Zero-Cost Collar Strike Selection

What “Zero-Cost” Means for a Collar

A collar combines a long position in the underlying asset, a long put, and a short call with the same expiration date. Selecting strikes for a zero-cost collar means choosing a put strike and a call strike so that the put premium paid equals the call premium received. The option transactions then have no net initial premium. The stock still has to be purchased, so the whole stock-and-options position is not costless.

Strike selection uses the premiums available for the possible options, not just the distance between each strike and the current stock price. A put with a higher strike generally costs more, while a call with a higher strike generally costs less, but an exact match depends on the actual quoted premiums. There may be no matching pair among the available strikes.

Definition: A zero-cost collar, in the premium sense, is a long position in one share of stock, a long put with strike $K_P$, and a short call with strike $K_C$, where both options have the same expiration and the initial put premium equals the initial call premium. The option overlay has zero net premium when $P_0(K_P)=C_0(K_C)$. A conventional collar has $K_P\leq K_C$, so the put strike provides a floor and the call strike provides a cap at expiration.

Here $P_0(K_P)$ denotes the put premium per share and $C_0(K_C)$ denotes the call premium per share. The short call contributes a premium received; the long put requires a premium paid. Thus the option overlay’s initial cash flow per share is $-P_0(K_P)+C_0(K_C)$. Matching the two premiums makes this amount zero. If the options cover a different number of shares, their total premiums—not merely their per-share quotes—must be compared.

Why the Expiration Value Has a Floor and a Cap

Assume the collar is held to the options’ common expiration, the position is one share of stock plus one put and one written call per share, and $K_P\leq K_C$. Let $S_T$ be the stock price at expiration. The stock is worth $S_T$; the put pays $\max(K_P-S_T,0)$; and the written call contributes $-\max(S_T-K_C,0)$. Therefore the total expiration value per share is

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

The formula can be simplified by considering the three possible price regions. If $S_T<K_P$, the put is worth $K_P-S_T$ and the call expires without value, so $V_T=S_T+(K_P-S_T)=K_P$. If $K_P\leq S_T\leq K_C$, both options expire without value and $V_T=S_T$. If $S_T>K_C$, the put expires without value and the written call costs $S_T-K_C$, so $V_T=S_T-(S_T-K_C)=K_C$. Consequently,

V_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}

This piecewise result explains the collar’s price limits: its expiration value per share cannot be below $K_P$ or above $K_C$. These are limits on the combined stock-and-options value at expiration, not necessarily limits on profit. To find profit, subtract the original stock cost and add or subtract any relevant cash flows. In a zero-cost collar, the matched option premiums cancel at initiation, but the purchase price of the stock does not.

Result: When $K_P\leq K_C$ and the matched options cover the same shares for the same expiration, the collar’s expiration value is the stock price restricted to the interval from $K_P$ to $K_C$. Below the put strike the value is $K_P$; between the strikes it is $S_T$; above the call strike it is $K_C$.

Selecting Strikes from Premium Quotes

A practical selection process is to list the available put strikes and call strikes, compare premiums on the same per-share or total-contract basis, and identify pairs for which the paid put premium equals the received call premium. After finding a match, confirm the strikes have the desired order and calculate the resulting expiration limits. The premium match establishes the zero initial option cash flow; it does not establish that the collar is suitable for every investor or that it eliminates investment risk.

Set the contract basis.
Confirm that the put and call have the same underlying, expiration, and coverage. Convert premiums to the same per-share or total-dollar basis.

Find an equal-premium pair.
For each candidate pair, calculate put premium paid minus call premium received. A zero result is the premium-matching condition.

Check the collar structure.
Verify $K_P\leq K_C$ for the usual floor-and-cap collar, then use those strikes to determine the expiration value in each price region.

Keep option cash flow separate from stock profit.
Zero net option premium does not mean zero stock cost. Subtract the stock basis separately if the question asks for profit or loss.

Worked Example: Matching a Put and Call for One Share

A hypothetical investor buys one share for $100$. For the same expiration, a put with strike $95$ costs $6$ per share and a call with strike $105$ can be sold for $6$ per share. The other quoted choices are a $90$-strike put at $4$ and a $110$-strike call at $4$. Select a zero-cost collar using the $95$-strike put, and calculate the expiration value and profit if $S_T$ is $88$, $102$, or $121$.

Method. Compare the premiums paid and received, then apply the piecewise expiration-value formula. The $95$-strike put costs $6$, and the $105$-strike call brings in $6$, so the option net premium is

$$ -6+6=0. $$

The collar has $K_P=95$ and $K_C=105$, with $95\leq105$. The stock cost is still $100$ per share, so profit is expiration value minus $100$.

If $S_T=88$: Since $88<95$, the put is worth $95-88=7$ and the call expires without value. The total expiration value is $88+7=95$. Profit is $95-100=-5$.

If $S_T=102$: Since $95\leq102\leq105$, both options expire without value. The expiration value is $102$, and profit is $102-100=2$.

If $S_T=121$: Since $121>105$, the written call costs $121-105=16$ and the put expires without value. The expiration value is $121-16=105$, and profit is $105-100=5$.

The expiration values are therefore $95$, $102$, and $105$, all within the collar’s floor and cap. The profits are $-5$, $2$, and $5$. This check confirms that the option overlay’s zero premium does not remove the downside between the stock purchase price and the put strike.

Comparing Candidate Strike Pairs

When several strikes are quoted, equal premiums are the selection criterion for a zero-cost overlay; equal distances from the current stock price are not. In practice, an investor may compare more than one matching pair if several exist. Each pair gives a different floor and cap. The premium condition alone does not rank those alternatives: the investor must interpret the resulting limits in light of the desired trade-off.

Worked Example: Finding a Match in a Strike Table

A hypothetical stock is priced at $76$. For one expiration, the available put quotes are: strike $68$, premium $1$; strike $70$, premium $2$; strike $72$, premium $3.50$; and strike $74$, premium $5.50$. The call quotes are: strike $78$, premium $7$; strike $80$, premium $5.50$; strike $82$, premium $3.50$; and strike $84$, premium $2$. Find a zero-cost collar among these quotes and calculate the expiration values if the stock ends at $65$, $79$, or $90$. The stock was purchased for $76$.

Method. Match a put premium to a call premium, then verify the strike order. The put with strike $72$ costs $3.50$; the call with strike $82$ can be sold for $3.50$. The net option premium is $-3.50+3.50=0$. The strikes satisfy $72\leq82$, so this is a conventional collar with floor $72$ and cap $82$.

At $S_T=65$, the put pays $72-65=7$, and the call expires without value. The combined expiration value is $65+7=72$; profit relative to the $76$ stock cost is $72-76=-4$.

At $S_T=79$, the price is between the strikes. Neither option has a positive expiration payoff, so the value is $79$ and the profit is $79-76=3$.

At $S_T=90$, the written call costs $90-82=8$ and the put expires without value. The combined expiration value is $90-8=82$; profit is $82-76=6$.

The resulting expiration values are $72$, $79$, and $82$, consistent with the selected floor and cap. The equal-distance idea would not identify the pair by itself; the premium table does.

Contract Multipliers and Total Premiums

Option quotes are often stated per share while one contract covers a specified number of shares. The zero-cost condition must be checked for the actual number of contracts. If a position has equal numbers of put and call contracts with the same multiplier, matching per-share premiums is sufficient. If quantities or multipliers differ, compare total dollars paid and received. A per-share match does not guarantee zero net premium when the position sizes differ.

Worked Example: Verifying a Match for Multiple Contracts

A hypothetical fund owns $300$ shares purchased for $64$ per share. Each option contract covers $100$ shares, so the fund uses three put contracts and writes three call contracts. For a common expiration, the put with strike $62$ costs $4.10$ per share and the call with strike $66$ can be sold for $4.10$ per share. Calculate the net option premium and the portfolio’s expiration value and profit if the stock ends at $58$, $63$, or $75$.

Method. Match both the per-share premiums and the covered share counts. The fund pays $4.10\times300=\$1{,}230$ for the puts and receives $4.10\times300=\$1{,}230$ for the calls. The net option premium is $\$1{,}230-\$1{,}230=\$0$. The stock basis is $64\times300=\$19{,}200$, and the collar strikes are $K_P=62$ and $K_C=66$.

At $S_T=58$, each share has a put payoff of $62-58=4$. The collar value per share is $58+4=62$; the portfolio value is $62\times300=\$18{,}600$. Profit is $\$18{,}600-\$19{,}200=-\$600$.

At $S_T=63$, the stock price lies between the strikes and both options expire without value. The portfolio value is $63\times300=\$18{,}900$, and profit is $\$18{,}900-\$19{,}200=-\$300$.

At $S_T=75$, each written call costs $75-66=9$ per share at expiration. The collar value per share is $75-9=66$; the portfolio value is $66\times300=\$19{,}800$. Profit is $\$19{,}800-\$19{,}200=\$600$.

The expiration portfolio values of $\$18{,}600$, $\$18{,}900$, and $\$19{,}800$ correspond to per-share values of $62$, $63$, and $66$. All fall within the selected limits. The dollar calculation also confirms that the three put contracts and three call contracts have exactly offsetting premiums.

Common Strike-Selection Errors

A frequent error is reversing the direction of the call cash flow. The investor pays the put premium but receives the premium for writing the call. If the put costs $4.10$ per share and the call receives $3.20$, the net option cash flow is $-4.10+3.20=-0.90$ per share: a debit of $0.90$, not a credit. Calling this pair zero-cost because the two quoted amounts are “close” is incorrect.

Another error is matching strikes based on symmetry around the current stock price. A put $5$ below the stock and a call $5$ above it need not have equal premiums. The correct test is equality of premiums for the actual quotes and position sizes. Finally, even after premiums match, do not call the total position costless: the stock purchase remains an initial cash outflow.

Key takeaway: A zero-cost collar is identified by equal total put premiums paid and call premiums received. Once the strikes are selected, the expiration value is bounded below by the put strike and above by the call strike, while profit must still account for the stock basis.

Check Your Understanding

For each problem, identify the premium condition and use the expiration-value regions carefully.

A hypothetical stock is at $72$. A $68$-strike put costs $2.40$, and a $78$-strike call can be sold for $2.40$. What is the net option premium per share, and what are the collar expiration values when $S_T=60$, $74$, and $83$? (Answer: Net option premium $\$0$; values $\$68$, $\$74$, and $\$78$, respectively.)
A stock was purchased for $80$. A $75$-strike put costs $5$, and an $85$-strike call receives $5$. Find the expiration profit per share when $S_T=70$, $82$, and $96$. (Answer: $-\$5$, $\$2$, and $\$5$, respectively.)
A hypothetical stock is at $80$. Available puts are: strike $70$, premium $2$; strike $74$, premium $4$; strike $76$, premium $5.50$. Available calls are: strike $84$, premium $5.50$; strike $88$, premium $3$; strike $90$, premium $2$. Select a zero-cost pair with $K_P\leq K_C$, and give its expiration values when $S_T=72$, $85$, and $95$. (Answer: $K_P=76$, $K_C=84$; values $\$76$, $\$84$, and $\$84$, respectively.)
An investor owns $200$ shares and buys two put contracts covering $100$ shares each at a premium of $3.60$ per share. The investor writes two calls with the same multiplier and receives $3.25$ per share. Calculate the net option premium in dollars and state whether the overlay is zero-cost. (Answer: $\$70$ net debit; it is not zero-cost.)
A share is bought for $75$. A put with strike $80$ costs $6$, and a call with strike $85$ receives $6$. Find the net option premium and the combined position’s expiration values when $S_T=70$, $82$, and $91$. (Answer: Net option premium $\$0$; values $\$80$, $\$82$, and $\$85$, respectively.)

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