Comparing the Terminal Price With the Strike
At expiration, a call gives its holder the right to buy the underlying asset for the strike price, \(K\). The key comparison is between \(S_T\), the underlying asset’s price at expiration, and \(K\). That comparison determines whether exercising the call has positive value. It does not depend on the premium originally paid for the option.
This tutorial treats prices at expiration and measures the call’s exercise value per unit of the underlying asset. Unless stated otherwise, assume that the contract allows exercise at expiration, the quoted terminal price is the price available for valuing the asset, and there are no transaction costs, taxes, or other exercise frictions. Because the terminal price is given, no probability distribution or interest-rate assumption is needed.
“In the money” means that exercising produces a positive difference between the asset’s market value and the amount paid to acquire it. “Out of the money” means that buying at the strike would cost more than buying at the market price. At the money, the two prices are equal, so exercise produces zero value before any costs.
Moneyness is a description of the relationship between the asset price and the strike. It is not a measure of the option holder’s overall gain or loss. In particular, a call can be in the money while the holder’s total investment has not been recovered. The premium is not included in either the moneyness classification or the expiration exercise value.
Exercise Value and the Exercise Region
Suppose the holder has one call on one unit of an asset. If the holder exercises at expiration, the holder pays \(K\) and receives an asset worth \(S_T\). The net value of that exchange is \(S_T-K\). If this amount is negative, the holder can decline to exercise instead. Therefore, the holder chooses between exercising for \(S_T-K\) and not exercising for zero.
Here \(x^+=\max(x,0)\) denotes the positive part of \(x\). The corresponding regions are:
| Terminal-price condition | Moneyness | Exercise value per unit | Exercise decision at expiration |
|---|---|---|---|
| \(S_T<K\) | Out of the money | 0 | Do not exercise |
| \(S_T=K\) | At the money | 0 | Exercise has no positive value; the holder is indifferent before costs |
| \(S_T>K\) | In the money | \(S_T-K\) | Exercise has positive value |
The strict exercise region—where exercise gives a positive value—is \(S_T>K\). At \(S_T=K\), exercising and not exercising both produce zero value in this idealized model. A holder may describe the call as at the money there, but equality is not a case of positive exercise value.
This result is for expiration. Before expiration, comparing the current asset price with \(K\) can describe current moneyness, but the difference between those prices is not generally the entire market value of the option. An option with time remaining may have value even when immediate exercise would produce zero. The expiration formula applies when the terminal price has been reached and no time remains.
Why the Positive-Part Formula Holds
The formula follows directly from the holder’s two available choices at expiration. The holder can exercise and obtain the asset for \(K\), or let the option expire without exercising. Under the stated assumptions, the first choice has net value \(S_T-K\), while the second has value zero. The holder chooses the larger amount:
If \(S_T<K\), then \(S_T-K<0\), so zero is larger and the holder does not exercise. If \(S_T=K\), both choices yield zero. If \(S_T>K\), then \(S_T-K>0\), so exercising is better than letting the call expire. This also establishes the moneyness classifications: the call is out of the money below the strike, at the money at the strike, and in the money above it.
Worked Examples
Worked Example: Classifying Three Terminal Prices
A hypothetical call has a strike of \(K=\$72\). At expiration, consider three possible terminal prices: \(\$68\), \(\$72\), and \(\$89\). Classify the call in each case and calculate its exercise value per unit. Assume one unit, no fees, and that each terminal price is given deterministically; no probability distribution or interest rate is needed.
Method. Compare each terminal price with \(\$72\), then apply \(\max(S_T-K,0)\).
Terminal price \(\$68\). Since \(\$68<\$72\), the call is out of the money. Exercising would mean paying \(\$72\) for an asset worth \(\$68\), so:
Terminal price \(\$72\). Since \(\$72=K\), the call is at the money. Exercising exchanges \(\$72\) for an asset worth \(\$72\):
Terminal price \(\$89\). Since \(\$89>\$72\), the call is in the money. The value of exercising is:
Check. Only the terminal price above the strike produces a positive exercise value, and that value is exactly \(\$17\) per unit. A common mistake is to report \(-\$4\) as the value when \(S_T=\$68\); the holder instead declines to exercise, so the exercise value cannot be negative.
Worked Example: A Premium Does Not Change Moneyness
A hypothetical investor holds one call with strike \(K=\$125\). The investor paid a premium of \(\$9.50\). At expiration, the underlying asset is worth \(\$131.25\). Determine the call’s moneyness and its exercise value per unit. Assume no fees and a deterministically given terminal price; no probability distribution or interest rate is required.
Method. Use the terminal price and strike for moneyness and exercise value. Keep the premium separate because it is not part of the expiration exercise-value formula.
First compare the prices:
Therefore, the call is in the money. Next calculate the amount by which the asset’s value exceeds the strike:
The exercise value is:
Check. Exercising requires paying \(\$125\) for an asset worth \(\$131.25\), a difference of \(\$6.25\). The \(\$9.50\) premium does not change the classification or this exercise value. A common mistake is to subtract the premium and call \(\$3.25\) the exercise value. That mixes the option’s expiration exercise value with a separate measure involving the purchase cost.
Worked Example: Exercise Value for a Multi-Unit Purchase
A hypothetical call gives its holder the right to buy 100 units of a commodity for \(K=\$96\) per unit at expiration. The commodity’s terminal price is \(S_T=\$100.25\) per unit. Determine the moneyness, the exercise value per unit, the total amount paid to exercise, and the total market value of the units received. Assume physical delivery, no fees, and a deterministically given terminal price; no probability distribution or interest rate is needed.
Method. Classify the call using the per-unit prices, find the per-unit exercise value, and then multiply the relevant per-unit amounts by 100.
First, compare terminal price and strike:
The call is in the money. Its exercise value per unit is:
Across 100 units, the total exercise value is:
The holder pays the strike for each of the 100 units:
The 100 units have a market value at expiration of:
Subtracting the exercise payment from the market value checks the total exercise value:
Check. The total from the per-unit calculation matches the value from comparing the total market value with the total strike payment. A common mistake is to report \(\$4.25\) as the value for all 100 units; it is the value per unit, so the total is \(\$425\).
Common Mistakes and Exam Checks
The most reliable approach is to classify first and calculate second. Write down \(S_T\) and \(K\), compare them, and only then evaluate the positive-part formula. This order helps prevent sign errors and keeps the boundary case clear.
- Subtracting in the wrong order. A call’s exercise value uses \(S_T-K\), not \(K-S_T\). The holder pays the strike and receives an asset worth the terminal price.
- Reporting a negative value. If \(S_T<K\), the calculation \(S_T-K\) is negative, but the call’s exercise value is zero because the holder can decline to exercise.
- Treating equality as in the money. If \(S_T=K\), the call is at the money and its exercise value is zero. Positive exercise value requires the strict inequality \(S_T>K\).
- Subtracting the premium from exercise value. The premium paid to acquire the option is a separate cash flow. It is not used to classify the call or calculate its exercise value at expiration.
- Confusing per-unit and total values. For a contract covering multiple units, first calculate the per-unit exercise value and then multiply by the number of units.
These distinctions matter whenever an exam problem asks specifically about moneyness or exercise at expiration. The terminal price and strike settle the question; a premium, a prior price, or the holder’s overall investment result does not replace that comparison. This material is aligned with the SOA Exam FM (Financial Mathematics) syllabus.
Check Your Understanding
For each problem, classify the call at expiration and calculate the requested exercise value. Assume no fees and use the stated terminal prices as given.
- A call has strike \(K=\$54\) and terminal price \(S_T=\$51.80\). State its moneyness and exercise value per unit. (Answer: out of the money; \(\$0\) per unit)
- A call has strike \(K=\$83.40\) and terminal price \(S_T=\$83.40\). State its moneyness and explain whether exercise has positive value. (Answer: at the money; no positive exercise value, \(\$0\) per unit)
- A call has strike \(K=\$47\), terminal price \(S_T=\$52.60\), and premium \(\$7\). Find its moneyness and expiration exercise value per unit; do not subtract the premium. (Answer: in the money; \(\$5.60\) per unit)
- A call covers 25 units, has strike \(K=\$110\) per unit, and expires when the underlying is worth \(\$108.50\) per unit. Find the total exercise value if the holder follows the optimal exercise decision. (Answer: out of the money; \(\$0\) total exercise value)
- A call covers 40 units at a strike of \(\$26.75\) per unit. At expiration, the underlying is worth \(\$29.20\) per unit. Find the call’s moneyness, exercise value per unit, total strike payment, total market value received, and total exercise value. (Answer: in the money; \(\$2.45\) per unit; \(\$1{,}070\) strike payment; \(\$1{,}168\) market value; \(\$98\) total exercise value)