Options Spreads

Options, Futures and Derivative Securities

Lorenzo Naranjo

Spring 2026

What Is an Option Spread?

An option spread combines options of the same type, all calls or all puts, on the same underlying asset with different strikes.
Option spreads can be directional, such as bull or bear spreads, or pay the spread if the stock price is within a certain range.

Bull Spread

A bull spread is a two-leg option strategy that consists in a long position in a call with strike K_{1} and a short position in a call with strike K_{2}, where K_{1} < K_{2}.

Payoff at maturity against the stock price S. A dashed long call with strike K1 and a dashed short call with strike K2 add up to the solid bull spread, which is zero below K1, rises one for one between K1 and K2, and stays flat at K2 minus K1 above K2.

Example 1: Bull Spread

A non-dividend paying stock currently trades at $50.
Call options with strikes K_{1} = \$40 and K_{2} = \$60 trade for $13.23 and $3.45, respectively.
The cost of the bull spread is 13.23 - 3.45 = $9.78.
The payoff and profit for different stock prices at maturity is:

Stock Price	30	50	70
Long Call	0	10	30
Short Call	0	0	-10
Payoff	0	10	20
Profit	-9.78	0.22	10.22

Bull Spread

The payoff of a bull spread can then be described as follows:

	S \leq K_{1}	K_{1} < S \leq K_{2}	S > K_{2}
Long Call	0	S - K_{1}	S - K_{1}
Short Call	0	0	-(S - K_{2})
Bull Spread	0	S - K_{1}	K_{2} - K_{1}

If K_{2} - K_{1} is small, the bull spread is like an all-or-nothing bet on the stock going above K_{2}.

Call Premium and the Strike Price

The previous analysis shows that the payoff of the bull spread is either zero or positive.
Thus, no-arbitrage implies that the cost of a bull spread cannot be negative, that is, C_{1} - C_{2} \geq 0
If not, you could build a bull spread with a negative cost!
This implies that a call with a lower strike cannot cost less than an otherwise equivalent call with a higher strike price: C_{1} \geq C_{2}

Bear Spread

A bear spread is a two-leg option strategy that consists in a long position in a put with strike K_{2} and a short position in a put with strike K_{1}, where K_{1} < K_{2}.

Payoff at maturity against the stock price S. A dashed long put with strike K2 and a dashed short put with strike K1 add up to the solid bear spread, which equals K2 minus K1 below K1, falls one for one between K1 and K2, and is zero above K2.

Example 2: Bear Spread

A non-dividend paying stock currently trades at $50.
Put options with strikes K_{1} = \$40 and K_{2} = \$60 trade for $1.28 and $10.53, respectively.
The cost of the bear spread is 10.53 - 1.28 = $9.25.
The payoff and profit for different stock prices at maturity is:

Stock Price	30	50	70
Long Put	30	10	0
Short Put	-10	0	0
Payoff	20	10	0
Profit	10.75	0.75	-9.25

Bear Spread

The payoff of a bear spread can then be described as follows:

	S \leq K_{1}	K_{1} < S \leq K_{2}	S > K_{2}
Long Put	K_{2} - S	K_{2} - S	0
Short Put	-(K_{1} - S)	0	0
Bear Spread	K_{2} - K_{1}	K_{2} - S	0

If K_{2} - K_{1} is small, the bear spread is like an all-or-nothing bet on the stock going below K_{1}.

Put Premium and the Strike Price

The payoff diagram of the bear spread shows that the strategy can either pay nothing, or a positive amount.
Thus, no-arbitrage implies that the cost of a bear spread cannot be negative, that is, P_{2} - P_{1} \geq 0
If not, you could build a bear spread with a negative cost!
This implies that a put with a higher strike cannot cost less than an otherwise equivalent put with a lower strike price: P_{2} \geq P_{1}

Butterfly

A butterfly is a three-leg option strategy that consists in a long call with strike K_{1}, short two calls with strike K_{2} and a long call with strike K_{3} where K_{1} < K_{2} < K_{3} and K_{2} = (K_{1} + K_{3}) / 2.

Payoff at maturity against the stock price S. Dashed lines show a long call with strike K1, two short calls with strike K2 and a long call with strike K3. Their sum, the solid butterfly, is a tent: zero below K1 and above K3, peaking at K2 minus K1 when S equals K2.

Example 3: Butterfly

A non-dividend paying stock currently trades at $50.
Call options with strikes K_{1} = \$45, K_{2} = \$50 and K_{3} = \$55 trade for $9.85, $7.12 and $5.01, respectively.
A butterfly with strikes K_{1}, K_{2} and K_{3} then costs 9.85 - 2(7.12) + 5.01 = $0.62.
Below are some possible butterfly payoffs and profits for different stock prices at maturity:

Stock Price	40	50	60
Payoff	0	5	0
Profit	-0.62	4.38	-0.62

Building a Butterfly with Put Options

The butterfly can also be obtained by buying puts with strikes K_{1} and K_{3}, and shorting two puts with strikes K_{2} = (K_{1} + K_{3})/2.

Payoff at maturity against the stock price S. Dashed lines show a long put with strike K3, two short puts with strike K2 and a long put with strike K1. Their sum is the same solid tent-shaped butterfly as with calls, peaking at K2 minus K1 when S equals K2.

Option Premium Convexity with Respect to the Strike Price

No-arbitrage implies that the price of a butterfly cannot be negative, that is, P_{1} - 2 P_{2} + P_{3} = C_{1} - 2 C_{2} + C_{3} \geq 0, which in turn implies that P_{2} \leq \dfrac{P_{1} + P_{3}}{2} \quad \text{and} \quad C_{2} \leq \dfrac{C_{1} + C_{3}}{2}.
If this was not the case, you could make a butterfly out of calls or puts with a negative price!

Condor

A condor consists in a long call with strike K_{1}, a short call with strike K_{2}, a short call with strike K_{3} and a long call with strike K_{4} where K_{1} < K_{2} < K_{3} < K_{4} and K_{2} - K_{1} = K_{4} - K_{3}.

Payoff at maturity against the stock price S. Dashed lines show long calls with strikes K1 and K4 and short calls with strikes K2 and K3. Their sum, the solid condor, is zero below K1 and above K4, rises to K2 minus K1 between K1 and K2, stays flat until K3, and falls back to zero at K4.

Example 4: Condor

A non-dividend paying stock currently trades at $50.
Call options with strikes K_{1} = \$40, K_{2} = \$45, K_{3} = \$55 and K_{4} = \$60 trade for $13.23, $9.85, $5.01 and $3.45, respectively.
A condor with strikes K_{1}, K_{2}, K_{3} and K_{4} then costs 13.23 - 9.85 - 5.01 + 3.45 = $1.82.
Below are some possible condor payoffs and profits for different stock prices at maturity:

Stock Price	40	50	60
Payoff	0	5	0
Profit	-1.82	3.18	-1.82