Skip to content

FRM Exam Part I · Trading Strategies

Box Spreads and Butterfly Spreads for FRM Part I

Updated 11 October 2026 · Fact-checked

A box spread combines a bull call spread and a bear put spread on the same strikes and expiry, locking in a riskless payoff of K2 − K1 at expiry. A butterfly spread uses three strikes to profit when the price stays near the middle strike. Solve by computing net cost and payoff at each price range.

Understand Box Spreads and Butterfly Spreads

Spreads combine options of the same type or the same expiry to shape a payoff. Once you can build a bull call spread and a bear put spread, box and butterfly spreads are just two ways to combine them.

A box spread uses two strikes, K1 < K2, and one expiry. You buy a bull call spread (long call at K1, short call at K2) and a bear put spread (long put at K2, short put at K1). At expiry the bull call spread pays S − K1 when S is between the strikes, and the bear put spread pays the rest, so the total is always K2 − K1, whatever the final price. It is a riskless payoff, like a zero-coupon bond. With European options the fair price is the present value of K2 − K1. If the box trades below that value, you buy it and earn a riskless profit. If it trades above, you sell it.

A butterfly spread uses three equally spaced strikes K1 < K2 < K3, where K2 is the midpoint. With calls: buy one call at K1, sell two calls at K2, buy one call at K3. You pay a small net premium. The maximum payoff is K2 − K1 when the final price equals K2. The payoff is zero if the price ends at or below K1, or at or above K3. So you profit if the price stays near K2, and your loss is limited to the net premium. It is a bet on low volatility.

You can build the same butterfly with puts (buy K1 put, sell two K2 puts, buy K3 put). By put-call parity the payoff is identical. A calendar spread is different: it uses the same strike but different expiries, such as selling a near-dated option and buying a longer-dated one. A butterfly uses one expiry and three strikes.

Key formulas to remember

Box spread payoff at expiry
Payoff = K2 − K1
Same for every final price S. Bull call spread plus bear put spread on strikes K1 < K2.
Box spread no-arbitrage value (European)
Value = (K2 − K1) × e^(−rT)
Use (K2 − K1) ÷ (1 + r)^T with discrete compounding. A price away from this value signals arbitrage.
Long call butterfly payoff
Payoff = max(S − K1, 0) − 2 × max(S − K2, 0) + max(S − K3, 0)
K2 = (K1 + K3) ÷ 2. Payoff is never negative.
Butterfly maximum payoff
Max payoff = K2 − K1, at S = K2
Max profit = K2 − K1 − net premium.
Butterfly breakevens
Lower = K1 + premium; Upper = K3 − premium
Profit only between these two prices. Maximum loss is the net premium.
Butterfly net premium
Cost = c(K1) − 2 × c(K2) + c(K3)
Positive for a long butterfly, given convex option prices in strike.

How to solve Box Spreads and Butterfly Spreads questions

Use this for any box or butterfly question. Work from the payoff, then compare with the price paid.

  1. 1Identify the strategy: two strikes with calls and puts means a box; three strikes with a 1, −2, 1 pattern means a butterfly.
  2. 2List each leg: long or short, call or put, strike, and premium paid or received.
  3. 3Compute the net premium: add what you pay and subtract what you receive.
  4. 4For a box, the payoff is K2 − K1. Discount it at the risk-free rate and compare with the net premium paid.
  5. 5For a butterfly, find the payoff in each range: S ≤ K1, K1 to K2, K2 to K3, and S ≥ K3.
  6. 6Subtract the net premium from the payoff to get profit. Find the maximum profit, maximum loss and breakevens.
  7. 7State the conclusion: arbitrage and the direction to trade for a box, or the market view (low volatility) for a butterfly.

Quickest way: Shortcut: payoff shape and discounting

When to use it: Use when the question asks for maximum profit, breakeven, or whether a box is mispriced.

  1. Box: payoff is always K2 − K1. Profit = PV(K2 − K1) − price paid. No need to analyse each leg.
  2. Butterfly: maximum payoff is K2 − K1 at the middle strike. Max profit = that minus net premium.
  3. Butterfly breakevens: K1 + premium and K3 − premium.
  4. Maximum loss on a long butterfly is the net premium. It occurs at or beyond the outer strikes.
  5. Check the strikes are equally spaced before using these shortcuts.

Common mistakes in Box Spreads and Butterfly Spreads

  • Forgetting to discount the box payoff

    The payoff K2 − K1 is known, so students compare it directly with the price.

    Fix: The payoff comes at expiry. Compare the price with (K2 − K1) discounted at the risk-free rate.

  • Selling one middle call instead of two in a butterfly

    Students focus on the 'three strikes' and miss the 1, −2, 1 ratio.

    Fix: Write the pattern buy 1, sell 2, buy 1 before you start.

  • Calling a butterfly a bet on high volatility

    Confusion with straddles, which profit from large moves.

    Fix: A long butterfly profits when the price stays near K2, so it is a low-volatility position. A straddle is the opposite.

  • Mixing up calendar and butterfly spreads

    Both are called spreads and both are low-volatility bets.

    Fix: Calendar: same strike, different expiries. Butterfly: same expiry, three strikes.

  • Quoting maximum profit without subtracting the premium

    Students stop at the payoff K2 − K1.

    Fix: Profit = payoff − net premium. Maximum profit = K2 − K1 − net premium.

  • Treating a box as riskless for American options

    The riskless result is memorised without conditions.

    Fix: The exact K2 − K1 payoff holds for European options at expiry. Early exercise of American options can break it.

Worked examples

Example 1

European options on a stock expire in 1 year. A box spread uses strikes $90 and $100. The risk-free rate is 5% continuously compounded. The box can be bought for $9.30. Is there an arbitrage? Give the profit in present value terms.

Show the solution
  1. Payoff at expiry = 100 − 90 = $10.
  2. Present value = 10 × e^(−0.05) = 10 × 0.951229 = $9.5123.
  3. The box costs $9.30, which is below $9.5123, so it is underpriced.
  4. Buy the box for $9.30 (borrow to fund it) and receive $10 at expiry.
  5. Profit in present value = 9.5123 − 9.30 = $0.2123.

Answer: Yes. Buy the box; the riskless profit is about $0.21 in present value.

Example 2

Call prices on a stock with the same expiry are: $50 strike $12, $55 strike $9, $60 strike $7. You buy a butterfly: long one $50 call, short two $55 calls, long one $60 call. Find the net cost, maximum profit, and breakevens.

Show the solution
  1. Net cost = 12 − 2 × 9 + 7 = 12 − 18 + 7 = $1.
  2. Maximum payoff = 55 − 50 = $5, at S = 55.
  3. Maximum profit = 5 − 1 = $4.
  4. Lower breakeven = 50 + 1 = $51.
  5. Upper breakeven = 60 − 1 = $59.
  6. Maximum loss = net cost = $1, if S ≤ 50 or S ≥ 60.

Answer: Net cost $1; maximum profit $4 at S = $55; breakevens $51 and $59; maximum loss $1.

Exam tips

  • Memorise the two results: box payoff = K2 − K1, butterfly maximum payoff = K2 − K1 (middle minus lower). Many questions need only these.
  • Read whether the question asks for payoff or profit. Profit needs the premium subtracted.
  • For box questions, check the compounding convention and time to expiry before discounting.
  • Expect conceptual questions: which view a butterfly expresses, or how it differs from a straddle or calendar spread.
  • If the options are puts, the same shapes apply. Put-call parity gives the same payoff.

Practice questions from Trading Strategies

Box Spreads and Butterfly Spreads in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Box Spreads and Butterfly Spreads: frequently asked questions

What is the payoff of a butterfly spread?

A long call butterfly pays zero if the final price is at or below K1 or at or above K3. It rises linearly to K2 − K1 at K2. Subtract the net premium to get profit.

Why does a box spread lock in a riskless payoff?

The bull call spread and bear put spread together always pay K2 − K1 at expiry, whatever the final price. The payoff is certain, so it should be worth the present value of K2 − K1.

How do I construct a butterfly spread with calls?

Buy one call at the lowest strike, sell two calls at the middle strike, and buy one call at the highest strike. All calls share one expiry, and the strikes are equally spaced.

What is the difference between a calendar spread and a butterfly spread?

A calendar spread uses the same strike with two different expiries. A butterfly uses one expiry and three strikes in a 1, −2, 1 ratio. Both generally profit from small price moves, but their payoff profiles differ.