when the question asks only for the option’s expiration cash flow. The premium is a separate cash flow at a different time. Keeping the time and scope of each requested amount clear prevents a payoff calculation from being mistaken for a profit calculation.

Key takeaway: For a written put, calculate the holder’s positive-part payoff and negate it, or use $S_T-K$ only when $S_T<K$ and zero otherwise. Then scale by the number of contracts and the units per contract. The result is the option’s cash flow at expiration, not a total investment profit.

Check Your Understanding

For each question, report the written put’s expiration cash flow only; exclude any premium unless the question asks otherwise.

A single written put covers one unit, has strike $\$36$, and expires when the underlying price is $\$29$. What is the writer’s cash flow? (Answer: $-\$7$.)
An account writes 5 put contracts, each representing 100 units, with strike $\$84$. The terminal price is $\$91$. Find the total expiration cash flow. (Answer: $\$0$.)
A written put has strike $\$125$, and one contract represents 50 units. The terminal price is $\$118$. Find the contract’s total expiration cash flow. (Answer: $-\$350$.)
A portfolio has 4 written puts with strike $\$30$ and 2 written puts with strike $\$38$. Each contract represents 10 units. At expiration the underlying price is $\$34$. Find the portfolio’s total cash flow. (Answer: $-\$80$.)
An account writes 3 contracts with strike $\$60$, each representing 25 units. The underlying price at expiration is $\$52$. The account received a premium of $\$3$ per unit earlier. What is the written puts’ expiration cash flow, excluding that earlier premium? (Answer: $-\$600$.)

Terminal price	Holder’s payoff per unit	Writer’s cash flow per unit
\(S_T<K\)	\(K-S_T\)	\(S_T-K\), a negative amount
\(S_T=K\)	0	0
\(S_T>K\)	0	0

What You'll Learn