Deriving Put Payoff from the Exercise Decision

Introduction

A put option gives its holder a choice at expiration: sell the underlying asset for the strike price, or decline to exercise. The payoff follows from comparing those two alternatives. If the market price is below the strike, selling at the strike is preferable; if the market price is above the strike, the holder can let the option expire.

The key point is that the holder has a right, not an obligation. The exercise decision therefore produces a payoff that is never negative. This tutorial derives that result directly from the expiration decision and shows how to apply it per unit and across contracts. It focuses on payoff at expiration, not the option’s premium or profit.

Definition: A put with strike price $K$ gives its holder the right, but not the obligation, to sell one unit of the underlying asset for $K$ at expiration. If the underlying asset’s market price at expiration is $S_T$, the holder’s payoff per unit is the amount received from the exercise decision, before accounting for any premium or other costs.

Comparing Exercise With Expiration

Assume a standard put that can be exercised at expiration, with no transaction costs, taxes, or settlement frictions. Under physical settlement, exercising means delivering one unit of the underlying asset and receiving $K$. The market value of that unit at expiration is $S_T$. Thus, relative to selling the asset at its market value, exercising has value $K-S_T$ per unit. If the holder does not exercise, the option expires and its payoff is zero.

The holder chooses the alternative with the greater payoff. When $S_T<K$, exercise gives a positive amount $K-S_T$, so the holder exercises. When $S_T>K$, exercise would give a negative amount, so the holder declines and receives zero. When $S_T=K$, exercise and expiration both give zero.

\text{Put payoff per unit at expiration} = \begin{cases} K-S_T, & S_T<K,\\ 0, & S_T\ge K \end{cases} = \max(K-S_T,0).

For a contract covering $q$ units, the total payoff is $q\max(K-S_T,0)$. If the holder owns $n$ identical contracts, each covering $q$ units, the total payoff is $nq\max(K-S_T,0)$. These formulas apply to the expiration payoff; they do not subtract the premium paid to acquire the option.

Why the Payoff Cannot Be Negative

The positive-part formula is a direct expression of the exercise choice. The holder compares exercising, which yields $K-S_T$ relative to a market sale, with not exercising, which yields zero. The better choice is the larger of those two amounts. Therefore, the payoff is $\max(K-S_T,0)$, not simply $K-S_T$ in every price scenario.

Find the exercise value.
For each unit, compare the strike proceeds $K$ with the market value $S_T$. The advantage of exercise over a market sale is $K-S_T$.

Compare with letting the option expire.
Not exercising gives a payoff of zero. The holder chooses between $K-S_T$ and zero.

Scale to the position.
Multiply the per-unit payoff by the number of units covered by the contracts, while retaining zero whenever exercise is not advantageous.

This reasoning also explains the boundary cases. At $S_T=K$, the holder receives $K$ by delivering an asset worth $K$, so exercise produces no economic advantage over selling at the market price. The option’s payoff is zero whether the holder exercises or allows it to expire. If $S_T>K$, exercise would sacrifice market value: the holder would exchange an asset worth more than $K$ for only $K$. Since exercise is optional, the holder avoids that outcome by not exercising.

Worked Example: A Put Below Its Strike

A hypothetical put has a strike price of $K=\$52.00$ per unit. At expiration, the underlying asset is worth $S_T=\$46.80$ per unit. The holder owns three contracts, each covering 100 units. Find the payoff per unit and the total payoff.

Method. Compare the exercise proceeds with the market value at expiration, then multiply the positive per-unit payoff by the total units covered.

Step 1: Find the exercise advantage per unit. Delivering a unit through exercise produces $\$52.00$, while selling it at the market price would produce $\$46.80$. Therefore,

K-S_T=\$52.00-\$46.80=\$5.20.

Because $\$5.20>0$, exercise is preferable to letting the put expire. The payoff is $\$5.20$ per unit.

Step 2: Count the units covered. Three contracts cover $3\times100=300$ units.

Step 3: Calculate total payoff.

300\times \$5.20=\$1{,}560.00.

Check. The holder receives $300\times\$52.00=\$15{,}600.00$ through exercise. The 300 units have a market value of $300\times\$46.80=\$14{,}040.00$. The difference is $\$15{,}600.00-\$14{,}040.00=\$1{,}560.00$, matching the calculated payoff. The common mistake avoided here is to report the per-unit payoff without multiplying by the units covered.

Worked Example: A Put Above Its Strike

A hypothetical put has a strike price of $K=\$87.50$ per unit. At expiration, the underlying asset is worth $S_T=\$94.25$ per unit. The holder owns four contracts, each covering 25 units. Determine the total expiration payoff and explain what happens if the holder mistakenly exercises.

Method. Evaluate the signed exercise value, compare it with the zero payoff from expiration, and choose the better alternative.

Step 1: Calculate the signed exercise value per unit.

K-S_T=\$87.50-\$94.25=-\$6.75.

This negative amount means that exercise would be worse than selling the asset at its market price. The holder instead lets the put expire, receiving a payoff of $\$0$ per unit.

Step 2: Find the total units. The contracts cover $4\times25=100$ units. The total payoff is $100\times\$0=\$0.00$.

Step 3: Check the consequence of exercising by mistake. Exercising would give up $\$6.75$ per unit compared with a market sale. Across 100 units, that disadvantage would be $100\times\$6.75=\$675.00$. The right to decline exercise prevents that negative amount from becoming the option’s payoff.

Check. Since $S_T=\$94.25$ exceeds $K=\$87.50$, the positive-part expression gives $\max(\$87.50-\$94.25,0)=\max(-\$6.75,0)=\$0$ per unit, as required.

Worked Example: Payoff at the Strike Boundary

A hypothetical put has a strike price of $K=\$125.00$ per unit, and the underlying asset is worth $S_T=\$125.00$ at expiration. The holder owns two contracts, each covering 50 units. Calculate the total payoff and compare exercise with expiration.

Method. Evaluate both available choices at the boundary $S_T=K$, then scale the resulting payoff.

Step 1: Compare exercise and market value per unit. Exercise provides $\$125.00$ in exchange for an asset worth $\$125.00$. The exercise advantage is

K-S_T=\$125.00-\$125.00=\$0.00.

Letting the put expire also gives a payoff of $\$0.00$ per unit. The holder is indifferent between exercising and not exercising in terms of expiration payoff.

Step 2: Calculate the number of units. The position covers $2\times50=100$ units.

Step 3: Calculate total payoff. Each unit has a zero payoff, so $100\times\$0.00=\$0.00$.

Check. The positive-part expression gives $\max(\$125.00-\$125.00,0)=\$0.00$ per unit. The result is unchanged by multiplying by 100. A common mistake is to treat exercise at the strike as producing a positive payoff of $K$; the asset delivered in exercise also has value $K$, so the payoff advantage is zero.

Common Mistake: Using the Signed Difference in Every Case

A tempting but incorrect rule is to use $K-S_T$ as the payoff for every expiration price. That expression measures the advantage of exercising over a market sale, but it is not itself the holder’s payoff when it is negative. For example, if $K=\$80$ and $S_T=\$86$, the incorrect calculation gives

K-S_T=\$80-\$86=-\$6.

The correction is to include the holder’s alternative of not exercising. That alternative has a payoff of zero, so the holder chooses $\max(-\$6,0)=\$0$. More generally, the negative branch of $K-S_T$ indicates that exercise is unattractive; it does not create a negative expiration payoff for the put holder.

A second exam trap is to confuse payoff with profit. The payoff derived here is determined by $K$, $S_T$, and the number of units. A premium paid earlier is not part of this expiration payoff calculation. Keep the exercise decision and any earlier cash flow distinct.

Key takeaway: At expiration, exercise a put only when selling at the strike is at least as valuable as selling at the market price. The payoff per unit is $\max(K-S_T,0)$; the total payoff is this amount multiplied by the units covered. This exercise-decision derivation is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, first compare exercise with letting the put expire, then account for the units covered.

A hypothetical put has $K=\$64.00$, and the underlying asset is worth $\$58.75$ at expiration. The holder owns three contracts covering 40 units each. Find the payoff per unit and total payoff. (Answer: $\$5.25$ per unit; $\$630.00$ total.)
A hypothetical put has $K=\$105.00$, and the underlying asset is worth $\$111.40$ at expiration. The holder owns two contracts covering 100 units each. Find the total payoff and the total disadvantage if the holder exercises instead of selling at the market price. (Answer: $\$0.00$ payoff; $\$1{,}280.00$ disadvantage from exercising.)
A hypothetical cash-settled put has $K=\$72.50$, and the underlying asset is worth $\$69.20$ at expiration. Its settlement amount is the same as the economic advantage of exercising a physically settled put. The holder owns five contracts covering 25 units each. Find the total payoff. (Answer: $\$412.50$.)
A hypothetical put has $K=\$90.00$, and the underlying asset is worth $\$90.00$ at expiration. The holder owns seven contracts covering 10 units each. State the exercise value per unit and total payoff. (Answer: $\$0.00$ per unit; $\$0.00$ total.)
A hypothetical put has $K=\$150.00$, and the underlying asset is worth $\$143.60$ at expiration. The holder owns four contracts covering 75 units each. Find the total payoff and verify it by comparing aggregate strike proceeds with the market value of the units. (Answer: $\$1{,}920.00$; strike proceeds $\$45{,}000.00$ less market value $\$43{,}080.00$.)

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