Why Position Size Changes the Dollar Result
An option payoff or profit diagram is usually stated per unit of the underlying. A traded position, however, may consist of several contracts, and each contract may represent more than one underlying unit. To find the position’s total dollar result, keep the per-unit calculation separate from the contract count and the contract multiplier.
This distinction matters even when the per-unit answer is correct. If a call produces a profit of $1.50 per underlying unit, that is not necessarily a $1.50 profit for the investor: the position might cover hundreds of units. The multiplier scales both the option payoff and the premium, while the number of contracts scales the position again.
For an option with strike \(K\), terminal underlying price \(S_T\), and premium \(\pi\) per underlying unit, define the buyer’s per-unit expiration payoff as \(g(S_T)\). For a call, \(g(S_T)=\max(S_T-K,0)\); for a put, \(g(S_T)=\max(K-S_T,0)\). The buyer’s per-unit profit, under the expiration-profit convention used here, is \(g(S_T)-\pi\). The writer’s per-unit payoff and profit are the negatives of the buyer’s amounts.
This convention compares the premium paid or received at initiation directly with the expiration payoff; it does not accumulate or discount the premium. It is appropriate for the payoff and profit calculations in this tutorial. If a problem instead asks for a value at another date or specifies financing, apply the stated interest convention separately.
How the Scaling Works
The calculation has three distinct quantities: the payoff or profit per underlying unit, the number of underlying units per contract, and the number of contracts. A reliable method computes these in that order. If the position contains several types of options, calculate each type’s total profit separately and then add the dollar amounts.
Use the strike, terminal price, and premium to determine payoff and profit for one underlying unit. Apply the option’s exercise payoff before subtracting or adding the premium.
Multiply the per-unit amount by the contract multiplier \(M\). The result is a dollar amount per contract.
Multiply the per-contract amount by the number of contracts. For a short position, use a negative signed count or reverse the buyer’s cash-flow signs.
Confirm that total profit equals total signed expiration payoff minus the net premium outflow, using the same units throughout.
Worked Examples
Worked Example: Several Long Call Contracts
An investor buys 4 call contracts. Each contract represents 100 shares. The strike is $50, the premium is $2.40 per share, and the terminal share price is $56.50. Find the total expiration payoff, initial premium paid, and profit. Ignore interest on the premium.
Method. Find payoff and profit per share first, then scale by \(100\) shares per contract and \(4\) contracts. This avoids treating the quoted per-share premium as if it were already the premium for all contracts.
The call payoff per share is \( \max(56.50-50,0)=\max(6.50,0)=\$6.50 \). The profit per share is \(6.50-2.40=\$4.10\).
The position covers \(4\times100=400\) shares. Therefore, total expiration payoff is \(400\times\$6.50=\$2,600.00\). The initial premium paid is \(400\times\$2.40=\$960.00\). Total profit is \(400\times\$4.10=\$1,640.00\).
Check. Total payoff less initial premium is \(\$2,600.00-\$960.00=\$1,640.00\), matching the scaled per-share profit. The common mistake avoided is multiplying by the contract count but forgetting the 100-share multiplier.
Worked Example: Short Put Contracts With a Nonstandard Multiplier
An investor writes 3 put contracts, each representing 50 units. The strike is $42, the premium received is $1.80 per unit, and the terminal underlying price is $39.50. Find the total signed expiration payoff, premium received, and profit. Ignore interest.
Method. Calculate the buyer’s put payoff per unit, then reverse the payoff sign for the writer. The writer receives the premium but is responsible for the exercise payoff.
The put payoff to its buyer per unit is \(\max(42-39.50,0)=\$2.50\). Thus, the writer’s signed payoff per unit is \(-\$2.50\). The writer’s profit per unit is premium received less the payoff: \(1.80-2.50=-\$0.70\).
The position represents \(3\times50=150\) units. The writer’s total signed expiration payoff is \(150\times(-\$2.50)=-\$375.00\). The premium received is \(150\times\$1.80=\$270.00\). The total profit is \(150\times(-\$0.70)=-\$105.00\).
Check. Premium received plus signed payoff is \(\$270.00-\$375.00=-\$105.00\). The loss is not $0.70 per contract: $0.70 is the loss per underlying unit, and each contract covers 50 units.
Worked Example: Adding Long and Short Call Positions
A trader buys 2 call contracts with strike $40 and premium $3.20 per share, and writes 1 call contract with strike $46 and premium $1.10 per share. All contracts represent 100 shares and have the same expiration. The terminal share price is $49. Find the combined total payoff, net initial premium outflow, and profit. Ignore interest.
Method. Calculate and scale each option position independently, including its own strike and premium, and then add the dollar results. This prevents a short position’s payoff from being incorrectly treated as a premium adjustment to the long call.
For each long call, payoff per share is \(\max(49-40,0)=\$9.00\). Profit per share is \(9.00-3.20=\$5.80\). The two contracts cover \(2\times100=200\) shares, so their total payoff is \(200\times\$9.00=\$1,800.00\), their premium paid is \(200\times\$3.20=\$640.00\), and their profit is \(200\times\$5.80=\$1,160.00\).
For the written call, the buyer’s payoff per share is \(\max(49-46,0)=\$3.00\), so the writer’s signed payoff per share is \(-\$3.00\). The writer’s profit per share is \(1.10-3.00=-\$1.90\). The contract covers 100 shares, giving total signed payoff \(-\$300.00\), premium received \(\$110.00\), and profit \(-\$190.00\).
Adding the positions gives total signed payoff \(\$1,800.00-\$300.00=\$1,500.00\). The net initial premium outflow is \(\$640.00-\$110.00=\$530.00\). Combined profit is \(\$1,160.00-\$190.00=\$970.00\).
Check. Total payoff less net premium outflow is \(\$1,500.00-\$530.00=\$970.00\). The result agrees with the sum of the two separately scaled profits.
Why the Scaling Formula Holds
The multiplier is a scaling factor because identical units of the same option have identical per-unit cash flows. If one unit has payoff \(g(S_T)\), then \(M\) units have payoff \(M g(S_T)\). For \(q\) identical contracts, there are \(qM\) signed units, so the total signed payoff is \(qM g(S_T)\). The same reasoning applies to the premium: the signed net premium outflow is \(qM\pi\).
When \(q<0\), both signed cash flows reverse: the position receives the premium and pays the option payoff. Consequently, the formula works for long and short positions without changing the per-unit buyer’s profit formula. For several different options, add their signed results:
Here each position \(j\) may have its own contract count, multiplier, strike, payoff function, and premium. The sum is valid when the component profits are all measured on the same basis and at the same horizon.
Quoting Units and Common Exam Traps
Always identify how the premium is quoted before multiplying. If the premium is quoted per underlying unit, multiply by the multiplier and contract count. If a problem instead gives a dollar premium per contract, multiply that amount by the number of contracts only; multiplying it by the contract multiplier again would count the underlying units twice.
| Quantity supplied | Conversion to a total position amount | Unit check |
|---|---|---|
| Payoff per underlying unit | Multiply by multiplier and contract count | Dollars per unit × units per contract × contracts |
| Premium per underlying unit | Multiply by multiplier and contract count | Dollars per unit × units per contract × contracts |
| Premium per contract | Multiply by contract count | Dollars per contract × contracts |
| Profit per contract | Multiply by contract count | Dollars per contract × contracts |
A frequent error is to multiply only the payoff by the multiplier but leave the premium at its per-unit amount. For example, in the first worked example, subtracting $2.40 from the total $2,600 payoff would mix a position-level amount with a per-share amount. The correct premium outflow is $960, so the correct profit is $2,600 minus $960, or $1,640.
- Do not confuse contract count with multiplier. Four contracts with a 100-unit multiplier represent 400 units, not 104 units.
- Do not scale only one side of the profit calculation. The premium and payoff must cover the same total number of units.
- Keep the writer’s signs straight. A written option receives premium but has the negative of the buyer’s expiration payoff.
- Do not add per-unit profits from different positions before scaling them. Convert each position to dollars first, then add.
Check Your Understanding
For each problem, use the stated expiration-profit convention and ignore interest unless stated otherwise.
- An investor buys 2 call contracts, each covering 100 units. The strike is $30, the premium is $1.25 per unit, and the terminal price is $33. Find the total profit. (Answer: $350.00)
- An investor writes 4 put contracts, each covering 25 units. The strike is $20, the premium received is $0.80 per unit, and the terminal price is $18.50. Find the writer’s total profit. (Answer: -$70.00)
- An investor buys 3 call contracts, each covering 50 units. The strike is $72, the premium is $2.40 per unit, and the terminal price is $70. Find the total payoff and total profit. (Answer: payoff $0.00; profit -$360.00)
- A position consists of 1 long call and 2 written calls, all with strike $25 and multiplier 10. The long call premium is $1.60 per unit, each written call premium is $1.10 per unit, and the terminal price is $29. Find the combined profit. (Answer: -$34.00)
- An investor buys 2 put contracts with strike $55, multiplier 100, and premium $2.30 per unit, and writes 3 put contracts with strike $50, multiplier 50, and premium $1.20 per unit. The common terminal price is $48. Find the combined total payoff and profit. (Answer: payoff $1,100.00; profit $820.00)