Maximum Gain And Loss In Vertical Spreads

Why Maximum Gain and Loss Depend on Both Strikes and Premium

A vertical spread combines options of the same type and expiration but different strikes. Its expiration payoff is limited by the distance between those strikes. Its profit, however, also depends on the net premium paid or received to establish the position. To find maximum gain and maximum loss correctly, keep those two ingredients separate: first determine the spread’s payoff, then account for its premium.

The methods below apply to standard one-unit vertical spreads held to expiration. They are aligned with the SOA Exam FM (Financial Mathematics) syllabus. We assume the options have the same underlying and expiration, strikes satisfy $K_1<K_2$, and transaction costs and taxes are ignored. We use the conventional option-profit measure: expiration payoff less the net premium, with financing of that premium ignored. If a problem instead asks for terminal wealth including interest on the initial premium, carry that premium forward at the stated rate before subtracting it.

Definition: The strike width is $W=K_2-K_1$. A net debit is the premium paid for the long option minus the premium received for the short option. A net credit is the premium received for the short option minus the premium paid for the long option. A spread’s profit at expiration is its signed option payoff at expiration minus its net debit, or plus its net credit.

A General Method for Finding the Extremes

Let $S_T$ be the underlying price at expiration. In the interval between the strikes, a vertical spread’s payoff changes one-for-one with $S_T$. Outside that interval, both options are either out of the money or the spread has reached its capped payoff. Therefore, profit is piecewise linear, and its extreme values occur in the flat regions outside the strikes—not at an unbounded price.

For a debit spread, let $D$ be the net debit and $W$ the strike width. The spread’s payoff ranges from 0 to $W$. Subtracting $D$ gives a maximum loss of $D$ and a maximum gain of $W-D$. For a credit spread, let $C$ be the net credit. Its profit ranges from $C$ down to $C-W$, so its maximum gain is $C$ and its maximum loss is $W-C$.

Spread	Premium type	Maximum gain and region	Maximum loss and region
Bull call: long $K_1$, short $K_2$	Debit $D$	$W-D$, when $S_T\ge K_2$	$D$, when $S_T\le K_1$
Bear put: long $K_2$, short $K_1$	Debit $D$	$W-D$, when $S_T\le K_1$	$D$, when $S_T\ge K_2$
Bear call: short $K_1$, long $K_2$	Credit $C$	$C$, when $S_T\le K_1$	$W-C$, when $S_T\ge K_2$
Bull put: short $K_2$, long $K_1$	Credit $C$	$C$, when $S_T\ge K_2$	$W-C$, when $S_T\le K_1$

At the strike boundaries, the adjoining formulas agree. For example, a bull call spread has the same maximum loss at $S_T=K_1$ as at every lower price, and the same maximum gain at $S_T=K_2$ as at every higher price. Thus, when a question asks where an extreme occurs, include the boundary in the relevant region.

Why the Width-and-Premium Formulas Hold

The formulas follow directly from the expiration payoffs of the options. For a bull call spread, one call is long at $K_1$ and one is short at $K_2$. Its signed payoff is zero below $K_1$, increases from zero to $K_2-K_1$ between the strikes, and is capped at $K_2-K_1$ above $K_2$. Subtracting the debit shifts every payoff down by $D$.

\text{Bull call profit}= \begin{cases} -D, & S_T\le K_1,\\ S_T-K_1-D, & K_1<S_T<K_2,\\ W-D, & S_T\ge K_2. \end{cases}

The middle expression increases as $S_T$ increases. Its value at $K_1$ is $-D$, and at $K_2$ it is $W-D$. It cannot be below $-D$ or above $W-D$ between those endpoints. This proves that the maximum loss is $D$ below or at $K_1$, and the maximum gain is $W-D$ above or at $K_2$. The other spreads have the same two possible payoff levels, but their profitable price direction differs according to which option is long and which is short.

Identify the position.
Record which strike is long and which is short. Do not infer the spread type from its name alone.

Calculate width and premium.
Compute $W=K_2-K_1$, then calculate the net debit or net credit using the position signs.

Find the capped payoffs.
A debit spread loses its debit when both options expire worthless and gains width less debit when its payoff reaches the full width. A credit spread gains its credit when both options expire worthless and loses width less credit when its liability reaches the full width.

State the price regions and check.
Use the spread’s direction to place each extreme on the correct side of the strikes. Check the endpoints and confirm the result is consistent with the width and premium.

Worked Applications

Worked Example: Maximum Gain and Loss on a Bull Call Spread

A trader buys a call with strike $42$ for a premium of $\$6.40$ and sells a call with strike $50$ for $\$2.10$. Both options have the same expiration. Find the maximum gain, maximum loss, and the regions where they occur. Also find the profit when $S_T=\$47$.

Method: calculate the debit and strike width, then evaluate the piecewise profit. The net debit is $\$6.40-\$2.10=\$4.30$. The width is $50-42=\$8.00$.

For $S_T\le42$, both calls expire worthless. The spread payoff is zero, so profit is $0-\$4.30=-\$4.30$. This is the maximum loss, and it occurs at every price at or below $\$42$.

For $42<S_T<50$, only the lower-strike call has positive payoff. Profit is $(S_T-42)-4.30=S_T-46.30$. At $S_T=47$, the payoff is $\$47-\$42=\$5.00$, and profit is $\$5.00-\$4.30=\$0.70$.

For $S_T\ge50$, the long call’s payoff exceeds the short call’s payoff by the full width, $\$8.00$. Profit is $\$8.00-\$4.30=\$3.70$. This is the maximum gain, occurring at every price at or above $\$50$.

Check: At $\$42$, the middle expression approaches a loss of $\$4.30$; at $\$50$, it gives a gain of $\$3.70$. The example’s maximum loss equals the debit, and maximum gain equals width less debit.

Worked Example: Maximum Gain and Loss on a Bull Put Spread

A trader sells a put with strike $72$ for $\$5.60$ and buys a put with strike $65$ for $\$2.35$. Determine the profit limits and their underlying-price regions. Find the profit if the underlying finishes at $\$68$.

Method: treat the position as a credit spread and subtract the put-spread liability from the credit. The net credit is $\$5.60-\$2.35=\$3.25$, and the strike width is $72-65=\$7.00$.

When $S_T\ge72$, both puts expire worthless. The trader keeps the $\$3.25$ credit, so this is the maximum gain, occurring at or above $\$72$.

When $65<S_T<72$, only the short put has a positive payoff. The spread liability is $72-S_T$, so profit is $3.25-(72-S_T)=S_T-68.75$. At $S_T=68$, the liability is $\$72-\$68=\$4.00$, giving profit $\$3.25-\$4.00=-\$0.75$.

When $S_T\le65$, both puts are in the money. The long lower-strike put offsets part of the short put’s liability, leaving a net liability equal to the width, $\$7.00$. Profit is $\$3.25-\$7.00=-\$3.75$. This is the maximum loss, occurring at or below $\$65$.

Check: The maximum loss is width less credit, $\$7.00-\$3.25=\$3.75$, and the profit at $\$68$ lies between that loss and the maximum gain.

Worked Example: Maximum Gain and Loss on a Bear Call Spread

A trader sells a call with strike $100$ for $\$8.20$ and buys a call with strike $112$ for $\$3.70$. Both options expire on the same date. Find the maximum gain and loss, their price regions, and the profit at $S_T=\$106$.

Method: calculate the net credit, then subtract the spread’s liability at expiration. The credit is $\$8.20-\$3.70=\$4.50$, and the width is $112-100=\$12.00$.

For $S_T\le100$, both calls expire worthless, so the trader keeps the credit. The maximum gain is $\$4.50$, occurring at or below $\$100$.

For $100<S_T<112$, the short call is in the money and the long call is not. The liability is $S_T-100$, so profit is $4.50-(S_T-100)=104.50-S_T$. At $S_T=106$, the liability is $\$6.00$, and profit is $\$4.50-\$6.00=-\$1.50$.

For $S_T\ge112$, both calls are in the money, but the long call offsets the short call above its strike. The net liability is the full width, $\$12.00$, so profit is $\$4.50-\$12.00=-\$7.50$. This is the maximum loss, occurring at or above $\$112$.

Check: Width less credit is $\$12.00-\$4.50=\$7.50$, matching the maximum loss. The profit declines across the middle region from $\$4.50$ to $-\$7.50$.

Common Mistake: Treating Width as the Loss

The strike width is the spread’s largest possible expiration payoff or liability, not automatically its maximum profit or maximum loss. Premium changes both profit limits. For a debit spread, writing the width as the maximum loss is incorrect: the maximum loss is the debit. For a credit spread, writing the entire width as the loss is also incorrect: the credit offsets part of that liability.

For example, in the bear call spread above, an incorrect calculation would report a maximum loss of $\$12.00$, equal to the width. The correct calculation is $\$12.00-\$4.50=\$7.50$, because the trader received $\$4.50$ at establishment. Similarly, the maximum gain is the credit, $\$4.50$, not the width. Always label the premium as a debit or credit before applying the formulas.

Key takeaway: A debit spread has maximum loss equal to its debit and maximum gain equal to strike width less debit. A credit spread has maximum gain equal to its credit and maximum loss equal to strike width less credit. The spread’s option positions determine which side of the strikes contains each extreme.

Check Your Understanding

For each spread, identify the maximum gain and loss and the regions where they occur. Use the conventional expiration-profit measure stated above.

A bull call spread buys the $30$ call for $\$7.10$ and sells the $38$ call for $\$2.60$. Give its maximum gain and loss with their regions. (Answer: Maximum gain $\$3.50$ when $S_T\ge38$; maximum loss $\$4.50$ when $S_T\le30$.)
A bear put spread buys the $90$ put for $\$9.40$ and sells the $80$ put for $\$3.10$. Give its maximum gain and loss with their regions. (Answer: Maximum gain $\$3.70$ when $S_T\le80$; maximum loss $\$6.30$ when $S_T\ge90$.)
A bull put spread sells the $55$ put for $\$4.80$ and buys the $48$ put for $\$2.05$. Give its maximum gain and loss with their regions. (Answer: Maximum gain $\$2.75$ when $S_T\ge55$; maximum loss $\$4.25$ when $S_T\le48$.)
A bear call spread sells the $120$ call for $\$10.20$ and buys the $130$ call for $\$6.10$. State its profit limits and the profit when $S_T=126$. (Answer: Maximum gain $\$4.10$ when $S_T\le120$; maximum loss $\$5.90$ when $S_T\ge130$; profit at $126$ is $\$4.10-\$6.00=-\$1.90$.)
A bear call spread has strikes $95$ and $110$, and its net credit is $\$6.25$. Find its maximum loss and its profit at $S_T=107$. (Answer: Maximum loss $\$8.75$ when $S_T\ge110$; at $107$, the liability is $\$107-\$95=\$12.00$, so profit is $\$6.25-\$12.00=-\$5.75$.)

Spread	Premium type	Maximum gain and region	Maximum loss and region
Bull call: long \(K_1\), short \(K_2\)	Debit \(D\)	\(W-D\), when \(S_T\ge K_2\)	\(D\), when \(S_T\le K_1\)
Bear put: long \(K_2\), short \(K_1\)	Debit \(D\)	\(W-D\), when \(S_T\le K_1\)	\(D\), when \(S_T\ge K_2\)
Bear call: short \(K_1\), long \(K_2\)	Credit \(C\)	\(C\), when \(S_T\le K_1\)	\(W-C\), when \(S_T\ge K_2\)
Bull put: short \(K_2\), long \(K_1\)	Credit \(C\)	\(C\), when \(S_T\ge K_2\)	\(W-C\), when \(S_T\le K_1\)

What You'll Learn