Designing a Collar Around a Floor and a Cap
A collar can be designed to limit the range of sale proceeds from a stock position. The put establishes a minimum terminal sale value, while the call limits the value available above its strike. The challenge is to select strikes that meet the investor’s stated targets and then account for the option premium: a collar’s gross floor and cap are not necessarily its net proceeds after the options are paid for.
The calculations below use one share of stock, one put, and one written call, with each option covering one share. Prices are hypothetical. Assume European options held to expiration, no transaction costs, no dividends or other cash flows, and no interest earned on or charged for the initial option cash flow. These assumptions let us compare the stock and option cash flows directly at expiration. If actual contract sizes, dividends, or financing are relevant, they must be included separately.
Before premiums, the stock-and-options position has a terminal sale value between \(K_P\) and \(K_C\). Below the put strike, exercising the put lets the holder sell at \(K_P\). Between the strikes, neither option changes the stock’s value. Above the call strike, assignment on the written call requires selling at \(K_C\). The premium is an initial cash flow, so it shifts the net proceeds but does not change these gross terminal strike levels.
How to Choose the Strikes
If the targets refer to gross sale proceeds, the design is direct. To guarantee at least a gross amount \(L\) per share, choose \(K_P \ge L\). To ensure gross proceeds do not exceed a cap \(U\) per share, choose \(K_C \le U\). The standard collar also requires \(K_P \le K_C\). Choosing \(K_P=L\) and \(K_C=U\) sets the floor and cap exactly, provided \(L \le U\).
If the targets instead refer to proceeds after the initial option cash flow, the premium must be included. With the stated assumption of no interest on that cash flow, net terminal proceeds equal the gross collar value minus \(D\). Therefore a net floor \(L\) requires \(K_P-D \ge L\), while a net cap \(U\) requires \(K_C-D \le U\). Since \(D\) depends on the chosen options, check both constraints using the premiums for the selected strikes.
A premium debit lowers both the net floor and the net cap by the same amount. A premium credit raises both. Thus, simply setting the put and call strikes equal to stated net targets can miss both requirements. In practice, option premiums may vary across candidate strikes, so the net targets should be checked for each proposed pair rather than adjusted by a guessed amount.
Determine whether the required floor and cap are gross sale values or net proceeds after the option premiums.
For gross targets, compare \(K_P\) with the floor and \(K_C\) with the cap. Confirm \(K_P \le K_C\).
Find \(D=p-c\). A negative result is a credit, not a debit.
Subtract \(D\) from the gross floor and cap. Compare the resulting position with the unhedged stock at relevant terminal prices.
Why the Collar Has a Floor and a Cap
At expiration, the long put pays \(\max(K_P-S_T,0)\), and the written call pays \(-\max(S_T-K_C,0)\). Adding these payoffs to the stock value gives the gross collar value. The regions follow from which option, if any, is in the money.
For example, when \(S_T<K_P\), the put pays \(K_P-S_T\) and the call expires without value. Adding the put payoff to the stock gives \(S_T+(K_P-S_T)=K_P\). Between the strikes, both options expire without value, so the position is worth \(S_T\). Above \(K_C\), the written call costs \(S_T-K_C\), leaving \(S_T-(S_T-K_C)=K_C\). This proves that the floor and cap arise from the option payoffs, rather than from an assumption that the stock itself stays within a particular range.
The net terminal proceeds, under the no-interest assumption, are \(V_T-D\). For a position originally purchased at \(S_0\), the profit including the option cash flow is \(V_T-S_0-D\). These are different quantities: proceeds measure what remains from the sale after the premium, while profit also accounts for the stock’s original purchase price.
Worked Example: Set a Gross Floor and Cap
An investor owns 100 shares purchased for $50 per share. To set a gross terminal floor of $46 and cap of $58, the investor buys a $46 put for $2.10 per share and writes a $58 call for $1.35 per share. Find the option cash flow, the net terminal range, and the profit at each endpoint. Assume the stated premiums are paid or received at the start and earn no interest.
Method. Use the target gross floor and cap as the put and call strikes, then subtract the net option debit from the expiration values. This avoids treating gross strike protection as net proceeds after premiums.
Initial option cash flow. The put costs $2.10 per share and the call brings in $1.35 per share, so \(D=2.10-1.35=\$0.75\) per share. For 100 shares the net debit is \(100(0.75)=\$75\).
Net proceeds. The gross terminal floor is $46 and the gross cap is $58. After subtracting $0.75 per share, the net terminal range is from \(46-0.75=\$45.25\) to \(58-0.75=\$57.25\) per share. For 100 shares, that is $4,525 to $5,725.
Endpoint profits. The stock cost was \(100(50)=\$5{,}000\). At the floor, profit is \(4{,}525-5{,}000=-\$475\). At the cap, profit is \(5{,}725-5{,}000=\$725\).
Check. Below $46, exercising the put produces $46 gross per share; above $58, assignment on the call limits gross proceeds to $58 per share. The $0.75 debit shifts both values down to the calculated net endpoints.
Measuring Protection and Surrendered Upside
A floor limits how far terminal proceeds fall, but the amount of protection depends on the stock price used for comparison. At a specified terminal price below the put strike, compare the collar’s gross value \(K_P\) with the unhedged stock value \(S_T\). The gross downside protection per share is \(K_P-S_T\); net proceeds must also reflect \(D\). Above the call strike, the gross value surrendered relative to unhedged stock is \(S_T-K_C\). If comparing net proceeds with unhedged stock and including the option cash flow, the difference is \(S_T-(K_C-D)=S_T-K_C+D\).
Within the strikes, the stock participates one-for-one in price changes before the premium adjustment. Below the put strike, further stock-price declines do not reduce the gross collar value. Above the call strike, further gains in the stock do not increase gross collar value. The collar therefore protects against losses beyond its floor in exchange for giving up gains beyond its cap, while the premium changes the net level of proceeds.
Worked Example: Check Net Targets and Downside Protection
A hypothetical fund holds 200 shares bought for $72 each. It considers buying a $66 put for $3.20 per share and writing an $82 call for $1.55 per share. Its stated requirements are a gross floor of at least $66 and a gross cap no higher than $82. Calculate the initial option debit, check those requirements, and find net proceeds and protection if the stock expires at $50.
Method. First check the strikes against the gross targets. Then calculate the premium-adjusted values. Do not subtract the option premium when checking a target explicitly stated as gross.
Initial option cash flow. The debit per share is \(D=3.20-1.55=\$1.65\). For 200 shares, the total debit is \(200(1.65)=\$330\).
Target check. The $66 put gives a gross floor of $66, meeting the minimum. The $82 call gives a gross cap of $82, meeting the maximum. Since \(66\le82\), the strikes form a valid collar.
Net floor and cap. Net terminal proceeds range from \(66-1.65=\$64.35\) to \(82-1.65=\$80.35\) per share. For 200 shares, the range is \(200(64.35)=\$12{,}870\) to \(200(80.35)=\$16{,}070\).
At \(S_T=\$50\). The put is exercised, producing gross value of $66 per share. Net proceeds are \(66-1.65=\$64.35\) per share, or $12,870 total. The unhedged shares would be worth \(200(50)=\$10{,}000\). Thus, net proceeds from the collar exceed the unhedged terminal value by \(12{,}870-10{,}000=\$2{,}870\). Before accounting for the premium, the put’s gross protection is \(66-50=\$16\) per share.
Check. The net proceeds also equal the gross value of \(200(66)=\$13{,}200\) less the $330 option debit, giving $12,870.
Premium-Adjusted Targets and a Common Trap
When a target is stated in net proceeds, checking strikes alone is insufficient. Suppose a selected collar has a $42 put, a $56 call, a put premium of $2.40, and a call premium of $1.10. Its debit is \(2.40-1.10=\$1.30\), so its net floor is $40.70 and its net cap is $54.70 per share. It meets a net floor of $40 and a net cap of $55, even though neither strike equals its corresponding net target.
A common mistake is to state the net floor as \(K_P+D\) when \(D\) is a positive debit. That adds the cost instead of accounting for it. For this collar, the incorrect calculation would be \(42+1.30=\$43.30\). The correct net floor is \(42-1.30=\$40.70\), because the premium was paid and reduces the cash retained. A credit has the opposite effect: if \(D<0\), subtracting \(D\) increases net proceeds.
Another important distinction is whether the requirement concerns the option-and-stock sale value or the investor’s profit. A $46 net floor does not mean the investor earns $46 of profit per share; profit also subtracts the stock’s original basis. State clearly whether a result is a gross terminal value, net proceeds after option premiums, or profit after the stock cost.
Check Your Understanding
For each problem, report per-share amounts unless a share count is specified. Assume options are held to expiration and initial option premiums earn no interest.
- A collar has a $27 put and a $35 call. The put premium is $1.20 and the call premium is $0.80. Find the net debit and the minimum and maximum net terminal proceeds. (Answer: Debit $0.40; net range $26.60 to $34.60 per share.)
- An investor holds 100 shares in a collar with a $48 put costing $2.75 and a $61 call written for $1.50. Find the total option debit and the total net proceeds at the floor and cap. (Answer: Debit $125; net floor $4,675 and net cap $5,975.)
- A proposed collar has a $53 put costing $4.20 and a $67 call written for $2.10. Does it meet a net floor of at least $50 and a net cap of no more than $65? (Answer: Yes; debit $2.10, net floor $50.90, and net cap $64.90 per share.)
- A collar has a $74 put and a $91 call; the put costs $3.60 and the call brings in $1.40. Find net proceeds and the difference from unhedged stock value at expiration prices of $68 and $104. (Answer: Debit $2.20; at $68, net proceeds are $71.80, which is $3.80 above the stock value; at $104, net proceeds are $88.80, which is $15.20 below the stock value.)
- A proposed collar uses a $40 put costing $2.10 and a $54 call written for $1.60. Check whether it meets a net floor of $38.50 and a net cap of $52.50, and state the amount by which each target is exceeded or left unmet. (Answer: Debit $0.50; net floor $39.50, exceeding the minimum by $1.00; net cap $53.50, exceeding the maximum by $1.00, so the collar does not meet both requirements.)