Why a Call’s Expiration Payoff Is a Choice
A call gives its holder the right, but not the obligation, to buy an underlying asset for a specified amount called the strike price. At expiration, the holder compares the value of exercising that right with the value of not exercising. The holder can choose the better alternative. This choice explains both the call payoff formula and why the payoff cannot be negative.
The calculation is about the option’s value at expiration, not the amount originally paid to acquire it. If exercising provides a positive net value, the holder exercises; if it does not, the holder lets the option expire. At the strike price, the holder is indifferent: either choice gives a zero payoff.
This definition assumes the holder may choose whether to exercise, and that exercising one call exchanges the strike price \(K\) for one unit of the asset. There are no transaction costs, taxes, or restrictions that alter the exercise comparison. A cash-settled call can instead pay the same positive difference directly. In either case, the formula gives the payoff at expiration per unit; multiply by the number of units covered to get the total.
Deriving the Payoff from the Exercise Decision
Imagine the two available actions at expiration. If the holder does not exercise, the option expires and produces no payoff. If the holder exercises, the holder pays \(K\) and receives an asset worth \(S_T\). The net value produced by that exchange is therefore \(S_T-K\). Because exercise is optional, the holder selects whichever action produces the larger value.
The result follows directly from the choice, not from an assumption that every call must be exercised. When \(S_T-K\) is positive, exercise produces more than walking away. When \(S_T-K\) is negative, exercise would give the holder an asset worth less than the amount paid, so the holder chooses the zero-payoff alternative. When the two amounts are equal, either action yields zero.
Use the value \(S_T\) at the option’s expiration, not its value when the contract was purchased.
Subtract the strike price from the expiration value: \(S_T-K\). This is the net value of paying the strike and receiving the asset.
Walking away gives zero. The holder’s payoff is the larger of zero and the exercise value.
If one call covers \(q\) units and there are \(n\) identical calls, multiply the per-unit payoff by \(nq\).
No interest-rate calculation or mortality assumption is needed for this expiration comparison. The values are measured at the same time, \(T\), so the strike is not discounted within the payoff formula. A probability distribution for \(S_T\) is also unnecessary when the terminal value is given; if the terminal value is uncertain, the formula still determines the payoff for each realized value.
Worked Example: Exercising a Call on a Single Asset
A hypothetical call gives its holder the right to buy one unit of an asset for \(K=\$83\) at expiration. The asset is worth \(S_T=\$91\) at that time. Determine whether exercising is preferable and calculate the expiration payoff.
Method and assumptions. Compare the exercise value with the zero payoff from not exercising. The contract covers one unit. The stated terminal value is treated as certain for this calculation; no interest rate, mortality assumption, or probability distribution is used.
Exercise value. Paying \(\$83\) for an asset worth \(\$91\) gives: $$ S_T-K=\$91-\$83=\$8. $$
Compare the alternatives. Exercising gives \(\$8\), while not exercising gives \(\$0\). Since \(\$8>\$0\), the holder exercises.
Payoff. For one unit, the expiration payoff is: $$ \max(\$91-\$83,0)=\max(\$8,0)=\$8. $$ The total payoff is \(\$8\), because the call covers one unit. This is the expiration payoff only; it does not subtract any amount paid earlier to acquire the option.
Check. The holder pays \(\$83\) and receives an asset worth \(\$91\), leaving a net value of \(\$8\), which agrees with the formula. The common mistake avoided here is reporting the asset’s full \(\$91\) value as the call payoff rather than subtracting the strike.
Reading the Formula at and Around the Strike
The strike price is the decision boundary because it is the value the holder must pay to receive the asset. The same formula handles all terminal values:
| Expiration value | Exercise comparison | Call payoff per unit |
|---|---|---|
| \(S_T>K\) | \(S_T-K>0\), so exercise gives more than walking away | \(S_T-K\) |
| \(S_T=K\) | \(S_T-K=0\), so exercise and walking away are equal | \(0\) |
| \(S_T<K\) | \(S_T-K<0\), so walking away gives more | \(0\) |
The payoff is never below zero because zero is always available by declining to exercise. This is a statement about the holder’s payoff at expiration under the stated assumptions. It does not mean the call was free to acquire, nor does it say that every amount associated with owning the option must be nonnegative. The premium is a separate cash flow and is not part of this exercise-versus-expiration calculation.
For contracts covering multiple units, calculate the per-unit payoff first and then apply the contract size. For example, if one call covers 25 units, its total payoff is \(25(S_T-K)^+\). If there are four identical calls, the total is \(4(25)(S_T-K)^+\). Applying the positive part before multiplying is equivalent to multiplying the per-unit payoff afterward, because the number of units is nonnegative.
Worked Example: Calls That Should Be Left Unexercised
A hypothetical investor holds three identical calls. Each call covers 40 units of an asset and has strike price \(K=\$118\). At expiration, each unit is worth \(S_T=\$114\). Find the payoff per unit, per call, and for all three calls.
Method and assumptions. Calculate the exercise value, compare it with zero, and then scale by the number of units. The stipulated terminal value is used directly; no interest rate, mortality assumption, or probability distribution is needed.
Exercise value per unit. $$ S_T-K=\$114-\$118=-\$4. $$ Exercising would mean paying \(\$118\) for an asset worth \(\$114\), a net value of \(-\$4\) per unit.
Choose the better alternative. The holder can instead decline to exercise for a payoff of \(\$0\). Since \(\$0>-\$4\), the holder leaves all three calls unexercised.
Scale the payoff. The payoff per unit is: $$ \max(-\$4,0)=\$0. $$ Each call covers 40 units, so the payoff per call is \(40(\$0)=\$0\). There are three calls, so the total payoff is \(3(\$0)=\$0\).
Check. The total number of covered units is \(3(40)=120\). Multiplying the negative exercise value by 120 would give \(-\$480\), but that is not the payoff: the holder is not forced to exercise. Applying the positive-part operation gives \(120\max(-\$4,0)=\$0\). The common mistake avoided is treating an unfavorable exercise as a compulsory purchase.
Contract Size and Settlement Form
The exercise comparison is unchanged by the number of units or by whether the contract settles through delivery or cash. Those details affect how the total is expressed, not the underlying decision. With \(n\) identical calls, each covering \(q\) units, the combined expiration payoff is:
For physical settlement, the holder who exercises pays the strike for each unit and receives the underlying asset. For cash settlement, the contract pays the positive difference between the asset value and strike for each covered unit. These descriptions give the same expiration payoff when the terms use the same underlying value and unit multiplier. At \(S_T=K\), the difference is zero under either form.
Worked Example: Cash-Settled Calls with a Unit Multiplier
A hypothetical cash-settled call is written on an index. Each contract pays the positive difference between the expiration index level and \(K=2{,}400\), multiplied by 50 dollars per index point. An investor holds two contracts, and the index finishes at \(S_T=2{,}438\). Calculate the payoff for one contract and for the position.
Method and assumptions. Find the positive index-point difference, apply the stated dollar multiplier, and then multiply by the number of contracts. The terminal index level is stipulated; no interest rate, mortality assumption, or probability distribution is used.
Exercise value in index points. $$ S_T-K=2{,}438-2{,}400=38\text{ points}. $$ The difference is positive, so the holder receives the cash-settlement amount rather than zero.
Payoff per contract. Each index point represents \(\$50\), so: $$ 38(\$50)=\$1{,}900. $$
Total payoff. Two contracts produce: $$ 2(\$1{,}900)=\$3{,}800. $$
Check. The position covers \(2(50)=100\) dollars per index point. Multiplying the 38-point positive difference by \(\$100\) per point gives \(38(\$100)=\$3{,}800\), matching the contract-by-contract calculation. The common mistake avoided is applying the index-point difference without the contract’s dollar multiplier.
Common Mistakes and a Reliable Check
- Reporting a negative payoff. A negative value of \(S_T-K\) is the result of exercising, not the holder’s chosen payoff. Compare it with zero and take the larger amount.
- Forgetting the strike. The asset’s terminal value is not the call payoff. Subtract \(K\) to account for the amount paid to obtain the asset.
- Multiplying too early and losing the choice. For a contract covering several units, use the positive part of the per-unit difference before scaling. Do not treat a negative exercise value as a required payment.
- Confusing payoff with profit. The expiration payoff comes from the exercise decision. An earlier premium is a separate cash flow and is not subtracted in this formula.
- Using the wrong time value. Use \(S_T\), the underlying’s value at expiration. The call’s payoff is determined at that date; do not substitute its value at an earlier date.
A useful final check is to ask: “Could the holder choose not to exercise?” If the answer is yes, the expiration payoff cannot be less than zero. Then check that any positive payoff equals the terminal asset value less the strike, multiplied by the number of covered units.
Check Your Understanding
For each problem, compare exercising with letting the call expire, then apply any contract multiplier.
- A call on one unit has strike \(K=\$61\), and the underlying is worth \(S_T=\$57\) at expiration. What is the payoff per unit? (Answer: \(\$0\).)
- A call with strike \(K=\$125\) covers 30 units. The underlying finishes at \(S_T=\$143\). What is the total expiration payoff? (Answer: \(\$540\).)
- An investor holds five calls, each covering 100 units, with strike \(K=\$72\). The underlying finishes at \(S_T=\$69.40\). What is the total payoff, and why is the negative exercise value not charged to the holder? (Answer: \(\$0\); the holder declines to exercise, so the negative exercise value is replaced by zero.)
- A hypothetical cash-settled call has strike \(K=4{,}100\), a multiplier of \(\$20\) per index point, and expires when the index is \(4{,}147\). If three contracts are held, what is the total payoff? (Answer: \(\$2{,}820\).)
- Two identical calls each cover 40 units and have strike \(K=\$97\). The underlying is worth \(S_T=\$103.25\) at expiration. Find the exercise value per unit, the payoff per call, and the total payoff. (Answer: \(\$6.25\) per unit, \(\$250\) per call, and \(\$500\) total.)