Why Collar Payoffs Need an Audit
A collar combines a stock position with options whose payoffs change at two strike prices. A small sign error can turn a capped position into one that grows without a cap; a region error can make a supposed floor appear where it does not belong. The reliable approach is to start with the actual positions, write their expiration payoffs, and then check the result at the strikes and in every region.
Assume throughout that the position holds one share of stock, one European put, and one European call, all settled at expiration. The standard collar for a stated floor and cap is long one share, long one put, and short one call. The put strike is the floor and the call strike is the cap, provided the strikes are ordered correctly. This tutorial audits expiration values; premiums and financing matter separately when calculating profit.
Write the Positions Before Choosing Regions
At expiration, the value of one share is \(S_T\). A long put pays \(\max(K_P-S_T,0)\), and a short call pays \(-\max(S_T-K_C,0)\). Therefore, the collar’s total expiration value is
For the standard ordering \(K_P \leq K_C\), the result in each region is:
At \(S_T=K_P\) or \(S_T=K_C\), the option at the strike has zero intrinsic value, so either adjacent expression gives the same result. The payoff is continuous at both boundaries. Continuity is a useful check, but it is not enough by itself: a portfolio with the wrong option sign can still be continuous while failing to have the intended cap or floor.
Why the Piecewise Formula Holds
The formula follows directly from which options finish in the money. Below \(K_P\), the put is exercised for \(K_P-S_T\), while the call expires without value. Adding the stock gives \(S_T+(K_P-S_T)=K_P\). Between the strikes, both options expire without value, leaving only \(S_T\). Above \(K_C\), the put expires without value and the short call costs \(S_T-K_C\); the stock less that amount is \(S_T-(S_T-K_C)=K_C\).
This also gives a fast diagnostic using slopes. As \(S_T\) rises, the standard collar value has slope 0 below the put strike, slope 1 between the strikes, and slope 0 above the call strike. A negative slope or a slope of 2 in the middle or upper region is a warning to recheck the option positions.
A Practical Collar Audit
When a proposed payoff or calculation is in doubt, use the positions—not a memorized picture—as the starting point. The checks below catch different types of error.
Long stock contributes \(+S_T\); a long put contributes \(+\max(K_P-S_T,0)\); a short call contributes \(-\max(S_T-K_C,0)\). Do not infer a sign from the word “collar.”
A put floor and call cap require \(K_P\leq K_C\). If the put strike is above the call strike, do not apply the standard three-region floor-and-cap formula.
Use a price below both strikes, a price between them, and a price above both. Substitute into the position-level payoff before simplifying.
Substitute \(S_T=K_P\) and \(S_T=K_C\). For correctly ordered strikes, verify the value never falls below \(K_P\) or rises above \(K_C\).
Expiration value is not profit. To calculate profit, compare the terminal value with the accumulated initial net cost under the stated financing assumption.
Worked Examples: Detecting Sign and Region Errors
Worked Example: Checking a Collar’s Boundaries and Profit
A hypothetical investor buys one share for $55, buys a put with strike $48 for a premium of $3, and sells a call with strike $62 for a premium of $2. The options expire in one year. The effective annual interest rate is 5%, and there are no other cash flows. Find the expiration value and profit if the stock finishes at $40, $55, or $70. Check the floor, cap, and values at both strikes.
Method. First calculate the position’s expiration value from the stock and option payoffs. Then accumulate the initial net cost for one year and subtract it from the expiration value. This avoids the common mistake of treating an expiration value as profit.
The strikes are correctly ordered because \(48<62\). The initial net cost is the stock cost plus the put premium less the call premium:
At an effective annual rate of 5% for one year, the accumulated cost is:
For \(S_T=\$40\), the stock is worth $40, the long put pays \(48-40=\$8\), and the short call pays $0. Thus \(V_T=40+8-0=\$48\), and profit is \(48-58.80=-\$10.80\).
For \(S_T=\$55\), both options expire without value. Thus \(V_T=55\), and profit is \(55-58.80=-\$3.80\).
For \(S_T=\$70\), the put pays $0 and the short call costs \(70-62=\$8\). Thus \(V_T=70+0-8=\$62\), and profit is \(62-58.80=\$3.20\).
At the put strike, the value is \(48+0-0=\$48\). At the call strike, it is \(62+0-0=\$62\). The three tested expiration values are $48, $55, and $62, consistent with a $48 floor and a $62 cap. Profits are respectively −$10.80, −$3.80, and $3.20.
Worked Example: Finding a Long-Call Sign Error
A proposed portfolio consists of one share, a long put with strike $44, and a long call with strike $58. A note describes it as a collar with a $44 floor and a $58 cap. Calculate its expiration value at stock prices $35, $50, and $75, and decide whether the stated cap is valid.
Method. Use the recorded positions exactly as written. The common error here is to assume that a portfolio must be a standard collar because it is described as one; the call’s actual long sign must be checked.
The portfolio value is \(S_T+\max(44-S_T,0)+\max(S_T-58,0)\). At \(S_T=\$35\), the put pays \(44-35=\$9\) and the call pays $0. The total is \(35+9+0=\$44\).
At \(S_T=\$50\), both options expire without value. The total is \(50+0+0=\$50\).
At \(S_T=\$75\), the put pays $0 and the long call pays \(75-58=\$17\). The total is \(75+0+17=\$92\).
The values are $44, $50, and $92. The portfolio has a $44 floor at low stock prices, but it does not have a $58 cap: above the call strike, its value is \(S_T+(S_T-58)=2S_T-58\), which continues to increase. The call must be short, not long, for the standard capped collar.
As an additional boundary check, the value at $44 is $44, and at $58 it is $58. These boundary values are continuous, yet the claimed cap still fails. Boundary checks must therefore be combined with a position-sign check and a high-price test.
Worked Example: Diagnosing Reversed Strike Order
A portfolio holds one share, a long put with strike $68, and a short call with strike $54. Determine its expiration value at $40, $60, and $80. Check its value at both strikes and explain why the usual floor-and-cap interpretation does not apply.
Method. The put strike exceeds the call strike, so partition the prices at $54 and $68 and calculate from the position payoffs. The error to avoid is using the standard collar formula without first checking strike order.
For \(S_T=\$40\), the put pays \(68-40=\$28\), while the call expires without value. The total is \(40+28-0=\$68\).
For \(S_T=\$60\), the put pays \(68-60=\$8\), and the short call costs \(60-54=\$6\). The total is \(60+8-6=\$62\).
For \(S_T=\$80\), the put expires without value, and the short call costs \(80-54=\$26\). The total is \(80+0-26=\$54\).
At the call strike \(S_T=\$54\), the put pays \(68-54=\$14\), so the total is \(54+14=\$68\). At the put strike \(S_T=\$68\), the call costs \(68-54=\$14\), so the total is \(68-14=\$54\). The portfolio is continuous at both strikes, but between them its value is \(68+54-S_T=122-S_T\), which declines as the stock price rises. It is not a standard collar with a lower put floor and higher call cap.
Common Mistakes and Exam Checks
A typical incorrect calculation for the standard collar is to add the call payoff rather than subtract it. For example, with a $50 put and a $65 call, a stock price of $80, and a short call, the correct value is \(80-(80-65)=\$65\). The incorrect long-call sign gives \(80+(80-65)=\$95\), contradicting the stated $65 cap. Write the position signs before substituting numbers.
- Using only option payoffs. A collar’s total expiration value includes the stock. Option payoff alone is not the value of the combined position.
- Confusing payoff with profit. Premiums and financing affect profit, not the stock-plus-options expiration value. State the financing rate and period if accumulating the initial cost.
- Assuming continuity proves the whole payoff is right. A wrong long-call position can be continuous at the strikes but uncapped. Check slopes, signs, and the stated limits as well.
- Sorting the strikes only after writing the regions. Identify which strike comes first. If \(K_P>K_C\), the standard floor-middle-cap formula is invalid.
- Ignoring exact-strike cases. At a strike, the option at the money has zero payoff. Substitution should agree with the expression from either adjacent region.
For the standard ordering, a final range check is especially efficient: for every \(S_T\), the expiration value must satisfy \(K_P\leq V_T\leq K_C\). If a computed value violates either inequality, revisit the option signs, the region, or whether the calculation is for value rather than profit.
Check Your Understanding
For each problem, write the positions with their signs before evaluating the expiration value.
- A portfolio holds one share, a long put at $30, and a short call at $40. Find its expiration value at stock prices $25, $35, and $50. (Answer: $30, $35, and $40, respectively.)
- A standard collar has a long put at $75 and a short call at $90. If the stock finishes at $72, find the stock value, put payoff, call payoff, and total expiration value. (Answer: $72, $3, $0, and $75, respectively.)
- An investor buys a share for $100, buys a put at $85 for $7, and sells a call at $120 for $4. The options expire in one year, the effective annual interest rate is 5%, and there are no other cash flows. If the stock finishes at $130, find expiration value and profit. (Answer: expiration value $120; profit $11.85.)
- A position is long one share, long a put at $45, and long a call at $60. Find its expiration values at stock prices $30, $52, and $75. Does it have a $60 cap? (Answer: $45, $52, and $90; no.)
- A position is long one share, long a put at $68, and short a call at $54. Find its expiration values at $40, $60, and $80, and state whether the value rises or falls between the strikes. (Answer: $68, $62, and $54; it falls between the strikes.)