Key Variables

• $S_0 = 50$: Initial stock price
• $S_T$: Stock price at expiration
• $K = 55$: Strike price of the call option
• $P = 1$: Premium received from selling the call option
• $n = 100$: Number of shares (one option contract)

Profit from Stock Holding

The profit or loss from holding the stock can be represented as: $\text{Profit}_{\text{stock}} = 100 \cdot (S_T - 50)$

Profit from Call Option

The profit from selling the call option can be represented as: $\text{Profit}_{\text{option}} = 100 \cdot 1 = 100$

However, if the stock price at expiration $S_T$ exceeds the strike price $K$, the call option will be exercised, and you will have to sell the stock at $K$. In this case, the loss from the call being exercised needs to be accounted for: $\text{Exercise Loss} = 100 \cdot \max(0, S_T - 55)$

Total Profit (or Loss)

The total profit or loss from the covered call strategy is the sum of the profit from holding the stock and the profit from selling the call option, minus any loss from the option being exercised: $\text{Total Profit} = \text{Profit}_{\text{stock}} + \text{Profit}_{\text{option}} - \text{Exercise Loss}$

Substituting the formulas: $\text{Total Profit} = 100 \cdot (S_T - 50) + 100 - 100 \cdot \max(0, S_T - 55)$

Simplifying, we get: $\text{Total Profit} = 100 \cdot [(S_T - 50) + 1 - \max(0, S_T - 55)]$

Piecewise Summation

To better visualize this, we can break it into a piecewise function: [ \text{Total Profit} = \begin{cases} 100 \cdot [(S_T - 50) + 1] & \text{if } S_T \leq 55 \ 100 \cdot [(55 - 50) + 1] & \text{if } S_T > 55 \end{cases} ]

Interpretation

• If the stock price $S_T$ is less than or equal to the strike price $K = 55$, you keep the stock and the premium. Your profit is the difference in stock price plus the premium: $\text{Total Profit} = 100 \cdot [(S_T - 50) + 1]$
• If the stock price $S_T$ exceeds the strike price $K = 55$, the option is exercised, and you sell the stock at $K = 55$. Your profit is the difference between the strike price and the initial stock price plus the premium: [ \text{Total Profit} = 100 \cdot [(55 - 50) + 1] = 100 \cdot [5 + 1] = 100 \cdot 6 = 600 ]

Example Calculation

Let's use the provided example values:

1. If $S_T \leq 55$: $\text{Total Profit} = 100 \cdot [(S_T - 50) + 1]$

2. If $S_T > 55$: [ \text{Total Profit} = 100 \cdot [(55 - 50) + 1] = 100 \cdot [5 + 1] = 100 \cdot 6 = 600 ]

Summary

• For $S_T \leq 55$: $\text{Total Profit} = 100 \cdot [(S_T - 50) + 1]$
• For $S_T > 55$: $\text{Total Profit} = 600$

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Do you want more details or have any questions?

Relative Questions

1. How does the expiration date of the call option affect the strategy?
2. What happens if the stock price is exactly the strike price at expiration?
3. How would increasing the premium $P$ to 2 change the total profit?
4. What are the tax implications of a covered call strategy?
5. How does volatility in the stock price affect the risk of the covered call strategy?
6. What alternative strategies can be used if the stock price is expected to rise significantly?
7. How does the dividend payment impact the covered call strategy?
8. What are the consequences of early exercise by the option holder?

Tip

Monitor the stock price regularly and be prepared to adjust your strategy if the stock price approaches the strike price.