Building A Collar From Stock And Options

What a Collar Combines

A stockholder who wants protection against a decline but is willing to limit gains can combine the stock with two options. The stock provides exposure to increases in the share price. A purchased put gives the holder a right to sell at a specified price, while a written call creates an obligation to sell at a specified price if exercised. Together, these positions create a collar.

The put strike establishes the protected lower level at expiration. The call strike establishes the upper level at which the stock’s value is capped. The construction depends on the positions and their coverage: the investor holds the stock, buys a put, and sells a call, with each option covering the same amount of stock.

Definition: A standard collar on one share consists of one long 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 $K_P<K_C$. All three positions have the same expiration date. The collar’s expiration value is the sum of the stock value, the put payoff, and the negative of the call payoff.

For this tutorial, assume the options are on the same underlying stock, have the same expiration, and use matching share coverage. Ignore dividends, transaction costs, taxes, and counterparty default. We are measuring the position’s gross value at expiration, not its profit. Option premiums and the stock’s purchase price are initial cash flows; they do not change the expiration payoff formula.

Constructing the Position and Its Bounds

The construction starts with the amount of stock to protect. For every share held, buy one put and write one call, using the same share-equivalent coverage. Choose the put strike below the call strike. If listed options each cover 100 shares, for example, 300 shares require three puts and three calls—not one of each.

Let $S_T$ be the stock price at expiration. The long stock is worth $S_T$ per share. The long put pays $\max(K_P-S_T,0)$. The short call contributes $-\max(S_T-K_C,0)$, since the call holder’s payoff is a liability to the call writer. Thus the combined expiration value per share is

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

The formula shows why the collar has a floor and a cap. When the stock finishes below the put strike, the put’s value offsets the stock’s decline below $K_P$. When the stock finishes above the call strike, the short call offsets the stock’s increase above $K_C$. Between the strikes, neither option has an expiration payoff, so the position retains the stock value.

Theorem: For a standard collar with $K_P<K_C$, the expiration value per share is never below $K_P$ and never above $K_C$. Its expiration floor is $K_P$, and its expiration cap is $K_C$. For $N$ matched shares, the corresponding total bounds are $NK_P$ and $NK_C$.

The bound can also be checked directly. If $S_T<K_P$, then $S_T+(K_P-S_T)=K_P$, and the call has no payoff because $S_T<K_C$. If $K_P\leq S_T\leq K_C$, both options have zero payoff and the stock value is $S_T$, which lies between the strikes. If $S_T>K_C$, the put has no payoff and $S_T-(S_T-K_C)=K_C$. These cases establish both bounds.

The conclusion is about the package’s expiration value. It is not a guarantee that the investor earns at least $K_P$ in profit: the stock may have been bought for more than $K_P$, and the put and call involve premiums. It also does not say that the collar’s market value must stay between the strikes before expiration.

Worked Examples

Worked Example: Finding the Floor and Cap Per Share

An investor holds one share currently priced at $52, buys a put with strike $46, and writes a call with strike $61. Both options expire on the same date and cover one share. Find the collar’s floor and cap, then calculate its gross expiration value if the stock finishes at $40, $55, or $68.

Method. Identify the long stock and matched options, then add their expiration payoffs. The current $52 price does not enter the expiration payoff calculation.

Floor and cap. The put strike is $K_P=\$46$, and the call strike is $K_C=\$61$. Therefore, the floor is $46 per share and the cap is $61 per share.

If $S_T=\$40$. The stock is worth $40. The put pays $\max(46-40,0)=\$6$, and the short call contributes $-\max(40-61,0)=\$0$. The total is $40+6+0=\$46$.

If $S_T=\$55$. The stock is worth $55. The put pays $\max(46-55,0)=\$0$, and the short call contributes $-\max(55-61,0)=\$0$. The total is $55+0+0=\$55$.

If $S_T=\$68$. The stock is worth $68. The put pays $\max(46-68,0)=\$0$, and the short call contributes $-\max(68-61,0)=-\$7$. The total is $68+0-7=\$61$.

The three values are $46, $55, and $61, all within the stated floor and cap. This check also confirms the short-call sign: above the call strike, the call reduces rather than increases the stockholder’s value. A common mistake is to add the call payoff as though the investor owned the call.

Worked Example: Matching Standard Option Contracts to Stock

A fund holds 300 shares of a hypothetical company. Each listed option contract covers 100 shares. The fund wants a collar with a $34 expiration floor and a $43 expiration cap. State the option positions and calculate the total expiration value if the stock finishes at $30, $39, or $48.

Method. First match the option contract coverage to the shares held; then apply the per-share collar formula and multiply by 300. The important trap is using one contract for 300 shares when each contract covers only 100.

To set the requested floor, buy puts with strike $K_P=\$34$. To set the requested cap, write calls with strike $K_C=\$43$. Since $300\div100=3$, the fund must buy three put contracts and write three call contracts. Both positions cover 300 shares, and $34<43$.

If $S_T=\$30$. Per share, the stock is worth $30 and the put pays $34-30=\$4$. The call has no payoff. The total per-share value is $30+4=\$34$, so the fund’s total value is $300\times34=\$10{,}200$.

If $S_T=\$39$. The stock finishes between the strikes, so neither option has an expiration payoff. The per-share value is $39, and the total is $300\times39=\$11{,}700$.

If $S_T=\$48$. The short call’s payoff per share is $-(48-43)=-\$5$; the put has no payoff. The total per-share value is $48-5=\$43$, and the fund’s total is $300\times43=\$12{,}900$.

The computed totals are $300\times\$34=\$10{,}200$, $300\times\$39=\$11{,}700$, and $300\times\$43=\$12{,}900$. The first and last equal the total floor and cap, respectively. This verifies both the strike choices and the contract count.

Worked Example: Selecting Strikes for a Stated Expiration Range

An investor owns 80 shares and wants the gross value of the stock-and-options position to be between $72 and $88 per share at expiration. The available options cover one share each. Construct the collar and find its total expiration value if the stock finishes at $65 or $93.

Method. Translate the desired per-share bounds directly into the put and call strikes, then check the position at the two outside prices. This avoids confusing the desired floor and cap with the stock’s current price.

Set $K_P=\$72$ by buying 80 puts, one for each share. Set $K_C=\$88$ by writing 80 calls, again one for each share. The investor holds 80 shares, so all three legs have equal coverage. The required ordering holds because $72<88$.

If $S_T=\$65$. Per share, stock value plus put payoff is $65+(72-65)=65+7=\$72$. The call payoff is zero. The total expiration value is $80\times72=\$5{,}760$.

If $S_T=\$93$. The put payoff is zero. Per share, the stock value less the short-call payoff is $93-(93-88)=93-5=\$88$. The total expiration value is $80\times88=\$7{,}040$.

The results reach the desired lower and upper bounds exactly. They are gross expiration values; they do not subtract the stock’s purchase cost or account for option premiums.

Construction Checks and Common Traps

Before using a collar formula, check the structure rather than relying on the word “collar” alone. A standard collar requires all of the following:

The investor is long the underlying stock.
The investor buys the put, rather than writes it.
The investor writes the call, rather than buys it.
The put strike is lower than the call strike.
The options have the same expiration and their coverage matches the stock position.

A sign error on the call can reverse the conclusion. Suppose the stock is worth $80 at expiration, the put strike is $60, and the call strike is $70. The put expires worthless. The written call costs the stockholder $80-70=\$10$ per share, so the correct collar value is $80-10=\$70$, the cap. The incorrect calculation $80+10=\$90$ treats the written call as if it were a long call and wrongly removes the cap.

A second trap is to call the floor a minimum profit. The floor is a bound on gross value at expiration for the matched package. To calculate profit, one would also need the initial stock outlay and option premiums, with cash flows placed at the appropriate times. Those amounts are separate from the payoff construction used here.

Key takeaway: Hold the stock, buy a put at the desired lower strike, and write a call at the desired higher strike, using matched share coverage. At expiration, the package value is bounded below by the put strike and above by the call strike.

Check Your Understanding

For each problem, identify the collar positions or calculate the requested gross expiration value. Treat option coverage as stated.

An investor holds one share, buys a put with strike $27, and writes a call with strike $36. What are the expiration floor and cap per share? (Answer: floor $27; cap $36.)
A collar has one share, a long put with strike $41, and a short call with strike $53. Find its gross expiration value per share when $S_T=\$38$. (Answer: $41.)
A portfolio holds 500 shares. Each option contract covers 100 shares. How many puts and calls are needed for a matched collar? (Answer: Buy 5 puts and write 5 calls.)
A collar consists of 120 shares, 120 long puts with strike $49, and 120 short calls with strike $62. Find its total gross expiration value if the stock finishes at $70. (Answer: $7,440.)
An investor holds 60 shares and wants a $31 per-share floor and a $40 per-share cap. Options cover one share each. State the strikes and option positions, then find the total value if the stock finishes at $35. (Answer: Buy 60 puts at strike $31 and write 60 calls at strike $40; total expiration value $2,100.)

What You'll Learn