Collar Profit And Breakeven With A Stock Basis

From Collar Payoff to Profit

A collar combines a stock position, a purchased put, and a written call. Its expiration payoff is bounded below and above by the option strikes, but payoff is not the same as profit: profit also depends on what the investor paid for the stock and on the options’ net premium. A collar can have the same strikes and expiration payoff as another collar yet have a different breakeven price because its initial costs differ.

This tutorial measures profit per share at expiration, in dollars. It uses the expiration payoff convention: the initial cash cost is subtracted without accumulating or discounting it. The stock pays no dividends, there are no transaction or financing costs, and each option position covers one share. The put and call are evaluated at expiration; early exercise is not part of the calculation. These assumptions matter: dividends, costs, or interest on the initial cash flow would change a full investment-return calculation.

Definition: Let $S_0$ be the stock purchase price, $S_T$ its price at expiration, $K_P$ the long put’s strike, and $K_C$ the short call’s strike, with $K_P<K_C$. Let $p$ and $c$ be the put and call premiums per share, respectively, quoted as positive amounts. Define the signed net option premium as $q=p-c$: a positive $q$ is a net payment, and a negative $q$ is a net receipt. The effective stock basis is $B=S_0+q$.

The basis $B$ is the net initial outlay per share under this convention: the investor buys the stock for $S_0$, pays $p$ for the put, and receives $c$ for writing the call. Thus the total is $S_0+p-c=S_0+q$. If the options produce a net credit, $B$ is less than the stock purchase price.

Profit in the Three Expiration Regions

At expiration, the stock is worth $S_T$. The put pays $\max(K_P-S_T,0)$, and the written call contributes $-\max(S_T-K_C,0)$. Subtract the initial outlay $B$ to obtain profit:

\Pi(S_T)=S_T+\max(K_P-S_T,0)-\max(S_T-K_C,0)-B.

The put is in the money below its strike, neither option pays between the strikes, and the written call costs the investor money above its strike. Expanding the payoff expression gives the piecewise profit formula:

\Pi(S_T)= \begin{cases} K_P-B, & S_T\leq K_P,\\ S_T-B, & K_P<S_T<K_C,\\ K_C-B, & S_T\geq K_C. \end{cases}

At $S_T=K_P$ and $S_T=K_C$, the adjacent expressions agree, so the formula has no jump at either strike. Below the put strike, the stock and put together are worth $K_P$, leaving a fixed floor profit of $K_P-B$. Between the strikes, neither option changes the stock’s value, so profit rises dollar for dollar with $S_T$. Above the call strike, the call offsets further stock-price increases, leaving a fixed capped profit of $K_C-B$.

Breakeven result: A breakeven price is an expiration price $S_T$ for which $\Pi(S_T)=0$. If $K_P<B<K_C$, there is one breakeven, $S_T=B$, in the middle region. If $B<K_P$, profit is positive in all three regions; if $B>K_C$, profit is negative in all three regions. At the boundary cases, $B=K_P$ gives zero profit for every $S_T\leq K_P$, while $B=K_C$ gives zero profit for every $S_T\geq K_C$.

Why the Basis Determines Breakeven

The breakeven formula follows directly from the region-by-region profit, rather than from the strikes alone. The two option strikes determine where the profit changes shape. The basis determines where that profit crosses zero.

Find the net initial outlay.
The stock costs $S_0$; the put costs $p$; the written call provides a receipt of $c$. Therefore $B=S_0+p-c$.

Use the expiration region.
Below $K_P$, the stock-plus-put value is $K_P$; between the strikes, the stock value is $S_T$; above $K_C$, the net position value is $K_C$.

Set profit equal to zero and check the region.
In the middle region, $S_T-B=0$ gives $S_T=B$. This is a valid unique breakeven only when $K_P<B<K_C$. In either outer region, profit is constant, so a zero there occurs across a whole region, not at a single price.

This method is aligned with the SOA Exam FM (Financial Mathematics) syllabus. The region check is essential: solving the middle expression gives a candidate, but it does not establish that the candidate lies between the strikes.

Worked Examples

Worked Example: A Net Debit With a Breakeven Between the Strikes

An investor buys one share for $S_0=\$52$, buys a put with strike $K_P=\$48$ for $p=\$2.40$, and writes a call with strike $K_C=\$60$ for $c=\$1.10$. Find the effective basis, all three profit expressions, the breakeven, and profit if $S_T$ is $\$44$, $\$56$, or $\$67$.

Method: calculate the signed premium first, then apply the piecewise profit formula. The net option premium is $q=p-c=\$2.40-\$1.10=\$1.30$, a debit. Thus $B=S_0+q=\$52+\$1.30=\$53.30$. Since $\$48<\$53.30<\$60$, the middle-region candidate is valid and is the unique breakeven.

For $S_T\leq \$48$, profit is $K_P-B=\$48-\$53.30=-\$5.30$. For $\$48<S_T<\$60$, profit is $S_T-\$53.30$. For $S_T\geq \$60$, profit is $K_C-B=\$60-\$53.30=\$6.70$.

At $S_T=\$44$, the stock-plus-put value is $\$44+(\$48-\$44)=\$48$, so profit is $\$48-\$53.30=-\$5.30$. At $S_T=\$56$, neither option pays, so profit is $\$56-\$53.30=\$2.70$. At $S_T=\$67$, the stock and written call together are worth $\$67-(\$67-\$60)=\$60$, so profit is $\$60-\$53.30=\$6.70$. The middle profit is zero at $S_T=\$53.30$, which is between the strikes. The calculations agree with the floor and cap.

Common mistake avoided: Counting the call premium as a cost would overstate the basis. The call writer receives $\$1.10$, so it must be subtracted from the put premium, not added to it.

Worked Example: A Net Credit and No Breakeven

An investor buys a share for $S_0=\$40$, buys a put with $K_P=\$42$ for $p=\$1.20$, and writes a call with $K_C=\$50$ for $c=\$2.00$. Calculate the basis and profit at expiration prices of $\$38$, $\$46$, and $\$54$. Determine whether there is a breakeven.

Method: treat the net credit as a negative premium. Here $q=p-c=\$1.20-\$2.00=-\$0.80$, so $B=S_0+q=\$40-\$0.80=\$39.20$. Because $B=\$39.20<K_P=\$42$, the basis is below the floor strike.

For $S_T\leq \$42$, profit is $K_P-B=\$42-\$39.20=\$2.80$. For $\$42<S_T<\$50$, profit is $S_T-\$39.20$. For $S_T\geq \$50$, profit is $K_C-B=\$50-\$39.20=\$10.80$.

At $S_T=\$38$, the stock-plus-put value is $\$38+(\$42-\$38)=\$42$, giving profit $\$42-\$39.20=\$2.80$. At $S_T=\$46$, neither option pays, giving $\$46-\$39.20=\$6.80$. At $S_T=\$54$, the stock and call together are worth $\$54-(\$54-\$50)=\$50$, giving $\$50-\$39.20=\$10.80$. All three profits are positive, and the piecewise expressions show that profit remains positive throughout. There is no breakeven price.

Common mistake avoided: A net credit does not make the basis $S_0+|q|$. The $\$0.80$ received reduces the outlay to $\$39.20$, which is why even the floor region has positive profit.

Worked Example: A Boundary Breakeven Across an Entire Region

A share costs $S_0=\$98.50$. The investor buys a put with strike $K_P=\$90$ for $p=\$2.40$ and writes a call with strike $K_C=\$100$ for $c=\$0.90$. Find the profit function and describe every breakeven price.

Method: compare the effective basis with each strike before solving for a crossing. The net premium is $q=\$2.40-\$0.90=\$1.50$. The basis is $B=\$98.50+\$1.50=\$100$, exactly equal to the call strike.

For $S_T\leq \$90$, profit is $\$90-\$100=-\$10$. For $\$90<S_T<\$100$, profit is $S_T-\$100$, which is negative throughout that open interval. For $S_T\geq \$100$, profit is $\$100-\$100=\$0$. Therefore every expiration price at or above $\$100$ is a breakeven; there is not a unique breakeven price.

As a check, at $S_T=\$100$ the middle expression reaches zero, and the capped expression is also zero. At $S_T=\$110$, the stock’s $\$110$ value is offset by the written call’s $\$10$ obligation, leaving $\$100$, equal to the initial basis. This confirms the zero-profit plateau.

Common mistake avoided: Reporting $\$100$ as the only breakeven ignores that profit stays at zero after the call caps the position’s value.

Common Sign and Region Errors

The most reliable check is to keep the initial cash flows separate until their signs are clear. A premium paid is an outflow; a premium received is an inflow. Only after calculating $q=p-c$ should it be combined with $S_0$. Then check which profit expression applies to the stated expiration price.

Worked Example: Correcting a Premium-Sign Error

Suppose the stock costs $\$52$, the put premium is $\$2.40$, and the written call premium is $\$1.10$, with strikes $\$48$ and $\$60$. A calculation incorrectly uses $B_{\text{wrong}}=\$52+\$2.40+\$1.10=\$55.50$. Identify the error and find the correct profit at $S_T=\$58$.

Method: reconstruct the initial cash flow. The investor pays $\$52$ for the share and $\$2.40$ for the put, then receives $\$1.10$ from the call. Therefore the net outlay is $\$52+\$2.40-\$1.10=\$53.30$, not $\$55.50$.

Since $\$48<\$58<\$60$, neither option pays, and the correct profit is $\$58-\$53.30=\$4.70$. The incorrect basis would imply profit of $\$58-\$55.50=\$2.50$, understating profit by $\$2.20$. That error counts the call receipt as a payment and reverses its cash-flow sign.

Do not call terminal payoff profit. Subtract the net initial outlay to calculate profit.
Do not treat both premiums as payments. The written call premium is received.
Do not assume the basis is between the strikes. If it is outside, a middle-region algebraic solution is not a valid breakeven.
Do not assume every breakeven is unique. When the basis equals an outer strike, the corresponding flat region has zero profit.

Key takeaway: Compute $B=S_0+p-c$, write profit in the three expiration regions, and only then locate breakeven. The middle-region breakeven is $S_T=B$ when that basis lies between the strikes; otherwise the profit may have no breakeven or a zero-profit plateau.

Check Your Understanding

For each collar, use the stated expiration-profit convention and identify the relevant region before concluding.

A share costs $\$31$. The investor pays $\$1.40$ for a put at $K_P=\$28$ and receives $\$0.60$ for a call at $K_C=\$36$. Find the basis and breakeven, and state the floor and cap profits. (Answer: $B=\$31.80$; unique breakeven $S_T=\$31.80$; floor profit $-\$3.80$; cap profit $\$4.20$.)
A share costs $\$55$. The put at $K_P=\$57$ costs $\$2.20$, and the call at $K_C=\$66$ brings in $\$2.70$. Find the basis and determine whether any expiration price breaks even. (Answer: $B=\$54.50$; no breakeven; profit is positive throughout, with floor $\$2.50$ and cap $\$11.50$.)
A share costs $\$82$; the put at $K_P=\$80$ costs $\$3.00$, and the call at $K_C=\$88$ brings in $\$1.75$. Find the basis, profit at $S_T=\$85$, and breakeven. (Answer: $B=\$83.25$; profit at $\$85$ is $\$1.75$; unique breakeven $S_T=\$83.25$.)
A share costs $\$104$; the put at $K_P=\$96$ costs $\$2.80$, and the call at $K_C=\$105$ brings in $\$1.30$. Find the basis and decide whether profit is ever zero. (Answer: $B=\$105.50>K_C$; no breakeven; floor profit is $-\$9.50$ and cap profit is $-\$0.50$.)
A share costs $\$47.20$; the put at $K_P=\$48$ costs $\$1.10$, and the call at $K_C=\$58$ brings in $\$0.30$. Determine all breakeven expiration prices and the cap profit. (Answer: $B=\$48$; every $S_T\leq \$48$ is a breakeven; cap profit is $\$10$.)

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