Describe how various spread strategies are used and compute their payoffs.
The idea
A spread combines options of the same type, calls or puts, with different strikes or maturities. A bull call spread is long a low-strike call and short a higher-strike call. It costs a net premium, the payoff is at most the strike difference, so profit is capped and the loss is limited to the net debit. A bear put spread is long a high-strike put and short a lower-strike put. A butterfly is long one call at the low strike, short two at the middle strike and long one at the high strike. It profits when the price ends near the middle. A calendar spread sells a near-dated option and buys a longer-dated one at the same strike, gaining from faster time decay if the price stays put. A box spread, a bull call spread plus a bear put spread on the same strikes, pays the strike difference for certain, so it is a loan. Always compute profit as payoff minus net premium.
Formulas
Payoff of a bull call spread at expiry
payoff = max(Sₜ − K₁, 0) − max(Sₜ − K₂, 0)
- Sₜ
- the price of the underlying at expiry
- K₁
- the lower strike, of the call bought
- K₂
- the higher strike, of the call sold
The payoff never exceeds K₂ − K₁. Profit is the payoff minus the net premium paid, which is the step most often forgotten.
Value of a box spread
box value = (K₂ − K₁) × e^(−r × T)
- K₁
- the lower strike
- K₂
- the higher strike
- r
- the continuously compounded risk-free rate
- T
- the time to expiry, in years
European options only: the box pays the strike difference for certain, so it is a loan. Any other price is an arbitrage.
A practice question
Which description matches a bull call spread?
- Short a call with a low strike and long a call with a higher strike, same expiryWhy not: That is a bear call spread, which profits when the price falls.
- Long a call and long a put with the same strike and expiryWhy not: That is a straddle, which is a volatility position, not a spread.
- Long a put with a high strike and short a put with a lower strikeWhy not: That is a bear put spread, whose payoff rises when the price falls.
- Long a call with a low strike and short a call with a higher strike, same expiry - correct
A bull call spread buys the low-strike call and sells the high-strike call to cut the cost; it gains as the price rises and is capped at the strike difference.
A harder one
A trader buys a 50 call at 4.20 and sells a 55 call at 1.70 on the same stock and expiry. If the stock is 54 at expiry, the profit per share on the spread is closest to:
- 4.00Why not: Ignores the net premium of 2.50 paid to set up the spread.
- -0.20Why not: Counts only the long call and ignores the short call's payoff-capping and premium received.
- 1.50 - correct
- 5.00Why not: Reports the maximum payoff of the spread, which is only reached at 55 or above; at 54 the short call is still out of the money.
Net debit = 4.20 - 1.70 = 2.50. Payoff at 54 = 54 - 50 = 4. Profit = 4 - 2.50 = 1.50.
The hardest tier
European options on a stock expire in one year. A trader considers a box spread with strikes 80 and 100 (long the 80 call, short the 100 call, long the 100 put, short the 80 put). The risk-free rate is 5% continuously compounded and the box can be bought for 18.60. The present value of the arbitrage profit from buying one box is closest to:
- 1.40Why not: Does not discount the guaranteed payoff of 20; the gain at expiry is 1.40 but its present value is smaller.
- 0.42 - correct
- 2.43Why not: Compounds the 20 payoff forward instead of discounting it, using exp(+rT).
- 0.98Why not: Reports the discount on the box's payoff (20 minus its PV), not the gap between its PV and the price paid.
The box pays 20 for certain, so it is worth 20e^(-0.05) = 19.025. Buying at 18.60 gives a present-value gain of 0.425.
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