What a Long Put Gives Its Holder
A put option gives its holder the right, but not the obligation, to sell the underlying asset for a specified price. At expiration, the holder compares that contractual sale price with the asset’s market price. The decision is straightforward: exercise when the contractual price is more favorable, and otherwise let the option expire without exercise.
This tutorial derives the resulting expiration payoff. The focus is the value produced by the exercise decision at expiration—not the option’s value before expiration and not the holder’s overall gain or loss after accounting for the price paid to acquire the option.
The formula assumes the option is held to expiration, the holder can choose whether to exercise, and the payoff is measured per unit of underlying. It describes the standard expiration payoff before any acquisition cost, transaction cost, tax, or contract multiplier is applied. A contract may cover more than one unit; in that case, the per-unit payoff must be scaled to the contract’s specified quantity.
Deriving the Payoff by Cases
The holder’s exercise choice depends on whether the strike price is greater than, equal to, or less than the market price at expiration. If the holder exercises, the holder sells an asset worth \(S_T\) for \(K\). The economic value of that exchange, per unit, is \(K-S_T\). If the holder does not exercise, the option expires and produces a payoff of zero. The holder chooses the more valuable of those two outcomes.
This maximum expression follows directly from the two available outcomes: exercise for \(K-S_T\), or abandon for zero. Applying it to each possible relationship between \(S_T\) and \(K\) gives the piecewise rule.
If \(S_T<K\), then \(K-S_T>0\). Exercising allows the holder to sell for more than the asset’s market price, so the payoff is \(K-S_T\).
If \(S_T=K\), then \(K-S_T=0\). Exercise and abandonment both produce a payoff of zero.
If \(S_T>K\), then \(K-S_T<0\). Selling for \(K\) would be worse than selling at the market price, so the holder does not exercise and the payoff is zero.
The two-case version combines equality with the above-strike case because both produce zero. The three-case explanation is often useful when reading a question carefully: equality is a boundary condition, not a case with a positive payoff.
Worked Example: Applying the Below-Strike and Above-Strike Cases
A hypothetical long put has a strike of \(K=\$52\) per share. At expiration, the underlying share price is first \(\$47.60\), and in a separate scenario is \(\$55.25\). Find the payoff per share in each scenario, then find the payoff for one contract covering 100 shares. Ignore transaction costs.
Method. Compare \(S_T\) with \(K\) in each scenario, apply the per-share payoff rule, and then multiply by the contract’s 100 shares.
Scenario 1: \(S_T=\$47.60\). Since \(\$47.60<\$52\), the put is exercised. The payoff per share is:
For one contract covering 100 shares, the payoff is \(100\times \$4.40=\$440.00\).
Scenario 2: \(S_T=\$55.25\). Since \(\$55.25>\$52\), selling at the strike would be less favorable than the market price. The holder does not exercise, so the payoff per share is \(\$0.00\), and the contract payoff is \(100\times \$0.00=\$0.00\).
Check. The first payoff is positive because the strike exceeds the market price by \(\$4.40\) per share. The second is zero rather than negative because the holder can abandon the option. A common mistake is to report \(K-S_T=-\$3.25\) in Scenario 2; the maximum-with-zero rule corrects that mistake.
From Per-Unit Payoff to Contract Payoff
The basic formula gives a payoff for one unit of the underlying. To calculate the total payoff for a position, identify how many underlying units the position represents. If a position contains \(n\) contracts and each contract covers \(m\) units, then it represents \(nm\) units, provided every contract has the same terms and expiration. Its total payoff is:
This scaling does not change the exercise decision. First determine whether \(K-S_T\) is positive; then multiply the nonnegative per-unit payoff by the number of units covered. For cash-settled contracts, the settlement amount may be specified as a dollar multiplier per index point rather than a number of deliverable shares. The same logic applies: calculate the positive difference in the contract’s quoted units, then apply its stated multiplier.
Worked Example: Multiple Contracts and a Zero-Payoff Scenario
A hypothetical investor holds three long put contracts on a fund. Each contract covers 100 shares and has a strike of \(K=\$80\) per share. Determine the total expiration payoff if the fund price is \(\$74.25\), and then if it is \(\$83.50\). The contract acquisition cost is not part of this calculation.
Method. Find the per-share payoff for each market-price scenario, determine the total number of shares covered, and scale the per-share amount.
The position covers \(3\times100=300\) shares.
Scenario 1: \(S_T=\$74.25\). Because \(\$74.25<\$80\), the per-share payoff is:
The total payoff is \(300\times \$5.75=\$1{,}725.00\).
Scenario 2: \(S_T=\$83.50\). Because \(\$83.50>\$80\), the holder does not exercise. The per-share payoff is \(\max(\$80.00-\$83.50,0)=\$0.00\), so the total payoff is \(300\times \$0.00=\$0.00\).
Check. The first total equals three contracts multiplied by 100 shares per contract multiplied by \(\$5.75\) per share. The second remains zero for all 300 shares. The common trap is to use the number of contracts alone as the multiplier and omit the 100 shares covered by each contract.
Worked Example: Cash-Settled Put With a Point Multiplier
A hypothetical cash-settled index put has a strike of 1,250 index points and pays \(\$20\) for each index point by which the index finishes below the strike. Find the expiration settlement if the index finishes at 1,218.5, and if it finishes at 1,250.0.
Method. Apply the long put rule in index points, then multiply any positive point difference by the stated dollar-per-point multiplier.
Scenario 1: index level \(S_T=1{,}218.5\). The index finishes below the strike, so the positive difference is \(1{,}250-1{,}218.5=31.5\) points. The settlement is \(31.5\times \$20=\$630.00\).
Scenario 2: index level \(S_T=1{,}250.0\). The difference is \(1{,}250.0-1{,}250.0=0\) points. The settlement is \(0\times \$20=\$0.00\).
Check. The first settlement is positive because the index is 31.5 points below the strike; the second is zero at equality. A common mistake is to treat the point difference itself as a dollar amount. The contract’s \(\$20\)-per-point multiplier is required to convert points into dollars.
Payoff Is Not the Same as the Amount Paid for the Option
The formula derived here measures what the long put delivers at expiration from exercise or cash settlement. It does not subtract the amount the holder paid to acquire the option. For example, if the underlying finishes below the strike, \(K-S_T\) is still the expiration payoff per unit even though the holder may have paid a cost for the option earlier. Keeping those quantities separate avoids mixing a contract’s expiration outcome with the cost of entering the position.
The formula also applies at expiration, when \(S_T\) is known. Before expiration, the underlying price can change and time remains before the exercise decision is final. The expiration rule alone does not determine an option’s value at an earlier date. Similarly, if a contract has a nonstandard multiplier or settlement convention, use the contract terms to translate the per-unit or per-point payoff into dollars.
| Expiration condition | Exercise decision | Payoff per unit |
|---|---|---|
| \(S_T<K\) | Exercise is favorable | \(K-S_T\) |
| \(S_T=K\) | Exercise or abandon | 0 |
| \(S_T>K\) | Abandon | 0 |
Check Your Understanding
For each problem, identify the relevant contract terms, apply the expiration cases, and scale the result when a multiplier is given.
- A hypothetical put has a strike of \(\$46\) per share and covers 50 shares. The share price at expiration is \(\$41.80\). What is the payoff per share and the total contract payoff? (Answer: \(\$4.20\) per share; \(\$210.00\) total.)
- A long put has a strike of \(\$125\), and its underlying finishes at \(\$129.40\). The contract covers 100 units. Find the per-unit and total payoff, and state the exercise decision. (Answer: \(\$0.00\) per unit; \(\$0.00\) total; do not exercise.)
- An investor owns two put contracts, each covering 40 shares, with strike \(\$72\). The share price at expiration is \(\$69.50\). Find the total payoff. (Answer: \(2\times40\times(\$72-\$69.50)=\$200.00\).)
- A cash-settled put has strike 3,400 index points and a settlement multiplier of \(\$5\) per point. The index finishes at 3,386. Find the settlement amount. (Answer: \((3{,}400-3{,}386)\times\$5=\$70.00\).)
- A put with strike \(\$58\) covers 25 shares. At expiration, the share price is exactly \(\$58\). Explain why the payoff is not positive and calculate the total payoff. (Answer: The strike-minus-market difference is zero, so the per-share payoff is \(\$0.00\) and the total payoff is \(\$0.00\).)