Protective Puts As Downside Collars Components

Why Combine a Share With a Put?

A long share position gains value when the share price rises, but it also loses value when the price falls. A long put adds a payoff if the share price at expiration is below the put’s strike price. Together, the share and put create a protective put: the put offsets some or all of the share’s decline below the strike, while the share remains exposed to price increases.

For this analysis, assume one share and one put on that share, with the same expiration date and strike price $K$. Let $S_0$ be the share price when the position is established, $S_T$ the share price at expiration, and $p$ the put premium per share paid initially. Ignore taxes, transaction costs, dividends, and interest unless stated otherwise. The payoff and expiration value below are per share; the examples will state when they use multiple shares or option contracts.

Definition: A protective put is a long position in an underlying share combined with a long put on that share. At expiration, the share is worth $S_T$, and the put’s payoff is $\max(K-S_T,0)$. Thus the combined expiration value per share is $S_T+\max(K-S_T,0)$. The premium $p$ is an initial cost, not part of the put’s expiration payoff.

Expiration Value and the Downside Floor

The put pays $K-S_T$ when $S_T<K$ and pays zero when $S_T\ge K$. Splitting the calculation at the strike gives the protective position’s expiration value:

V_T=S_T+\max(K-S_T,0) = \begin{cases} K, & S_T<K,\\ S_T, & S_T\ge K. \end{cases} = \max(S_T,K).

When the share finishes below the strike, the share and put together are worth $K$ per fully protected share: the share contributes $S_T$, and the put contributes $K-S_T$. When the share finishes at or above the strike, the put expires without value and the share is worth $S_T$. The expiration-value floor is therefore $K$, assuming every share is covered by a matching put.

This floor is a statement about expiration value, not profit. The initial outlay per share is $S_0+p$. Under a simple dollar-profit convention that does not accumulate the initial outlay with interest, the profit is $V_T-(S_0+p)$. Consequently, below the strike, that profit is $K-S_0-p$, which can be negative even though the expiration value cannot fall below $K$.

Key distinction: The put’s expiration payoff is $\max(K-S_T,0)$. The protective position’s expiration value is $S_T+\max(K-S_T,0)$. Subtract the initial share price and put premium only when calculating profit; do not subtract the premium from the put payoff or mistake the profit floor for the value floor.

Why the Floor Holds

The floor follows directly from the put’s contractual payoff. If $S_T<K$, then $K-S_T$ is positive, so the combined value is

$$ S_T+(K-S_T)=K. $$

If $S_T\ge K$, the put payoff is zero and the combined value is $S_T$, which is at least $K$. Thus in either region $V_T\ge K$, and below the strike it equals $K$. This argument depends on the put and share matching in quantity, underlying, and expiration. If fewer puts are held than shares, only the covered portion has this floor.

Find the share value.
For $q$ shares, calculate $qS_T$.

Find the put payoff.
For $n$ put contracts, multiplier $m$, and strike $K$, calculate $nm\max(K-S_T,0)$.

Add expiration values.
The position’s total expiration value is the share value plus put payoff.

Subtract the appropriate cost to find profit.
For simple dollar profit, subtract $qS_0+nmp$. If the problem asks for terminal profit after interest, accumulate that initial outlay to expiration first.

Worked Calculations

Worked Example: Value, Floor, and Profit Below the Strike

An investor buys one share for $S_0=\$62$ and one matching put with strike $K=\$55$, paying $p=\$4.25$ per share. At expiration, the share price is $S_T=\$38$. Find the put payoff, protective position’s expiration value, and simple dollar profit. Compare the profit with holding the share alone.

Method. Calculate the put payoff first, add it to the share’s expiration value, then subtract the initial outlay. No interest accumulation is specified, so use simple dollar profit.

Put payoff: Since $S_T<K$, the put pays $K-S_T=\$55-\$38=\$17$ per share.

Expiration value: The share is worth $\$38$, so the position is worth $\$38+\$17=\$55$ per share. This equals the strike-price floor.

Initial outlay and profit: The initial outlay is $S_0+p=\$62+\$4.25=\$66.25$. Therefore, simple dollar profit is $\$55-\$66.25=-\$11.25$ per share.

Comparison with the share alone: The share-alone profit is $\$38-\$62=-\$24$. The protective put’s profit is higher by $-\$11.25-(-\$24)=\$12.75$. Equivalently, the difference is put payoff minus premium: $\$17-\$4.25=\$12.75$. The premium reduces the protective position’s profit relative to its expiration value before premium, but it does not make its profit simply the share-alone profit minus the premium.

Check: The combined value is $\$55$, as $\max(\$38,\$55)=\$55$, and subtracting the $\$66.25$ outlay gives the stated loss of $\$11.25$.

Worked Example: Partial Coverage With Put Contracts

An investor buys 250 shares at $S_0=\$48$ and two put contracts, each covering 100 shares, with $K=\$44$. The premium is $p=\$2.10$ per share covered. At expiration, $S_T=\$31$. Find total expiration value and simple dollar profit, and identify which shares have a protected floor.

Method. Count the protected shares from the contract multiplier before calculating the put payoff. Two contracts cover $2(100)=200$ shares, not all 250 shares.

Share value: All 250 shares are worth $250(\$31)=\$7{,}750$.

Put payoff: Each covered share pays $K-S_T=\$44-\$31=\$13$. The two contracts therefore pay $200(\$13)=\$2{,}600$.

Total expiration value: Add the share value and put payoff: $\$7{,}750+\$2{,}600=\$10{,}350$. This also equals $200(\$44)+50(\$31)=\$8{,}800+\$1{,}550=\$10{,}350$: the 200 covered shares are worth the strike, while the 50 uncovered shares remain worth the market price.

Initial outlay and profit: The shares cost $250(\$48)=\$12{,}000$. The puts cost $200(\$2.10)=\$420$. Total initial outlay is $\$12{,}420$, so simple dollar profit is $\$10{,}350-\$12{,}420=-\$2{,}070$.

Check: The 200 covered shares have value $\$8{,}800$, and the uncovered shares have value $\$1{,}550$; together they give the same $\$10{,}350$ expiration value. Only the covered shares have a strike-price floor.

Worked Example: Profit With Interest on the Initial Outlay

An investor buys one share for $S_0=\$100$ and a matching put with $K=\$92$, paying $p=\$3$. The position is held for one year. The effective annual interest rate is $i=4\%$, and $S_T=\$80$. Find the expiration value and the terminal profit after accumulating the initial outlay at the stated rate.

Method. Calculate the expiration value from the share and put payoffs. For terminal profit, compare that value with the future value of the initial cash outlay, using annual compound interest for one year.

Put payoff and expiration value: The put pays $\max(\$92-\$80,0)=\$12$. The share is worth $\$80$, so the position’s expiration value is $\$80+\$12=\$92$.

Initial outlay and its accumulated value: The initial outlay is $\$100+\$3=\$103$. At an effective annual rate of $4\%$ for one year, its accumulated value is $\$103(1.04)=\$107.12$.

Terminal profit: Subtract the accumulated outlay from the expiration value: $\$92-\$107.12=-\$15.12$.

Check: If interest on the outlay were ignored, simple dollar profit would instead be $\$92-\$103=-\$11$. The $\$4.12$ difference is the interest accumulated on the initial $\$103$. The expiration value remains $\$92$ under either profit convention.

Premiums, Comparisons, and an Exam Trap

For one share and one matching put, the simple dollar profit is

\max(S_T,K)-S_0-p.

The premium reduces profit relative to the same position’s expiration value before the premium is accounted for. But comparing the protective put with holding only the share requires including both the put payoff and the premium. Under the simple dollar-profit convention, the difference is

\bigl(S_T+\max(K-S_T,0)-S_0-p\bigr) - (S_T-S_0) = \max(K-S_T,0)-p.

Common mistake: Below the strike, it is tempting to say the protective put’s profit equals the share-alone profit minus the premium. That omits the put payoff. For example, if $S_0=\$62$, $K=\$55$, $p=\$4.25$, and $S_T=\$38$, the share-alone profit is $-\$24$. The incorrect calculation $-\$24-\$4.25=-\$28.25$ is not the protective put’s profit. The correction is to add the put payoff of $\$17$ and subtract the premium: $-\$24+\$17-\$4.25=-\$11.25$. The difference from the share alone is $\$17-\$4.25=\$12.75$, not simply a reduction of $\$4.25$.

A second trap is to call $K$ the profit floor. With simple dollar profit, the floor for a fully covered share is $K-S_0-p$, which may be negative. If the problem instead asks for terminal profit after interest, accumulate $S_0+p$ at the specified effective rate and term before subtracting it from the expiration value. State which convention is being used and do not mix the two.

Check Your Understanding

Use the stated simple dollar-profit convention unless a question specifies interest accumulation. Treat all premiums as amounts per share covered.

An investor pays $S_0=\$74$ for a share and $p=\$2.50$ for a matching put with $K=\$68$. If $S_T=\$50$, find the put payoff, expiration value, and simple dollar profit per share. (Answer: payoff $\$18$; expiration value $\$68$; profit $-\$8.50$ per share.)
A share costs $S_0=\$35$, and a matching put with $K=\$40$ costs $p=\$6$. If $S_T=\$52$, find the expiration value and simple dollar profit. (Answer: expiration value $\$52$; profit $\$11$ per share.)
An investor buys 150 shares at $S_0=\$90$ and one put contract covering 100 shares, with $K=\$85$ and $p=\$4$. If $S_T=\$70$, find total expiration value and simple dollar profit. (Answer: expiration value $\$12{,}000$; profit $-\$1{,}900$.)
A share and matching put cost $S_0=\$120$ and $p=\$5$, respectively. The put has $K=\$110$, and $S_T=\$95$ after one year. At an effective annual rate of $4\%$, find the expiration value and terminal profit after accumulating the initial outlay. (Answer: expiration value $\$110$; terminal profit $-\$20$.)
An investor buys 300 shares at $S_0=\$25$ and three put contracts, each covering 100 shares, with $K=\$28$ and $p=\$1.20$. If $S_T=\$20$, find the total expiration value and simple dollar profit. (Answer: expiration value $\$8{,}400$; profit $\$540$.)

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