Why Breakpoints Matter
A vertical spread combines options of the same type and expiration date but different strike prices. Its payoff changes formula at the strikes: below one strike, neither option may pay; between strikes, one option may pay; beyond both strikes, both options may contribute. Writing the function by these regions makes the spread’s shape, maximum gain, maximum loss, and break-even price easier to verify.
A key distinction is between payoff and profit. Payoff is determined by the option positions at expiration. Profit also accounts for the premium paid or received at the start and, if the problem asks for a terminal profit, the value of that initial cash flow at expiration.
For all examples on this page, assume European-style options held to expiration, no transaction costs or taxes, and a deterministic effective annual interest rate \(i\) for \(T\) years. If a spread costs \(D\) at time 0, its terminal financing cost is \(D(1+i)^T\). If it generates a credit \(C\) at time 0, the credit’s terminal value is \(C(1+i)^T\). Thus terminal profit equals payoff minus the carried cost, or payoff plus the carried credit. If a question instead defines profit without interest on the initial premium, use that convention rather than adding a financing adjustment.
Call Spreads: Build the Function at Each Strike
For a call with strike \(K\), the expiration payoff to its holder is \(\max(S_T-K,0)\). A bull call spread buys the lower-strike call and sells the higher-strike call. A bear call spread does the reverse. On either spread, the payoff is zero below \(K_L\), changes linearly between the strikes, and is constant once \(S_T\) is at or above \(K_H\).
The bull call’s payoff rises from zero to the strike width \(K_H-K_L\); the bear call’s payoff falls from zero to the negative of that width. These payoff limits do not yet account for the initial premium.
Worked Example: Bull Call Spread With a Financed Debit
A hypothetical one-year bull call spread buys a call with strike \(K_L=40\) for a premium of $8 and sells a call with strike \(K_H=55\) for $3. The effective annual interest rate is \(i=4\%\). Find its piecewise payoff, terminal profit, break-even price, maximum gain, and maximum loss.
Method and financing. The initial net debit is \(D=8-3=\$5\). Carried for one year, it is \(5(1.04)^1=\$5.20\). The payoff breakpoints are 40 and 55.
Payoff by region. For \(S_T\le40\), both calls expire without value, so payoff is $0. For \(40<S_T<55\), only the long call has positive value, giving payoff \(S_T-40\). For \(S_T\ge55\), the difference between call values is \(55-40=\$15\).
The break-even must be inside the middle region. There, set \(S_T-40-5.20=0\), giving \(S_T=45.20\), which is between 40 and 55. Below 40, profit is \(-\$5.20\), the maximum loss. At or above 55, profit is \(15-5.20=\$9.80\), the maximum gain. Checking at \(S_T=50\), payoff is \(50-40=\$10\) and profit is \(10-5.20=\$4.80\), consistent with the increasing profit in the middle region.
Common mistake avoided: The payoff break-even is 40, where the long call begins to have value, but the profit break-even is 45.20 because the carried debit must also be recovered.
Put Spreads: Reverse Which Side of the Strikes Pays
A put with strike \(K\) pays its holder \(\max(K-S_T,0)\) at expiration. A bull put spread sells the higher-strike put and buys the lower-strike put; it is typically opened for a credit. A bear put spread buys the higher-strike put and sells the lower-strike put; it is typically opened for a debit.
The bull put payoff is bounded below by \(-(K_H-K_L)\) and above by zero. The bear put payoff is bounded below by zero and above by the strike width. The premium then shifts each payoff function up or down to produce profit.
Worked Example: Bear Put Spread With a Financed Debit
A hypothetical one-year bear put spread buys a put with strike \(K_H=55\) for $9 and sells a put with strike \(K_L=40\) for $3. The effective annual interest rate is \(i=5\%\). Find the payoff and profit by region, along with the break-even price and profit limits.
Method and financing. The net debit is \(D=9-3=\$6\), so its terminal value is \(6(1.05)=\$6.30\). The strike width is \(55-40=\$15\).
For \(S_T\le40\), both puts are in the money and their values differ by the strike width, so the spread payoff is $15. For \(40<S_T<55\), only the long 55 put pays, giving \(55-S_T\). For \(S_T\ge55\), neither put pays.
In the middle region, set \(55-S_T-6.30=0\). This gives \(S_T=48.70\), which lies between the strikes. At or below 40, profit is \(15-6.30=\$8.70\), the maximum gain. At or above 55, profit is \(-\$6.30\), the maximum loss. As a check, at \(S_T=50\), the payoff is \(55-50=\$5\), so profit is \(5-6.30=-\$1.30\); this is below break-even, as expected for a bear put spread.
Common mistake avoided: The higher-strike put is long and the lower-strike put is short. Reversing these positions changes the spread’s direction and the signs of its payoff.
Break-Even and Profit Limits for All Four Spreads
Once the payoff has been written by region, profit follows by adding or subtracting the carried initial premium. The break-even price is found by setting the profit—not the option payoff—to zero in the region where the zero occurs. The table summarizes the usual debit or credit orientation; \(W=K_H-K_L\), and the carried premium amounts are measured at expiration.
| Spread | Typical initial cash flow | Maximum gain | Maximum loss | Break-even price |
|---|---|---|---|---|
| Bull call | Debit \(D\) | \(W-D(1+i)^T\) | \(D(1+i)^T\) | \(K_L+D(1+i)^T\) |
| Bear call | Credit \(C\) | \(C(1+i)^T\) | \(W-C(1+i)^T\) | \(K_L+C(1+i)^T\) |
| Bull put | Credit \(C\) | \(C(1+i)^T\) | \(W-C(1+i)^T\) | \(K_H-C(1+i)^T\) |
| Bear put | Debit \(D\) | \(W-D(1+i)^T\) | \(D(1+i)^T\) | \(K_H-D(1+i)^T\) |
The table presumes ordinary, nonnegative net credits or debits and a break-even lying between the strikes. For example, the bull call break-even formula comes from the middle-region profit \(S_T-K_L-D(1+i)^T\). It is valid there only if \(K_L+D(1+i)^T\) lies between \(K_L\) and \(K_H\). If it does not, do not apply the middle-region formula blindly: use the piecewise profit function and determine whether a zero exists in any region. Likewise, the stated gain and loss formulas presume the premium does not exceed the spread width in a way that reverses the usual endpoint ordering.
The bull put break-even follows from middle-region profit \(S_T-K_H+C(1+i)^T\), and the bear put break-even from \(K_H-S_T-D(1+i)^T\). For a bear call, middle-region profit is \(K_L-S_T+C(1+i)^T\). These equations also show why the vertical distance between strikes limits the option payoff, while the initial premium determines how much of that payoff remains as profit.
Derivation: Why the Piecewise Call Formula Works
The bull call spread holds one long call at \(K_L\) and one short call at \(K_H\). Its payoff is the long call payoff minus the higher-strike call payoff. Evaluate that expression in each region determined by the strikes:
If \(S_T\le K_L\), then \(S_T-K_L\le0\) and \(S_T-K_H\le0\). Both positive-part payoffs are zero, so the spread payoff is zero.
If \(K_L<S_T<K_H\), then \(S_T-K_L>0\) but \(S_T-K_H<0\). Only the lower-strike call pays, so the spread payoff is \(S_T-K_L\).
If \(S_T\ge K_H\), both calls pay. Subtracting the short call’s payoff from the long call’s payoff gives \((S_T-K_L)-(S_T-K_H)=K_H-K_L\), a constant.
The same region-by-region method proves the other spread formulas: substitute the long and short positions into \(\max(K-S_T,0)\) for puts, and evaluate on the intervals \(S_T\le K_L\), \(K_L<S_T<K_H\), and \(S_T\ge K_H\). At a strike, the adjacent formulas agree, so the payoff is continuous.
Exam Checks and Common Traps
For a spread with one option long and one short, a reliable solution starts by recording the positions, not just the names “bull” or “bear.” Then mark both strikes, write the payoff in each region, and only afterward add the premium’s terminal value. This order prevents the most frequent errors:
- Using payoff as profit. A zero payoff does not mean zero profit when a debit or credit was exchanged initially.
- Reversing the short leg. A bull call buys the lower strike and sells the higher; a bear call sells the lower and buys the higher. For puts, identify the actual long and short strikes directly.
- Ignoring the capped region. Beyond both strikes, both options may be in the money, but their values offset except for the fixed strike width.
- Using a break-even outside its region. Solve in the sloping segment, then verify the resulting price lies between the strikes. Otherwise inspect the full piecewise profit.
- Mixing annual rates and periods. If the premium is carried for \(T\) years at an effective annual rate \(i\), use \((1+i)^T\), not \(1+iT\) unless simple interest is explicitly specified.
The vertical spread’s bounded payoff is useful for comparing an option position’s direction and range of outcomes. In exam work, a quick endpoint check is especially effective: evaluate profit below both strikes and above both strikes, then compare those values with the slope and the break-even in the middle. The resulting checks should agree with the positions and the premium.
Check Your Understanding
For each problem, treat the stated premiums as time-0 amounts and compute terminal profit using the stated effective annual rate. Assume no transaction costs or taxes.
- A one-year bull call spread buys the 30-strike call for $6 and sells the 45-strike call for $2. The effective annual rate is 5%. Find the break-even price, maximum gain, and maximum loss. (Answer: Break-even \(34.20\); maximum gain \(\$10.80\); maximum loss \(\$4.20\).)
- A one-year bear call spread sells the 70-strike call for $8 and buys the 85-strike call for $2. The effective annual rate is 5%. Find its terminal profit below both strikes, above both strikes, and its break-even price. (Answer: Below 70, \(\$6.30\); at or above 85, \(-\$8.70\); break-even \(76.30\).)
- A one-year bull put spread sells the 50-strike put for $8 and buys the 40-strike put for $3. The effective annual rate is 4%. Find its maximum gain, maximum loss, and break-even price. (Answer: Maximum gain \(\$5.20\); maximum loss \(\$4.80\); break-even \(44.80\).)
- A one-year bear put spread buys the 80-strike put for $11 and sells the 65-strike put for $4. The effective annual rate is 3%. Find the break-even price, maximum gain, and maximum loss. (Answer: Break-even \(72.79\); maximum gain \(\$7.79\); maximum loss \(\$7.21\).)
- A one-year bull call spread buys the 25-strike call for $5 and sells the 35-strike call for $2. The effective annual rate is zero. Find its profit at expiration prices 20, 30, and 40, and identify the break-even price. (Answer: Profits are \(-\$3\), \(\$2\), and \(\$7\), respectively; break-even \(28\).)