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Magazine Chapter 10: Problem 19
Three put options on a stock have the same expiration date and strike prices of \(\$ 55, \$ 60\) and \(\$ 65 .\) The market prices are \(\$ 3, \$ 5,\) and \(\$ 8,\) respectively. Explain how a butterfly spread can be created. Construct a table showing the profit from the strategy. For what range of stock prices would the butterfly spread lead to a loss?
Short Answer
Expert verified
A butterfly spread profits between stock prices of $55 and $60; loses outside this range.
Step by step solution
01
Understanding the Butterfly Spread
A butterfly spread with put options involves buying higher and lower strike put options and selling two put options with a middle strike price. In this case, the strategy will be as follows:
- Buy 1 put option with a strike price of $55 (costing $3)
- Sell 2 put options with a strike price of $60 (each at $5, total income of $10)
- Buy 1 put option with a strike price of $65 (costing $8)
The total cost to set up the strategy will be the sum of the costs related to buying and selling the options.
02
Calculating Net Cost
Calculate the net cost of the butterfly spread by considering the costs and income from the options:
- Cost to buy the $55 strike put: $3
- Cost to buy the $65 strike put: $8
- Income from selling two $60 strike puts: $10
Total cost = $3 (for $55 put) + $8 (for $65 put) - $10 (income from $60 puts) = $1.
03
Constructing the Profit Table
To determine the payoff and profit at different stock prices, we need to calculate the value of the options at expiration for each possible price scenario and subtract the initial cost:
| Stock Price at Expiry | Payoff from $55 Put | Payoff from $60 Puts (2) | Payoff from $65 Put | Total Payoff | Profit/Loss |
|-----------------------|---------------------|--------------------------|---------------------|--------------|--------------|
| Below $55 | 0 | 0 | 1 x $10 = $10 | $10 | $9 (profit) |
| $55 to $60 | $60 - Stock Price | 0 | 1 x $10 = $10 | $10 + ($60 - Stock Price) | $ 9 + ($60 - Stock Price) |
| $60 to $65 | 0 | 2 x ($60 - Stock Price) | 0 | Payoffs cancel each other | $9 - $1 = $8 (max) at $60 |
| Above $65 | 0 | 0 | 0 | 0 | $8 (loss) |
04
Analyzing the Profit or Loss Range
By examining the profit table, a butterfly spread leads to a loss when the stock price goes above $60 up to $65, causing the middle put options to expire in-the-money, reducing overall payoff due to a capitalized net cost. The range of stock prices where the butterfly will result in a loss is between $55 to $60.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
put options
A put option provides the holder the right, but not the obligation, to sell a particular stock at a specified strike price before the option's expiration date. If the stock price falls below the strike price, the holder can exercise the option to sell the stock at the higher strike price, earning a profit. Put options are a type of financial derivative and are often used for hedging or speculation.
For instance, in our butterfly spread example, you purchase put options at a low strike price ($55) and a high strike price ($65) while selling puts with a middle strike price ($60). This arrangement capitalizes on the scenarios where the stock price reaches or moves within certain bands, potentially delivering a diverse range of outcomes, from losses to profits. By carefully selecting these strike prices, you can create financial strategies, like the butterfly spread, to balance risk and reward effectively.
strike price
The strike price is a key component of an options contract. It is the price at which the holder of the option can buy (in the case of a call option) or sell (in the case of a put option) the underlying asset. In our butterfly spread example, the strike prices are set at $55, $60, and $65. These prices dictate the potential profits and losses at option expiration.
- The lower strike price of $55 represents the price at which you hold the right to sell one put option.
- The middle strike price of $60 is tied to two sold puts, where you jot income but also the potential for bearish exposure if the stock price hovers around this middle strike.
- The highest strike price of $65 involves purchasing another put option, helping to cap maximum losses when the price rises above $60.
expiration date
The expiration date is a crucial aspect of options trading, dictating the specific time frame in which the option must be exercised or it becomes void. With a butterfly spread or any options strategy, timing is critical, as options are inherently losing value over time due to theta decay.
For our butterfly spread configuration, all of the options share the same expiration date. This synchronization allows for the strategic balance between buying and selling options at different strike prices to create the spread, focusing on price movements within a defined band at expiration.
Timing your trades around the expiration date is important, especially when taking into account market conditions and expected price movements, as options can become worthless once expired, turning potential gains into complete losses if not properly managed.
stock price
Stock price signifies the current market value of a company's stock. It's a major factor in an options strategy, like a butterfly spread, where the goal is to predict and strategically plan for stock price movements within set strike prices.
Let鈥檚 observe how a butterfly spread relates to stock price movements:
- A stock price below the lowest strike ($55) or above the highest strike ($65) usually leads to a maximum loss due to premium costs.
- When the stock price is near the middle strike ($60), potential profit is optimized. Two middle puts generate income if they expire out-of-the-money, balancing the costs of buying the edge options.
- Within the range of $55 to $65, managing stock price movements through your spread can lead to an adjusted payoff profile with minimized risk and defined loss limits.
