Terminal-Value Notation and Option Position Signs

Introduction

An option payoff diagram is meaningful only when its perspective is clear. A line above zero might represent money received by the option holder, the writer, or a portfolio containing both; those are not interchangeable interpretations. It is also important to distinguish the cash flow at expiration from the profit after accounting for the premium paid or received earlier.

This tutorial establishes the basic language and sign conventions for options. The calculations use a cash-equivalent payoff at expiration: the amount the option is worth to its holder upon exercise, before including the premium. The next step, terminal profit, includes the premium carried forward to expiration under an explicitly stated interest assumption.

Definition: The expiration price, written $S_T$, is the market price of the underlying asset at the option’s expiration time $T$. The strike price, written $K$, is the price specified in the contract for buying or selling that asset. The premium is the amount the option holder pays the writer to enter the contract; unless stated otherwise, it is quoted per unit of the underlying asset and paid at time 0.

Holder, Writer, and Contract Position

A long option position means that the investor bought and holds the option. The long position is the holder of the contractual right. A short option position means that the investor wrote or sold the option and is the writer, who has the corresponding obligation if the holder exercises. “Long” and “short” identify the position in the option contract, not necessarily a position in the underlying asset.

A call gives its holder the right to buy the underlying asset at $K$. At expiration, exercising is economically beneficial when the market price $S_T$ is greater than $K$: the holder can pay $K$ for an asset worth $S_T$. A put gives its holder the right to sell the underlying asset at $K$. Exercising is economically beneficial when $S_T$ is less than $K$: the holder can sell for $K$ an asset worth $S_T$. If exercise would not be beneficial, the holder can let the option expire.

The standard option payoff at expiration is the holder’s cash-equivalent value from the exercise decision. It excludes the premium, because that payment occurred at time 0. The writer’s payoff is the negative of the holder’s payoff for the same contract: any exercise value received by one side is owed by the other side. This zero-sum relationship concerns the option contract’s expiration payoff; it does not mean that the two investors’ total profits must have equal magnitudes if they have other cash flows.

Position	At time 0	At expiration
Long option (holder)	Pays the premium	Receives the option payoff, if any
Short option (writer)	Receives the premium	Owes the holder’s option payoff, if any

Option premiums are commonly quoted per underlying unit, while a contract may cover many units. If there are $n$ contracts and each covers $M$ units, the position covers $Q=nM$ units. Multiply a per-unit payoff or premium by $Q$ to obtain the total cash amount. The contract multiplier must be taken from the problem statement; it should not be silently assumed to be 100.

Convention: A payoff diagram plots the expiration payoff of the specified position against $S_T$, unless its vertical axis is explicitly labelled as profit. A long position’s payoff is measured from the holder’s perspective; a written position’s payoff is measured from the writer’s perspective. Premiums are not part of payoff, but they are part of profit.

From Expiration Payoff to Terminal Profit

A premium paid at time 0 and a payoff received at time $T$ occur at different times. To compare them as a terminal profit, carry the premium to expiration using the stated accumulation assumption. In the examples below, the effective annual interest rate $i$ is deterministic, applies to the full period, and allows funds to be invested or borrowed at that rate. There are no taxes, transaction costs, or other cash flows. For a term of $T$ years, one dollar at time 0 accumulates to $(1+i)^T$ dollars at expiration.

\text{terminal profit} = \text{expiration payoff} - \text{initial net premium accumulated to }T

For a long position, the initial net premium is an amount paid, so its accumulated value is subtracted from the expiration payoff. For a writer, the premium is received, so it contributes positively to terminal profit; the writer’s expiration payoff is already negative when the holder receives a positive payoff. The same convention applies to a position containing multiple options: first find the net initial premium, then accumulate that amount. A net receipt is a negative net premium, so subtracting its accumulated value adds it to profit.

Why the Sign Convention Holds

The writer and holder have opposite expiration cash flows because the option contract transfers value between them. The premium is a separate time-0 transfer in the opposite direction: the holder pays it and the writer receives it. Carrying that initial transfer to time $T$ puts both cash-flow components on the same date.

\begin{aligned} \text{holder terminal profit} &= \text{holder expiration payoff} - \text{premium}(1+i)^T,\\ \text{writer terminal profit} &= -\text{holder expiration payoff} + \text{premium}(1+i)^T\\ &= -\text{holder terminal profit}. \end{aligned}

The final equality holds for the two sides of the same contract under the same carrying-rate convention and with no additional cash flows. It provides a useful check: if the computed terminal profits for the holder and writer of one option do not sum to zero, a sign or cash-flow timing error has occurred.

Worked Example: Long Call Payoff and Terminal Profit

An investor buys 2 call contracts, each covering 40 units. The strike is $K=\$75$, and the premium is $\$4.20$ per unit, paid at time 0. At expiration one year later, $S_T=\$82$. Assume a deterministic effective annual rate of $i=5\%$, with no other cash flows. Find the holder’s total expiration payoff and terminal profit, and check them from the writer’s perspective.

Method. First determine the exercise value per unit, then apply the total number of units. Keep the expiration payoff separate from the accumulated premium.

Step 1: Total units. The contracts cover $Q=2(40)=80$ units.

Step 2: Expiration payoff. Since $S_T=\$82$ exceeds $K=\$75$, the call holder can pay $\$75$ for an asset worth $\$82$. The value is $\$82-\$75=\$7$ per unit. Therefore, the total holder payoff is $80(\$7)=\$560$.

Step 3: Accumulated premium. The initial premium is $80(\$4.20)=\$336$. At 5% for one year, its terminal value is $\$336(1.05)=\$352.80$.

Step 4: Terminal profit. The holder’s terminal profit is $\$560-\$352.80=\$207.20$. The writer’s payoff is $-\$560$, and the writer’s premium receipt accumulates to $\$352.80$, giving writer profit $-\$560+\$352.80=-\$207.20$.

Check and common mistake avoided. The two profits sum to zero, as required for opposite sides of this contract. Subtracting the unaccumulated $\$336$ from the expiration payoff would mix cash flows from different dates; the stated terminal-profit convention requires the $\$352.80$ terminal value.

Worked Example: Written Put With a Contract Multiplier

An investor writes 3 put contracts, each covering 25 units. The strike is $K=\$120$, the premium is $\$5.60$ per unit, and $S_T=\$111$ after one year. Assume a deterministic effective annual rate of $i=4\%$, with no other cash flows. Find the writer’s total expiration payoff and terminal profit.

Method. Calculate the holder’s exercise value first, reverse its sign for the writer’s payoff, and then add the accumulated premium received.

Step 1: Total units. The position covers $Q=3(25)=75$ units.

Step 2: Holder’s exercise value and writer’s payoff. Since $S_T=\$111$ is below $K=\$120$, the put holder can sell each unit for $\$120$ rather than its market value of $\$111$. The holder’s value is $\$120-\$111=\$9$ per unit, or $75(\$9)=\$675$ in total. The writer’s expiration payoff is the opposite: $-\$675$.

Step 3: Premium receipt at expiration value. The writer receives $75(\$5.60)=\$420$ at time 0. Accumulated for one year, this becomes $\$420(1.04)=\$436.80$.

Step 4: Writer’s terminal profit. The writer owes the $\$675$ payoff and has the $\$436.80$ terminal value of the premium receipt. Thus, terminal profit is $-\$675+\$436.80=-\$238.20$. The holder’s terminal profit is $+\$238.20$.

Check and common mistake avoided. The holder’s and writer’s payoffs have opposite signs, and their terminal profits sum to zero. A common error is to report writer profit as $-\$675+\$420=-\$255$. That uses the time-0 premium as though it were already a time-$T$ amount; accumulating it at 4% gives the correct $-\$238.20$.

Reading a Diagram for a Portfolio

A diagram may show a single position or a portfolio of positions. For a portfolio, identify each option’s holder or writer, calculate each position’s payoff using its own strike and number of units, and add the signed payoffs. To convert the portfolio’s payoff diagram to terminal profit, include the portfolio’s net premium with the same sign convention. If a diagram’s vertical axis is not labelled clearly, do not assume whether it shows payoff or profit.

Worked Example: Interpreting a Two-Option Payoff Diagram

An investor buys 1 call contract and 1 put contract on the same asset. Each covers 20 units, and both have strike $K=\$100$. The call premium is $\$6$ per unit and the put premium is $\$4$ per unit. Both premiums are paid at time 0. The term is one year, $i=5\%$ effective annually, and there are no other cash flows. Find the portfolio payoff and terminal profit when $S_T=\$90$, $\$100$, and $\$115$.

Method. Find each contract’s holder payoff at each expiration price, add the payoffs, and subtract the accumulated cost of both premiums. This identifies a payoff diagram as a holder’s measure, rather than confusing it with profit.

Step 1: Initial premium and terminal value. The call costs $20(\$6)=\$120$, and the put costs $20(\$4)=\$80$. The total initial premium is $\$200$, which accumulates to $\$200(1.05)=\$210$ at expiration.

Step 2: When $S_T=\$90$. The call is not worth exercising, so its payoff is $\$0$. The put is worth $\$100-\$90=\$10$ per unit, giving $20(\$10)=\$200$. Portfolio payoff is $\$0+\$200=\$200$; terminal profit is $\$200-\$210=-\$10$.

Step 3: When $S_T=\$100$. Neither option gives an exercise advantage at the strike, so both payoffs are $\$0$. Portfolio payoff is $\$0$, and terminal profit is $\$0-\$210=-\$210$.

Step 4: When $S_T=\$115$. The call is worth $\$115-\$100=\$15$ per unit, so its payoff is $20(\$15)=\$300$. The put is not worth exercising, so its payoff is $\$0$. Portfolio payoff is $\$300$, and terminal profit is $\$300-\$210=\$90$.

Check and common mistake avoided. The portfolio’s payoff values at the three prices are $\$200,\ \$0,\ \$300$, while its profit values are $-\$10,\ -\$210,\ \$90$. Each profit equals the corresponding payoff less the same $\$210$ terminal premium cost. Plotting the payoff values as though they were profits would omit the premium and shift the entire profit diagram upward by $\$210$.

Common Sign and Timing Traps

Using the writer’s perspective for a holder’s diagram. A positive payoff to the holder is a negative payoff to the writer. State the position before assigning signs.
Including premium in payoff. Payoff is the expiration value from the option contract. Profit accounts for the premium as well.
Forgetting the contract multiplier. A quoted amount per unit must be multiplied by the number of contracts and units per contract to find the total cash flow.
Mixing dates. When terminal profit is requested, carry the time-0 premium to expiration using the specified rate and term. Do not add or subtract a time-0 amount directly from a time-$T$ payoff.
Leaving the diagram’s vertical axis ambiguous. A payoff diagram and a profit diagram can have the same shape but different vertical positions because profit includes the premium.

Key takeaway: Identify $S_T$, $K$, the premium, the position, and the total number of units before calculating. A payoff is the signed option value at expiration; terminal profit adds the appropriately accumulated premium cash flow. These conventions are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the position’s perspective and distinguish expiration payoff from terminal profit. Assume no other cash flows.

An investor buys 4 call contracts, each covering 15 units. The strike is $\$40$, the premium is $\$2.80$ per unit, and $S_T=\$46$ after one year. The effective annual rate is 5%. Find the holder’s total expiration payoff and terminal profit. (Answer: payoff $\$360$; terminal profit $\$183.60$)
An investor writes 2 put contracts, each covering 30 units. The strike is $\$50$, the premium is $\$3.25$ per unit, and $S_T=\$44$ after one year. The effective annual rate is 4%. Find the writer’s total expiration payoff and terminal profit. (Answer: payoff $-\$360$; terminal profit $-\$157.20$)
An investor buys 5 call contracts, each covering 12 units. The strike is $\$90$, the premium is $\$7.50$ per unit, and $S_T=\$88$ after one-half year. The effective annual rate is 6%. Find the holder’s expiration payoff and terminal profit, rounding money to the nearest cent. (Answer: payoff $\$0$; terminal profit $-\$463.30$)
An investor writes 1 call contract covering 40 units. The strike is $\$70$, the premium is $\$4.50$ per unit, and $S_T=\$76$ after two years. The effective annual rate is 3%. Find the writer’s expiration payoff and terminal profit, rounding money to the nearest cent. (Answer: payoff $-\$240$; terminal profit $-\$49.04$)
An investor buys 2 put contracts, each covering 50 units, and writes 1 call contract covering 100 units. Both options have strike $\$105$. The put premium is $\$6$ per unit and the call premium received is $\$5.50$ per unit. At $S_T=\$95$ after one year, the effective annual rate is 4%. Find the portfolio’s net expiration payoff and terminal profit. (Answer: payoff $\$1{,}000$; terminal profit $\$948$)

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