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FRM Exam Part I · Properties of Options

Option Trading Strategies and Spreads for FRM Part I

Updated 11 October 2026 · Fact-checked

Option strategies combine a position in the underlying with options, or combine several options, to shape the payoff. To solve a question, list each leg, add the payoffs at expiry, subtract the net premium paid or add the premium received, then read off profit, breakeven and maximum loss.

Understand Option Trading Strategies and Spreads

A single option gives you a simple payoff: limited loss for the buyer and one-sided gain. Strategies let you reshape that payoff to match a view on price level or volatility.

There are three families. Stock plus option strategies change the risk of a share you hold. A covered call is long stock plus a short call. A protective put is long stock plus a long put. Spreads use two or more options of the same type (all calls or all puts) with different strikes or maturities. Combinations mix calls and puts, such as straddles and strangles.

A bull spread profits when the price rises, but both gain and loss are capped. With calls, you buy the low-strike call and sell the high-strike call. It costs money up front. A bear spread is the mirror image: it profits when the price falls. With puts, you buy the high-strike put and sell the low-strike put. A bull spread with calls and a bear spread with calls are mirror images of each other, so the sign of the payoff flips.

A butterfly spread bets on low volatility. With calls: buy one call at the low strike K1, sell two calls at the middle strike K2, buy one call at the high strike K3, where K2 is halfway between K1 and K3. It pays most if the price ends at K2. A straddle is a long call and a long put at the same strike. It profits from a large move in either direction. A strangle does the same with an out-of-the-money put and call at different strikes. It is cheaper than a straddle but needs a bigger move to profit.

In every case, the payoff at expiry is the sum of the legs. Profit is payoff minus the net cost. Draw the legs, add them, and the shape appears.

Key formulas to remember

Covered call profit at expiry
Profit = (S_T − S_0) − max(S_T − K, 0) + c
Long stock, short call with strike K, premium c received. Maximum profit = K − S_0 + c. Breakeven = S_0 − c.
Protective put profit at expiry
Profit = (S_T − S_0) + max(K − S_T, 0) − p
Long stock plus long put with strike K and premium p paid. Maximum loss = S_0 − K + p. Breakeven = S_0 + p.
Bull call spread (K1 < K2)
Payoff = max(S_T − K1, 0) − max(S_T − K2, 0); Max profit = (K2 − K1) − net premium; Max loss = net premium; Breakeven = K1 + net premium
Buy the K1 call, sell the K2 call. Net premium is positive because the K1 call costs more.
Bear put spread (K1 < K2)
Payoff = max(K2 − S_T, 0) − max(K1 − S_T, 0); Max profit = (K2 − K1) − net premium; Max loss = net premium; Breakeven = K2 − net premium
Buy the K2 put, sell the K1 put.
Butterfly spread with calls (K2 = (K1 + K3) ÷ 2)
Payoff = max(S_T − K1, 0) − 2 × max(S_T − K2, 0) + max(S_T − K3, 0); Max payoff = K2 − K1 at S_T = K2
Net premium is positive. Max profit = (K2 − K1) − net premium. Max loss = net premium. Breakevens = K1 + net premium and K3 − net premium.
Straddle (same strike K)
Payoff = |S_T − K|; Profit = |S_T − K| − (c + p); Breakevens = K ± (c + p)
Long call and long put. Maximum loss = c + p, at S_T = K.
Strangle (put strike K1 < call strike K2)
Payoff = max(K1 − S_T, 0) + max(S_T − K2, 0); Breakevens = K1 − (c + p) and K2 + (c + p)
Max loss = c + p, for any S_T between K1 and K2.

How to solve Option Trading Strategies and Spreads questions

Use this method for any strategy question, whether it asks for profit, breakeven, maximum loss or the market view.

  1. 1Write down every leg: long or short, call or put, strike, and premium.
  2. 2Compute the net premium. Premiums paid are costs; premiums received are inflows. Note whether the strategy is a net debit or net credit.
  3. 3Split the stock price into zones using the strikes. In each zone, work out which options are exercised.
  4. 4Add the payoffs of all legs in each zone. Subtract the net debit, or add the net credit, to get profit.
  5. 5Find the key values: maximum profit, maximum loss and breakevens. Check each breakeven falls inside the zone where you solved it.
  6. 6Match the shape to the view: bullish, bearish, low volatility or high volatility. Check that your answer fits.
  7. 7Sanity check: a spread has capped gain and loss; a long straddle has capped loss and unlimited upside.

Quickest way: Shape-first shortcut

When to use it: Use when the question asks for maximum profit, maximum loss or breakeven and the strikes are given.

  1. For a vertical spread, the strike gap K2 − K1 is the maximum payoff. Max profit = gap − net debit. Max loss = net debit.
  2. For a butterfly, the maximum payoff is the gap between adjacent strikes (K2 − K1). Max loss is the net debit.
  3. For a straddle or strangle, the maximum loss is the total premium. Breakevens sit that distance outside the strikes.
  4. For a covered call, the upside is capped at the strike. For a protective put, the downside is floored at the strike.
  5. Do the arithmetic once, then check the sign: debit strategies lose the premium at worst.

Common mistakes in Option Trading Strategies and Spreads

  • Forgetting to subtract the net premium when finding breakevens.

    Students stop at the payoff diagram and treat payoff as profit.

    Fix: Always work in profit. Breakeven is where payoff equals net premium paid.

  • Mixing up bull and bear spreads built from calls.

    Both use two calls, so the structures look alike.

    Fix: A bull call spread buys the lower strike and sells the higher strike. A bear call spread does the reverse. The bull spread costs money; the bear call spread brings in a credit.

  • Using the wrong number of options in a butterfly.

    Students forget the middle strike has two short options.

    Fix: Remember 1 long, 2 short, 1 long. The payoff must sum to zero beyond the outer strikes.

  • Saying a covered call has unlimited upside.

    The long stock has unlimited upside, so students stop there.

    Fix: The short call gives away gains above the strike. Max profit = K − S_0 + c.

  • Treating a strangle as more profitable than a straddle.

    Strangles are cheaper, so they seem a better deal.

    Fix: A strangle costs less but needs a larger price move to break even. It also has a wider zone of maximum loss.

  • Confusing protective put maximum loss with the premium alone.

    Students think of the put as insurance costing only p.

    Fix: Max loss = S_0 − K + p. If the strike is below the purchase price, you absorb that gap too.

Worked examples

Example 1

A stock trades at $50. You buy a call with strike $50 for $4 and sell a call with strike $60 for $1. Find the maximum profit, maximum loss and breakeven at expiry.

Show the solution
  1. This is a bull call spread: long the K1 = 50 call, short the K2 = 60 call.
  2. Net premium = 4 − 1 = $3 paid (a debit).
  3. Maximum payoff = K2 − K1 = 60 − 50 = $10, reached when S_T ≥ 60.
  4. Maximum profit = 10 − 3 = $7.
  5. Maximum loss = net premium = $3, when S_T ≤ 50.
  6. Breakeven = K1 + net premium = 50 + 3 = $53.

Answer: Maximum profit $7, maximum loss $3, breakeven $53.

Example 2

A trader buys a call with strike $90 for $9 and a put with strike $90 for $6 on the same stock. What are the breakevens, and what is the profit if the stock ends at $110?

Show the solution
  1. This is a long straddle at K = 90.
  2. Total premium = 9 + 6 = $15.
  3. Upper breakeven = 90 + 15 = $105.
  4. Lower breakeven = 90 − 15 = $75.
  5. At S_T = 110 the call pays 110 − 90 = $20 and the put pays 0.
  6. Profit = 20 − 15 = $5.

Answer: Breakevens are $75 and $105. The profit at $110 is $5.

Exam tips

  • Always draw a quick profit diagram in your head before computing. The shape tells you which options are exercised.
  • Read the view the question gives: low volatility points to a butterfly or short straddle; large move points to a long straddle or strangle.
  • Check the sign of the premium: spreads with calls cost money when you buy the lower strike.
  • Questions often ask for breakeven. Use profit, not payoff, and include both legs' premiums.
  • Put-call parity links call and put spreads. Bull put and bull call spreads have the same payoff shape at expiry, but different premiums.

Practice questions from Properties of Options

Option Trading Strategies and Spreads in other exams

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

Option Trading Strategies and Spreads: frequently asked questions

What is the difference between a bull spread and a bear spread?

A bull spread profits when the underlying rises, and a bear spread profits when it falls. Both cap gain and loss. A bull call spread buys the low-strike call and sells the high-strike call, while a bear put spread buys the high-strike put and sells the low-strike put.

How do you build a butterfly spread with calls?

Buy one call at a low strike, sell two calls at the middle strike and buy one call at a high strike. The middle strike should sit halfway between the outer strikes. It profits most if the price ends at the middle strike.

When should you use a straddle instead of a strangle?

Use a straddle when you expect a big move and want profit to start sooner. Use a strangle when you want a cheaper position and can accept needing a larger move. Both lose the premium if the price stays near the strikes.

What is the payoff of a covered call and a protective put?

A covered call is long stock and short call. It earns the premium but caps the upside at the strike. A protective put is long stock and long put. It sets a floor on losses at the strike, at the cost of the premium.