Bear Call Spreads With Limited Risk

Introduction

A bear call spread is a two-call position designed to benefit when the underlying price stays low or falls. The trader sells a call at a lower strike and buys a call at a higher strike, with the same underlying asset and expiration date. The short call brings in premium; the long call costs premium but limits the short call’s risk at high expiration prices.

The strategy’s payoff changes across three price regions: below the lower strike, between the strikes, and above the higher strike. Calculating each option separately before combining the positions is a reliable way to track signs and see why both the gain and loss are capped.

Definition: A bear call spread consists of one short call with strike $K_1$ and one long call with strike $K_2$, where $K_1<K_2$, on the same underlying asset with the same expiration date. If the premiums per unit are $c_1$ and $c_2$, respectively, the initial net credit per unit is $C=c_1-c_2$.

Assume the contracts have matching terms and quantities, and ignore transaction costs, taxes, and early exercise. Let $S_T$ be the underlying price at expiration. The payoff of a long call with strike $K$ is $\max(S_T-K,0)$; the short call has the negative of that payoff. In this tutorial, profit at expiration means the signed option payoff plus the initial net credit, with no interest earned or charged on that credit. If a problem specifies that the credit is invested or financed, use its specified accumulated value instead.

Payoff Across the Three Price Regions

The short call creates an obligation when $S_T>K_1$. The long call creates a positive payoff when $S_T>K_2$. Because $K_1<K_2$, the options begin to have nonzero payoffs at different prices.

\text{Net option payoff at }T =-\max(S_T-K_1,0)+\max(S_T-K_2,0).

The minus sign on the first term is essential: the lower-strike call is sold, not bought. The second term is positive because the higher-strike call is owned. The region-by-region result is:

Expiration price	Short lower-strike call	Long higher-strike call	Net option payoff	Profit including credit
$S_T\le K_1$	0	0	0	$C$
$K_1<S_T<K_2$	$-(S_T-K_1)$	0	$-(S_T-K_1)$	$C-(S_T-K_1)$
$S_T\ge K_2$	$-(S_T-K_1)$	$S_T-K_2$	$-(K_2-K_1)$	$C-(K_2-K_1)$

At either strike boundary, the relevant call’s intrinsic value is zero, so the neighboring formulas agree. For example, at $S_T=K_2$, the net option payoff is $-(K_2-K_1)$, whether calculated from the middle-region expression at its boundary or from the upper-region expression.

Why the Loss Is Capped

When $S_T\ge K_2$, both calls have positive intrinsic value. The short call costs $S_T-K_1$, while the long call pays $S_T-K_2$. Subtracting the obligation from the owned call’s payoff cancels the $S_T$ terms:

$$ -(S_T-K_1)+(S_T-K_2)=K_1-K_2=-(K_2-K_1). $$

Thus, once the expiration price reaches the higher strike, a further increase does not make the spread’s option loss larger. The higher-strike call offsets the additional loss on the short call dollar for dollar. Below $K_1$, neither call pays at expiration, and the credit remains as profit. Between the strikes, the short call creates an increasing loss while the long call is still out of the money.

Theorem: Let $C=c_1-c_2$ be a positive net credit and $W=K_2-K_1$ be the strike width. Under the stated profit convention, the maximum gain per unit is $C$, attained when $S_T\le K_1$. The maximum loss per unit is $W-C$, attained when $S_T\ge K_2$, provided $0<C<W$.

The condition $0<C<W$ is the ordinary limited-risk credit-spread case: it makes the maximum loss positive and larger than zero, while the maximum gain is the credit. If a problem supplies prices outside that condition, calculate the stated cash flows rather than assuming the usual maximum-gain and maximum-loss description.

Worked Examples

Worked Example: Price Below the Lower Strike

A trader sells one call with strike $K_1=\$50$ for a premium of $\$8.40$ per share and buys one call with strike $K_2=\$60$ for $\$4.10$ per share. Both options expire on the same date. Find the spread’s profit per share if the underlying expires at $\$45$, and verify the maximum gain.

Method. First calculate the net credit, then calculate each option’s expiration payoff using the direction of the position.

The initial credit is $C=\$8.40-\$4.10=\$4.30$. Since $\$45<\$50$, both calls expire out of the money. The short call’s signed payoff is $-\max(\$45-\$50,0)=\$0$. The long call’s payoff is $\max(\$45-\$60,0)=\$0$.

The net option payoff is $\$0+\$0=\$0$, so profit is $\$4.30+\$0=\$4.30$ per share. The strike width is $\$60-\$50=\$10$, and $0<\$4.30<\$10$, so the maximum gain is the $\$4.30$ credit. The result matches because the expiration price is below the lower strike.

Common mistake avoided: Do not subtract the cost of the long call twice. It is already included when calculating the net credit of $\$4.30$.

Worked Example: Price Between the Strikes

A bear call spread has a short call at $K_1=\$72$, sold for $\$11.60$, and a long call at $K_2=\$80$, purchased for $\$7.10$. Find the profit per share if $S_T=\$76$, and determine whether the spread has a gain or a loss.

Method. Since $\$72<\$76<\$80$, use the middle region. Only the short call has positive intrinsic value.

The initial net credit is $\$11.60-\$7.10=\$4.50$. The short call’s expiration obligation is $\$76-\$72=\$4.00$, so its signed payoff is $-\$4.00$. The long call’s payoff is $\max(\$76-\$80,0)=\$0$.

The net option payoff is $-\$4.00+\$0=-\$4.00$. Profit is $\$4.50-\$4.00=\$0.50$ per share, a gain. As a check, the strike width is $\$8$, and the maximum loss would be $\$8-\$4.50=\$3.50$ per share, reached only at or above $\$80$. The calculated $\$0.50$ gain is consistent with a price between the strikes.

Common mistake avoided: Do not treat the long call as having a payoff of $\$4$. Its strike is $\$80$, so it expires worthless at $\$76$.

Worked Example: Price Above the Higher Strike

A trader sells a call at $K_1=\$40$ for $\$9.20$ and buys a call at $K_2=\$55$ for $\$3.70$. Find the profit per share at $S_T=\$63$, then verify that this equals the spread’s maximum loss.

Method. Calculate the net credit and both option payoffs. Because the expiration price exceeds both strikes, both options have positive intrinsic value.

The credit is $C=\$9.20-\$3.70=\$5.50$. The short call obligation is $\$63-\$40=\$23$, giving a signed payoff of $-\$23$. The long call payoff is $\$63-\$55=\$8$. Therefore the net option payoff is $-\$23+\$8=-\$15$, and profit is $\$5.50-\$15=-\$9.50$ per share.

The width is $W=\$55-\$40=\$15$. The maximum loss is $W-C=\$15-\$5.50=\$9.50$ per share, so the computed loss is exactly the maximum. This agreement also checks that the higher-strike call has been included.

Common mistake avoided: Ignoring the long call would give an incorrect profit of $\$5.50-\$23=-\$17.50$. The purchased call pays $\$8$, reducing the net loss to $\$9.50$.

Common Exam Traps

Reversing the legs. A bear call spread sells the lower-strike call and buys the higher-strike call. Reversing these positions changes the payoff pattern.
Using the wrong sign. The short call’s payoff is negative; the long call’s payoff is positive. Calculate each position separately before adding.
Confusing payoff and profit. The net option payoff excludes the initial premiums. Add the net credit once to get profit under the stated convention.
Assuming the long call pays below its strike. Its payoff is zero for $S_T\le K_2$, including prices between $K_1$ and $K_2$.
Forgetting the cap. Above $K_2$, the long call offsets further increases in the short call’s obligation. The net option payoff stays at $-W$, not at the short call’s full loss.

Record the positions.
Write “short $K_1$, long $K_2$” and check that $K_1<K_2$.

Find the credit.
Subtract the higher-strike call premium from the lower-strike call premium: $C=c_1-c_2$.

Select the expiration-price region.
Compare $S_T$ with both strikes before calculating intrinsic values.

Combine signed payoffs and check.
Add the short and long option payoffs, then add the credit. For a standard credit spread, check the extrema $C$ and $C-(K_2-K_1)$.

This spread is useful when a trader expects the underlying price not to rise substantially by expiration and wants a defined maximum loss rather than an uncovered short call. The trade-off is explicit: the credit caps the gain, and the difference between strike width and credit caps the loss. In an exam calculation, stating the position, region, signed payoff, and premium credit separately makes the result easier to verify.

Check Your Understanding

For each problem, calculate profit using the stated no-interest-on-credit convention.

A spread sells the $K_1=\$30$ call for $\$5.40$ and buys the $K_2=\$38$ call for $\$2.80$. What is the profit per share at $S_T=\$25$? (Answer: $\$2.60$ gain per share)
A spread sells a $K_1=\$64$ call for $\$7.80$ and buys a $K_2=\$70$ call for $\$4.60$. Find its profit per share at $S_T=\$67$. (Answer: $\$0.20$ gain per share)
A spread sells a $K_1=\$90$ call for $\$13.25$ and buys a $K_2=\$105$ call for $\$7.75$. Find the profit per share at $S_T=\$110$, and state the maximum loss per share. (Answer: $-\$9.50$ profit; maximum loss $\$9.50$ per share)
A spread sells a $K_1=\$48$ call for $\$6.20$ and buys a $K_2=\$56$ call for $\$3.10$. What is the profit per share at $S_T=\$52.50$? (Answer: $-\$1.40$ profit per share)
Two identical spreads each sell a $K_1=\$100$ call for $\$14$ and buy a $K_2=\$112$ call for $\$9$. Each option contract represents 100 shares. Find the total profit or loss at $S_T=\$118$, ignoring fees. (Answer: $\$1{,}400$ loss)

Expiration price	Short lower-strike call	Long higher-strike call	Net option payoff	Profit including credit
\(S_T\le K_1\)	0	0	0	\(C\)
\(K_1<S_T<K_2\)	\(-(S_T-K_1)\)	0	\(-(S_T-K_1)\)	\(C-(S_T-K_1)\)
\(S_T\ge K_2\)	\(-(S_T-K_1)\)	\(S_T-K_2\)	\(-(K_2-K_1)\)	\(C-(K_2-K_1)\)

What You'll Learn