Why Build an Expiration Payoff Table?
An expiration payoff table turns an option position into a set of explicit calculations. Each row gives a possible underlying price at expiration; the columns show the option’s contractual payoff and, if required, the position’s profit after the premium. This format makes it easier to compare outcomes on either side of the exercise price and to catch sign or premium errors.
The table is a calculation tool, not a forecast. The selected underlying prices are hypothetical scenarios. For a call, the exercise decision changes at the exercise price \(K\); for a put, it changes there as well, but in the opposite direction. At exactly \(K\), either option has zero expiration payoff.
The convention above reports profit per underlying unit and does not accrue interest on the premium. It also excludes commissions, taxes, and other transaction costs. When a contract represents \(m\) underlying units, multiply both the per-unit payoff and the per-unit profit by \(m\) to obtain the contract totals. If there are \(n\) identical contracts, multiply those totals by \(n\) as well.
For example, a long call’s payoff is never negative: the holder can decline to exercise. Its profit can nevertheless be negative because the holder paid a premium. A short call’s signed payoff is never positive: the writer owes the amount paid by the long holder when the call finishes in the money. The writer’s profit includes the premium received, but may become negative if the obligation exceeds that premium.
A Reliable Method for Filling the Table
Keep the steps separate. First calculate the option payoff from \(S_T\) and \(K\), without considering the premium. Then apply the sign for the position. Only after that should you add or subtract the premium to calculate profit. This prevents a premium from being mistaken for part of the contractual payoff.
Write down whether the position is a call or put, long or short, the exercise price, the premium, and the underlying prices to be evaluated.
For a call, use \(\max(S_T-K,0)\). For a put, use \(\max(K-S_T,0)\). The maximum function ensures that the holder does not exercise when doing so would be disadvantageous.
A short position has the negative of the long payoff at the same \(S_T\). Thus, long and short payoffs are opposites row by row.
Subtract the premium for a long position or add the premium for a short position. Multiply per-unit results by the contract multiplier, and then by the number of contracts, if totals are requested.
Why the Payoff Formulas Have This Shape
At expiration, the holder of a call has the right, but not the obligation, to buy the underlying for \(K\). If \(S_T>K\), buying at \(K\) and valuing the underlying at \(S_T\) gives a payoff of \(S_T-K\). If \(S_T\leq K\), exercising does not produce a positive payoff, so the holder does not exercise and the payoff is zero. Combining the two cases gives the maximum expression:
A put gives its holder the right to sell for \(K\). When \(S_T<K\), selling for \(K\) rather than the market price produces \(K-S_T\); when \(S_T\geq K\), the holder does not exercise for a positive payoff. This gives \(\max(K-S_T,0)\). A short position has the opposite signed payoff because the writer owes the amount the holder receives. Profit then follows from cash flows: the long holder pays the premium initially, while the short writer receives it.
Worked Tables for Calls
Worked Example: Long Call Below, At, and Above the Exercise Price
A hypothetical call has \(K=\$50\) and a premium of \(c=\$4\) per share. Construct the per-share payoff and profit table for \(S_T=\$42,\$50,\$63\). Then find the profit for one contract covering 100 shares in each scenario. Assume no transaction costs and use the stated convention that the premium is not accrued.
Method and payoff calculations. For a long call, the payoff per share is \(\max(S_T-50,0)\). At \(S_T=\$42\), this is \(\max(42-50,0)=\max(-8,0)=\$0\). At \(S_T=\$50\), it is \(\max(0,0)=\$0\). At \(S_T=\$63\), it is \(\max(13,0)=\$13\).
Profit calculations. Subtract the \(\$4\) premium from each payoff: \(\$0-\$4=-\$4\), \(\$0-\$4=-\$4\), and \(\$13-\$4=\$9\) per share.
| Underlying price \(S_T\) | Long call payoff per share | Long call profit per share | Profit for 100 shares |
|---|---|---|---|
| \(\$42\) | \(\$0\) | \(-\$4\) | \(-\$400\) |
| \(\$50\) | \(\$0\) | \(-\$4\) | \(-\$400\) |
| \(\$63\) | \(\$13\) | \(\$9\) | \(\$900\) |
Check and trap avoided. At or below \(K\), the call is not exercised, so the payoff is zero; the premium still makes the profit negative. Above \(K\), the \(\$13\) payoff exceeds the \(\$4\) premium by \(\$9\). Multiplying the per-share profit by 100 gives the contract profit. A common error is to report the \(\$13\) payoff as the profit, omitting the premium.
Worked Example: Short Call With a Contract Multiplier
A hypothetical investor writes one call with \(K=\$80\), receives \(c=\$6\) per share, and writes a contract covering 50 shares. Construct a table at \(S_T=\$68,\$80,\$95\), showing the short payoff and profit per share and for the contract. Ignore transaction costs and premium interest.
Method and payoff calculations. First calculate the long call payoff, then reverse its sign for the writer. At \(\$68\), \(\max(68-80,0)=\$0\), so the short payoff is \(\$0\). At \(\$80\), \(\max(80-80,0)=\$0\), so the short payoff is also \(\$0\). At \(\$95\), the long payoff is \(\max(95-80,0)=\$15\), so the short payoff is \(-\$15\) per share.
Profit calculations. The writer’s profit is premium received plus signed payoff. Thus, the three per-share profits are \(\$6+\$0=\$6\), \(\$6+\$0=\$6\), and \(\$6-\$15=-\$9\). Multiplying each amount by 50 gives the contract results.
| \(S_T\) | Short call payoff per share | Short call profit per share | Profit for 50 shares |
|---|---|---|---|
| \(\$68\) | \(\$0\) | \(\$6\) | \(\$300\) |
| \(\$80\) | \(\$0\) | \(\$6\) | \(\$300\) |
| \(\$95\) | \(-\$15\) | \(-\$9\) | \(-\$450\) |
Check and trap avoided. Each short payoff is the negative of the corresponding long payoff. In the last row, the writer receives \(\$6\) but owes \(\$15\) per share, leaving a loss of \(\$9\) per share. Do not use \(\$15\) as a positive payoff for the writer; that is the holder’s payoff.
Worked Tables for Puts
Worked Example: Long Put Across the Exercise Price
A hypothetical long put has \(K=\$32\) and premium \(p=\$3\) per share. Build the payoff and profit table at \(S_T=\$20,\$32,\$39\). Give results per share and for one 100-share contract. Ignore transaction costs and do not accrue interest on the premium.
Method and payoff calculations. For a long put, use \(\max(32-S_T,0)\). At \(\$20\), the payoff is \(\max(32-20,0)=\$12\). At \(\$32\), it is \(\max(0,0)=\$0\). At \(\$39\), it is \(\max(-7,0)=\$0\).
Profit calculations. Subtract the \(\$3\) premium. The per-share profits are \(\$12-\$3=\$9\), \(\$0-\$3=-\$3\), and \(\$0-\$3=-\$3\). For the 100-share contract, multiply each result by 100.
| \(S_T\) | Long put payoff per share | Long put profit per share | Profit for 100 shares |
|---|---|---|---|
| \(\$20\) | \(\$12\) | \(\$9\) | \(\$900\) |
| \(\$32\) | \(\$0\) | \(-\$3\) | \(-\$300\) |
| \(\$39\) | \(\$0\) | \(-\$3\) | \(-\$300\) |
Check and trap avoided. The put has a positive payoff only below \(K\). At \(S_T=\$20\), the difference \(K-S_T\) is \(\$12\), not \(\$20\); the put does not pay the entire underlying price. A frequent error is to reverse the put subtraction and use \(S_T-K\), which would incorrectly assign zero payoff to the in-the-money case.
Worked Example: Short Put and the Premium Offset
A hypothetical investor writes one put with \(K=\$120\), receives \(p=\$8\) per share, and the contract covers 25 shares. Find the short payoff and profit at \(S_T=\$105,\$120,\$138\). Report per-share amounts and contract profits. Assume no transaction costs or premium interest.
Method and payoff calculations. The long put payoff is \(\max(120-S_T,0)\), and the short payoff is its negative. At \(\$105\), the long payoff is \(\max(15,0)=\$15\), so the short payoff is \(-\$15\). At \(\$120\), the payoff is zero for both positions. At \(\$138\), the long payoff is \(\max(-18,0)=\$0\), so the short payoff is zero.
Profit calculations. Add the received \(\$8\) premium to each signed payoff. Per share, the profits are \(\$8-\$15=-\$7\), \(\$8+\$0=\$8\), and \(\$8+\$0=\$8\). Multiplying by 25 gives contract profits of \(-\$175,\$200,\$200\).
| \(S_T\) | Short put payoff per share | Short put profit per share | Profit for 25 shares |
|---|---|---|---|
| \(\$105\) | \(-\$15\) | \(-\$7\) | \(-\$175\) |
| \(\$120\) | \(\$0\) | \(\$8\) | \(\$200\) |
| \(\$138\) | \(\$0\) | \(\$8\) | \(\$200\) |
Check and trap avoided. Below the exercise price, the writer’s obligation is \(\$15\) per share and the premium offsets \(\$8\) of it, leaving a \(\$7\) loss. At and above \(K\), the put expires without a positive holder payoff, so the writer retains the premium. Do not subtract the premium from the short position’s payoff as though the writer paid it.
Reading and Checking a Completed Table
A useful table includes enough information to distinguish contractual payoff from profit. A row should not silently switch between a per-share amount and a contract total. Label the units in the column headings, and apply the multiplier only after the per-unit amounts have been calculated. If a problem gives multiple contracts, multiply the contract result by the number of contracts last.
The table’s pattern provides a quick check. A long call payoff is zero at and below \(K\), then increases dollar for dollar above \(K\). A short call payoff has the reverse sign. A long put payoff decreases dollar for dollar as \(S_T\) rises below \(K\), and is zero at and above \(K\); a short put again has the reverse sign. At \(S_T=K\), all four signed payoffs equal zero. Profits at that price are not generally zero because the premium remains.
Break-even prices offer another consistency check for profit columns. For a long call, profit per unit is zero when \(S_T-K=c\), or \(S_T=K+c\). For a long put, it is zero when \(K-S_T=p\), or \(S_T=K-p\), provided that price is feasible. The corresponding short positions have the same break-even prices because their profits are the negatives of the long profits when the same premium and contract terms are used. These checks do not replace the row-by-row calculation; they help identify a misplaced premium or sign.
A common trap is to treat the payoff as if it already included the premium. The contractual payoff depends on \(S_T\) and \(K\); profit also depends on the premium. Another is to assume “at the money” means zero profit. At \(S_T=K\), the expiration payoff is zero, but a long position’s profit is the negative premium and a short position’s profit is the premium received. A final trap is multiplying only one column by the contract size. If the requested figures are contract totals, scale both payoff and profit, not just one of them.
Check Your Understanding
For each problem, use the stated convention: profit equals signed payoff plus the initial premium cash flow, with no premium interest or transaction costs.
- A long call has \(K=\$44\) and premium \(\$2.50\) per share. Give the payoff and profit per share at \(S_T=\$38,\$44,\$51\). (Answer: Payoffs \(\$0,\$0,\$7\); profits \(-\$2.50,-\$2.50,\$4.50\).)
- A short put has \(K=\$70\), premium received \(\$5\) per share, and one 100-share contract. Find the contract profit at \(S_T=\$59,\$70,\$82\). (Answer: \(-\$600,\$500,\$500\).)
- A long put has \(K=\$125\) and premium \(\$9\) per share. Give the payoff and profit per share at \(S_T=\$103,\$125,\$141\). (Answer: Payoffs \(\$22,\$0,\$0\); profits \(\$13,-\$9,-\$9\).)
- A short call has \(K=\$36\), premium received \(\$1.80\) per share, and two contracts covering 100 shares each. Find the total profit across both contracts at \(S_T=\$30,\$36,\$42.50\). (Answer: \(\$360,\$360,-\$940\).)
- A long call has \(K=\$90\) and premium \(\$7\) per share. Construct the profit values at \(S_T=\$76,\$90,\$97,\$110\), and identify the break-even row. (Answer: Profits \(-\$7,-\$7,\$0,\$13\) per share; break-even at \(S_T=\$97\).)