Options Spreads
Options and Futures
Lorenzo Naranjo
Spring 2024
What Is an Option Spread?
- An option spread is an option strategy in which the payoff is limited.
- 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 \(K_{1} < K_{2}\).
Example 2: 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:
| 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:
| 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 \(K_{1} < K_{2}\).
Example 3: 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:
| 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:
| 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 must cost more 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 \(K_{2} = (K_{1} + K_{3}) / 2\).
Example 4: 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 straddle payoffs and profits for different stock prices at maturity:
| 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 \(K_{2} = (K_{1} + K_{3})/2\).
Option Premium Convexity with Respect 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 \(K_{2} - K_{1} = K_{4} - K_{3}\).
Example 5: 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 straddle payoffs and profits for different stock prices at maturity:
| Payoff |
0 |
5 |
0 |
| Profit |
-1.82 |
3.18 |
-1.82 |
