What a Long Put Payoff Diagram Shows
A long put gives its holder the right, but not the obligation, to sell an underlying asset for a specified exercise price. At expiration, the asset’s market price determines whether using that right produces a positive payoff. A payoff diagram makes this relationship visible: it shows how the amount received from the option changes as the underlying price at expiration changes.
The graph in this tutorial is for the option payoff at expiration, not a forecast of the underlying price. Its horizontal axis is the underlying price at expiration, and its vertical axis is the long put payoff. Unless a contract multiplier is specified, the payoff is measured per share of underlying asset.
If \(S_T<K\), exercising the put allows the holder to sell for \(K\) an asset worth \(S_T\), so the payoff per share is \(K-S_T\). If \(S_T\geq K\), selling at the exercise price would not produce a positive payoff relative to the market price, so the holder does not exercise and the payoff is zero. At exactly \(S_T=K\), either exercising or not exercising gives a payoff of zero.
Why the Graph Has This Shape
The piecewise formula follows directly from the holder’s choice at expiration. Below the exercise price, the value of selling at \(K\) rather than at the market price is the difference \(K-S_T\). At or above \(K\), that difference is zero or negative, and the holder can avoid exercising. The maximum in the definition records this choice: the payoff cannot be negative.
The formula also specifies the graph’s important coordinates. At \(S_T=0\), the payoff is \(K\), so the graph begins at \((0,K)\). At \(S_T=K\), the payoff is zero, giving the boundary point \((K,0)\). For every price above \(K\), the graph remains at zero. Between zero and \(K\), it is a straight declining segment because \(K-S_T\) decreases one-for-one as \(S_T\) increases.
To sketch the diagram, label both axes, mark the exercise price on the horizontal axis, and calculate a few payoffs. Plot the point at zero underlying price, the point at the exercise price, and one or more points on each side of the exercise price. Connect the points below \(K\) with a straight line, then draw the graph along the horizontal axis for prices at or above \(K\). Underlying prices are nonnegative, so the economically relevant graph does not extend to the left of zero.
Use price per share and payoff per share, unless a contract multiplier is specified.
The boundary between the positive-payoff segment and the zero-payoff segment is at \(S_T=K\).
For each stated expiration price, calculate \(\max(K-S_T,0)\), then apply any contract multiplier.
Draw the declining straight segment from \((0,K)\) to \((K,0)\), followed by a horizontal zero-payoff segment.
Worked Examples
Worked Example: Plotting a Put with a $50 Exercise Price
A hypothetical investor holds one put on one share of an asset. The exercise price is \(K=\$50\). Find the payoff at expiration prices of \(\$0,\$10,\$25,\$40,\$50,\) and \(\$65\), then describe the plotted graph. The option is evaluated at expiration; no probability distribution or interest-rate assumption is needed.
Method. Apply \(P_T=\max(50-S_T,0)\) at each stated price. Since the contract covers one share, no multiplier is needed.
At \(S_T=\$0\), \(P_T=\max(50-0,0)=\max(50,0)=\$50\).
At \(S_T=\$10\), \(P_T=\max(50-10,0)=\max(40,0)=\$40\).
At \(S_T=\$25\), \(P_T=\max(50-25,0)=\max(25,0)=\$25\).
At \(S_T=\$40\), \(P_T=\max(50-40,0)=\max(10,0)=\$10\).
At \(S_T=\$50\), \(P_T=\max(50-50,0)=\max(0,0)=\$0\).
At \(S_T=\$65\), \(P_T=\max(50-65,0)=\max(-15,0)=\$0\).
The coordinates are \((0,50),(10,40),(25,25),(40,10),(50,0)\), and \((65,0)\), with prices and payoffs in dollars per share. Connect the points below \(\$50\) with a straight declining line. From \((50,0)\) onward, draw a horizontal line at zero. The values check against the rule: each increase in price below \(\$50\) reduces the payoff, while prices at or above \(\$50\) give zero payoff.
Common mistake avoided: At \(\$65\), calculating \(50-65=-\$15\) and plotting a negative payoff would ignore the holder’s choice not to exercise. The maximum with zero sets the payoff to zero instead.
Worked Example: Applying a Contract Multiplier
A hypothetical exchange-traded put has an exercise price of \(K=\$84\) and covers 100 shares. Find the contract payoff when the underlying expires at \(\$61,\$84,\) and \(\$103\), and give the corresponding graph coordinates in contract dollars. The option is evaluated only at expiration; no probability distribution or interest-rate assumption is required.
Method. First calculate the per-share payoff, then multiply by 100. This keeps the graph’s units consistent with the requested contract payoff.
At \(S_T=\$61\), the per-share payoff is \(\max(84-61,0)=\max(23,0)=\$23\). The contract payoff is \(100\times 23=\$2{,}300\).
At \(S_T=\$84\), the per-share payoff is \(\max(84-84,0)=\$0\). The contract payoff is \(100\times 0=\$0\).
At \(S_T=\$103\), the per-share payoff is \(\max(84-103,0)=\max(-19,0)=\$0\). The contract payoff is \(100\times 0=\$0\).
For a graph of contract payoff, plot \((61,2300),(84,0)\), and \((103,0)\), where the horizontal values are dollars per share and the vertical values are dollars per contract. To establish the left endpoint as well, use \(S_T=0\): the per-share payoff is \(\max(84-0,0)=\$84\), and the contract payoff is \(100\times84=\$8{,}400\), giving \((0,8400)\). Connect the positive-payoff points in a straight line to \((84,0)\); the graph remains at zero beyond \(\$84\).
Common mistake avoided: Plotting \(\$23\) as the contract payoff at \(S_T=\$61\) would report the per-share amount on a graph labelled in contract dollars. The multiplier must be applied to the payoff, not to the underlying price.
Worked Example: Building a Contract Payoff Table Before Sketching
A hypothetical put covers 25 shares and has an exercise price of \(K=\$120\). Construct contract-payoff points for expiration prices of \(\$0,\$30,\$90,\$120,\) and \(\$150\). State the axis units and the shape. The prices are fixed scenarios, so no distribution or interest-rate assumption is used.
Method. Calculate \(\max(120-S_T,0)\) per share at each scenario, then multiply each result by 25.
At \(S_T=\$0\), the per-share payoff is \(\max(120-0,0)=\$120\); the contract payoff is \(25\times120=\$3{,}000\).
At \(S_T=\$30\), the per-share payoff is \(\max(120-30,0)=\$90\); the contract payoff is \(25\times90=\$2{,}250\).
At \(S_T=\$90\), the per-share payoff is \(\max(120-90,0)=\$30\); the contract payoff is \(25\times30=\$750\).
At \(S_T=\$120\), the per-share payoff is \(\max(120-120,0)=\$0\); the contract payoff is \(25\times0=\$0\).
At \(S_T=\$150\), the per-share payoff is \(\max(120-150,0)=\max(-30,0)=\$0\); the contract payoff is \(25\times0=\$0\).
Plot underlying price per share on the horizontal axis and contract payoff in dollars on the vertical axis. The coordinates are \((0,3000),(30,2250),(90,750),(120,0)\), and \((150,0)\). The first four lie on the declining segment from \((0,3000)\) to \((120,0)\); the last lies on the zero-payoff segment. The check is that the payoff is zero at the exercise price and stays zero above it.
Common mistake avoided: The graph should not continue below zero after the exercise price. Although \(120-150=-30\), the holder is not required to exercise, so the payoff at \(\$150\) is zero.
Reading and Checking the Diagram
A correctly labelled diagram contains more information than a collection of payoff values. The location of the boundary identifies the exercise price; the height at an underlying price of zero identifies the per-share exercise price; and the flat segment shows that the payoff cannot fall below zero. If the vertical axis is in contract dollars instead of dollars per share, both the vertical intercept and every positive payoff are scaled by the number of shares covered.
The graph represents payoff only. It does not subtract the amount paid to acquire the put or include other costs. Therefore, it should not be labelled as a graph of net profit or interpreted as a complete measure of an investor’s overall financial outcome. For this diagram, use the contractual payoff formula and the units requested.
| Expiration-price region | Per-share payoff | Graph segment |
|---|---|---|
| \(S_T=0\) | \(K\) | Left endpoint at \((0,K)\) |
| \(0<S_T<K\) | \(K-S_T\) | Declining straight segment above zero |
| \(S_T=K\) | \(0\) | Boundary point \((K,0)\) |
| \(S_T>K\) | \(0\) | Horizontal segment on the price axis |
A useful final check is to test one point in each region: one price below \(K\), the price \(K\) itself, and one price above \(K\). The first must give a positive payoff, and the last two must give zero. Also check that the axes use compatible units: a per-share payoff belongs on a per-share vertical scale, while a contract payoff includes the multiplier.
Check Your Understanding
For each problem, use the payoff at expiration and pay attention to the requested units.
- A one-share put has \(K=\$46\). Find the payoff at \(S_T=\$31,\$46,\) and \(\$58\), and give the three graph coordinates. (Answer: Payoffs \(\$15,\$0,\$0\); coordinates \((31,15),(46,0),(58,0)\), in dollars per share.)
- A put with \(K=\$95\) covers 10 shares. Find the contract payoff at \(S_T=\$40,\$80,\$95,\) and \(\$110\). (Answer: \(\$550,\$150,\$0,\$0\), respectively.)
- A per-share payoff graph begins at \((0,36)\) and reaches zero at an underlying price of \(\$36\). Identify \(K\), then calculate the payoff at \(S_T=\$21\) and \(S_T=\$49\). (Answer: \(K=\$36\); payoffs \(\$15\) and \(\$0\) per share.)
- A put covers 100 shares and has \(K=\$64\). Find the contract payoff at \(S_T=\$57\), then state the graph coordinate using contract dollars vertically. (Answer: \(\$700\); coordinate \((57,700)\).)
- Two one-share puts have exercise prices \(K_1=\$30\) and \(K_2=\$45\). For each put, calculate the payoff at \(S_T=\$20,\$35,\) and \(\$50\). (Answer: For \(K_1=\$30\), payoffs are \(\$10,\$0,\$0\); for \(K_2=\$45\), payoffs are \(\$25,\$10,\$0\), respectively.)