Spreads at expiry: add up the hockey sticks
Buy a call and sell a higher-strike call: cheaper, an earlier break-even, and a capped profit. The payoff is three straight lines, and every vertical spread and collar is read the same way.
- Compute a bull call spread's P&L at any expiry price
- State its maximum profit, maximum loss and break-even
- Compare it with the outright call and find where the two cross
- Recover the upper strike from the maximum profit
The intuition
You buy the $100 call for $6.00 and sell the $110 call for $2.50. The spread costs $3.50, against $6.00 for the call alone. Below $100 both calls expire worthless and you lose the $3.50. Between $100 and $110 the spread gains a dollar for each dollar of stock. Above $110 the call you sold pays out everything the one you bought earns, so the profit is capped at the $10 width less the $3.50 debit: $6.50.
The stock breaks even at $103.50 instead of $106.00, and a stock that goes nowhere loses $3.50 instead of $6.00. The price of that is everything above $110. If you think the stock will rise to about $110 but not much further, you have given up gains you did not expect to see. That is the whole case for a spread, and the whole case against it is the cap.
Net debit D = C(K1) − C(K2). P&L per share at expiry = min(max(S_T − K1, 0), K2 − K1) − D. Maximum loss = D, below K1. Maximum profit = (K2 − K1) − D, above K2. Break-even = K1 + D. The spread beats the outright call below K2 + C(K2), and loses to it above.
Why it works
- The conventions here: a bull call spread, long the lower strike K1 and short the upper K2, same expiry. Premiums are Black-Scholes at a zero rate, rounded to cents. Payoffs are at expiry, per share, ignoring the premium's time value. One contract is 100 shares.
- Three straight lines. Flat at −D below K1; slope one between the strikes; flat at (K2 − K1) − D above K2. Add the long call's hockey stick to the short call's inverted one and subtract the debit.
- The debit cannot exceed the width. A long call spread pays at most K2 − K1 at expiry, so it cannot cost more than that today; the maximum profit is never negative.
- Break-even moves down. The outright call needs K1 + C(K1); the spread needs only K1 + D, because the sold call paid for part of the bought one.
- Where the outright wins. Above K2 the spread is capped while the outright keeps gaining, dollar for dollar. They cross where the extra upside equals the premium collected on the short call: K2 + C(K2).
- The same reading for every structure. A collar is long stock, long put, short call. A risk reversal is short put, long call. Add the hockey sticks, then subtract or add the net premium.
| Net debit | $3.50 |
| Maximum loss, below $100 | $3.50 |
| Maximum profit, above $110: $10 − $3.50 | $6.50 |
| Break-even: $100 + $3.50 | $103.50 (the outright call: $106.00) |
| Stock at $108: min(8, 10) − 3.50 | $4.50 a share; the outright call $2.00 |
| Where the outright overtakes the spread: $110 + $2.50 | $112.50 |
On 10 contracts at $108 the spread makes $4,500 and the outright $2,000. At $120 the spread still makes $6,500; the outright makes $14,000.
The formulas
What the spread costs.
Three lines: flat, slope one, flat.
The three numbers every spread is quoted by.
The upper strike, recovered from the maximum profit.
Worked example
Value each leg at expiry, net them, subtract the debit, scale by the contracts. The follow-up places the expiry price on the diagram.
See it move
Same spread. Move the stock at expiry, widen the strikes, and change the volatility and days that set the premiums.
- Move the stock up. The spread's P&L rises dollar for dollar between the strikes and then stops; the outright call keeps rising.
- Widen the strikes. The upper strike, the maximum profit, the debit and the break-even all rise; the premium collected on the short call falls.
- Raise the volatility or the days. Both premiums rise; watch the debit, which can go either way because both legs move.
Run it backwards
A colleague paid a known debit for a spread with a known lower strike and says the most it can make is a stated amount. What is the upper strike?
Maximum profit = width − debit, so the width is maximum profit plus debit, and the upper strike is the lower strike plus the width.
The three quoted numbers of any spread, debit, maximum profit and break-even, each recover a strike or the other number.
Traps
Say it in the interview
“Why would you buy a call spread instead of the call?”
Check yourself
5 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- Debit = C(K1) − C(K2). Max loss = debit; max profit = width − debit; break-even = K1 + debit.
- Three lines: flat, slope one, flat. Add the hockey sticks.
- The spread beats the outright below K2 + C(K2) and loses above it.
- Choose the spread when you do not expect the upside you are selling.