Collar Payoffs Across Three Price Regions

Introduction

A collar combines a stock position with a protective put and a short call. At expiration, the put limits the effect of a falling stock price on the combined position, while the short call offsets further increases above its strike. To calculate the result correctly, track each position’s payoff separately and then add the amounts using their signs.

This tutorial derives the collar’s expiration payoff across the three relevant price regions. It focuses on payoff at expiration—not the initial cost of establishing the position or the overall profit after premiums.

Definition: A standard one-unit stock collar consists of one share of stock, one long put on that share with strike $K_p$, and one short call on that share with strike $K_c$, where $K_p<K_c$. For an underlying price $S_T$ at expiration, the collar’s total payoff is the sum of the stock’s value, the put payoff, and the short call payoff. Assume the options are European, refer to the same underlying and expiration, and each option covers one share. The payoff formula does not include option premiums, dividends, or other cash flows.

Build the Payoff From Its Three Positions

At expiration, a share of stock is worth $S_T$. A long put with strike $K_p$ pays $\max(K_p-S_T,0)$. A short call with strike $K_c$ has payoff $-\max(S_T-K_c,0)$: the call buyer receives a positive amount when the stock is above the call strike, so the short position has a negative payoff of the same size.

Adding these three components gives the collar payoff for one share:

C_T=S_T+\max(K_p-S_T,0)-\max(S_T-K_c,0).

The maximum expressions switch on or off at their respective strikes. This creates three regions: below the put strike, from the put strike through the call strike, and above the call strike. The endpoints can be included in either adjoining region because the formulas agree there.

Below the put strike.
If $S_T<K_p$, the put is exercised for $K_p-S_T$, and the call expires without value. The stock and put together pay $S_T+(K_p-S_T)=K_p$.

Between the strikes.
If $K_p\leq S_T\leq K_c$, both options expire without value. The stock position contributes $S_T$, so the total payoff is $S_T$.

Above the call strike.
If $S_T>K_c$, the put expires without value and the short call payoff is $-(S_T-K_c)$. The stock and call together pay $S_T-(S_T-K_c)=K_c$.

C_T= \begin{cases} K_p, & S_T<K_p,\\ S_T, & K_p\leq S_T\leq K_c,\\ K_c, & S_T>K_c. \end{cases}

The first region produces a constant payoff equal to the put strike; the middle region passes through the stock price; and the last region produces a constant payoff equal to the call strike. For one share and one option of each kind, the payoff therefore stays between the two strikes.

Why the Three-Region Formula Holds

The piecewise result follows directly from which options have positive expiration payoffs. For example, when $S_T<K_p$, the put payoff is $K_p-S_T$ and the call payoff is zero. Substituting those amounts into the combined payoff gives:

$$ C_T=S_T+(K_p-S_T)-0=K_p. $$

When $K_p\leq S_T\leq K_c$, both option payoffs are zero, so $C_T=S_T$. When $S_T>K_c$, the short call contributes $-(S_T-K_c)$, giving:

$$ C_T=S_T+0-(S_T-K_c)=K_c. $$

At the put strike, the below-strike expression gives $K_p$, and the middle expression gives $S_T=K_p$. At the call strike, the middle expression gives $S_T=K_c$, and the above-strike expression gives $K_c$. Thus the payoff is continuous at both boundaries. This boundary check is a useful way to detect a misplaced sign or strike.

Theorem: For a one-share collar with $K_p<K_c$, the expiration payoff is $K_p$ when $S_T\leq K_p$, is $S_T$ when $K_p\leq S_T\leq K_c$, and is $K_c$ when $S_T\geq K_c$. The overlapping endpoint descriptions agree. If the position has $q$ matching shares, puts, and short calls, multiply the one-share payoff by $q$.

Worked Examples

Worked Example: Calculate Payoff in All Three Regions

A hypothetical investor holds one share, one put with strike $K_p=\$42$, and one short call with strike $K_c=\$58$. Find the collar payoff at expiration for $S_T=\$35$, $\$50$, and $\$71$. Assume each option covers one share. The method is to evaluate the three position payoffs and add them; this avoids treating the short call as a positive payoff.

At $S_T=\$35$: the stock is worth $\$35$. The put pays $\max(42-35,0)=\$7$. The short call pays $-\max(35-58,0)=\$0$. Therefore, the total payoff is $\$35+\$7+\$0=\$42$.

At $S_T=\$50$: the stock is worth $\$50$. The put pays $\max(42-50,0)=\$0$, and the short call pays $-\max(50-58,0)=\$0$. The total payoff is $\$50+\$0+\$0=\$50$.

At $S_T=\$71$: the stock is worth $\$71$. The put pays $\max(42-71,0)=\$0$. The short call pays $-\max(71-58,0)=-\$13$. The total payoff is $\$71+\$0-\$13=\$58$.

The piecewise formula gives the same results: the payoff is the put strike below $\$42$, the stock price between $\$42$ and $\$58$, and the call strike above $\$58$. The three payoffs are $\$42$, $\$50$, and $\$58$, respectively.

Worked Example: Scale the Payoff to Multiple Units

A hypothetical fund holds 40 shares, 40 long puts with strike $K_p=\$76$, and 40 short calls with strike $K_c=\$91$. Each option covers one share. Calculate the total expiration payoff if $S_T=\$68$, $\$84$, or $\$103$. First calculate per share, then multiply by 40. This method avoids multiplying only the stock value while forgetting to scale the options.

At $S_T=\$68$: per share, the stock is worth $\$68$, the put pays $76-68=\$8$, and the call pays zero. The per-share total is $\$68+\$8=\$76$. Across 40 matching units, the payoff is $40(\$76)=\$3{,}040$.

At $S_T=\$84$: both options expire without value because $\$76<\$84<\$91$. The per-share total is $\$84$, and the position payoff is $40(\$84)=\$3{,}360$.

At $S_T=\$103$: the put expires without value. The short call pays $-(103-91)=-\$12$ per share, so the per-share total is $\$103-\$12=\$91$. Across 40 units, the payoff is $40(\$91)=\$3{,}640$.

The total payoff is $\$3{,}040$, $\$3{,}360$, and $\$3{,}640$ in the three cases. Each result equals 40 times the applicable strike or stock price, as required by the piecewise formula.

Worked Example: Infer the Expiration Price From a Payoff

A hypothetical position consists of 15 matching collar units, with put strike $K_p=\$30$ and call strike $K_c=\$40$. At expiration, the total payoff is $\$525$. Find $S_T$, or determine whether it is uniquely identified. Assume each option covers one share. The method is to convert the total payoff to a per-unit amount, then test the three regions. This avoids assuming every payoff corresponds to a unique stock price.

The per-unit payoff is $\$525/15=\$35$. Below or at the put strike, the per-unit payoff would be $\$30$, not $\$35$. Above or at the call strike, it would be $\$40$, not $\$35$. Therefore the stock price must lie in the middle region, where the per-unit payoff equals $S_T$.

In that region, $S_T=\$35$. It satisfies $30\leq35\leq40$, so the middle-region assumption is consistent. The expiration price is uniquely $\$35$, and the position payoff checks as $15(\$35)=\$525$.

Common Mistakes and What the Payoff Does Not Include

A collar payoff is the value of the positions at expiration, with option exercise or assignment represented by the option payoff formulas. It is not, by itself, the investor’s profit. In particular, the formula above does not subtract the put premium or add the call premium, and it does not include the amount paid for the shares. Those cash flows are distinct from expiration payoff.

The most common sign error is to add the call payoff as if the investor owned the call. The position is short the call, so its payoff is negative when $S_T>K_c$. For example, with $S_T=\$70$ and $K_c=\$60$, the short call contributes $-(\$70-\$60)=-\$10$, not $+\$10$. If the stock is worth $\$70$, the stock and short call together contribute $\$70-\$10=\$60$, the call strike.

Another error is to omit the stock’s value and report only the option payoffs. For $S_T<K_p$, the put alone pays $K_p-S_T$, which is not the collar payoff: the stock contributes $S_T$ as well, bringing the total to $K_p$. A final check is to examine both extreme price regions. A correct one-unit collar payoff is constant at $K_p$ below the put strike and constant at $K_c$ above the call strike.

Key takeaway: Add the stock value, the long put payoff, and the negative of the long call payoff. The resulting one-unit collar payoff is the put strike below the lower strike, the stock price between the strikes, and the call strike above the higher strike. Match the quantities of all three positions before scaling the result.

Check Your Understanding

For each question, assume European options on the same underlying with the stated expiration, one option per matching share, and no premiums or other cash flows included in the requested payoff.

A one-unit collar has $K_p=\$24$, $K_c=\$33$, and $S_T=\$19$. What is its expiration payoff? (Answer: $\$24$)
A one-unit collar has $K_p=\$47$, $K_c=\$62$, and $S_T=\$55$. State the stock, put, and short call payoffs and their total. (Answer: Stock $\$55$, put $\$0$, short call $\$0$; total $\$55$)
A one-unit collar has $K_p=\$81$, $K_c=\$96$, and $S_T=\$109$. Calculate the short call payoff and the total collar payoff. (Answer: Short call payoff $-\$13$; total payoff $\$96$)
A portfolio has 25 matching collar units with $K_p=\$38$, $K_c=\$52$, and $S_T=\$44$. Find the total expiration payoff. (Answer: $\$1{,}100$)
A position has 10 matching collar units with $K_p=\$70$ and $K_c=\$86$. Its total expiration payoff is $\$860$. Is $S_T$ uniquely determined? If so, find it. (Answer: Yes; the per-unit payoff is $\$86$, so $S_T\geq\$86$, but the exact stock price is not uniquely determined.)

What You'll Learn