Let's format the equations clearly:

Key Variables

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

Profit from Stock Holding

The profit or loss from holding the stock can be represented as: $\text{Profit}_{\text{stock}} = n \cdot (S_T - S_0)$

Profit from Call Option

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

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: $\text{Exercise Loss} = n \cdot \max(0, S_T - K)$

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: $\text{Total Profit} = n \cdot (S_T - S_0) + n \cdot P - n \cdot \max(0, S_T - K)$

Simplifying, we get: $\text{Total Profit} = n \cdot [(S_T - S_0) + P - \max(0, S_T - K)]$

Piecewise Summation

To better visualize this, we can break it into a piecewise function: [ \text{Total Profit} = \begin{cases} n \cdot [(S_T - S_0) + P] & \text{if } S_T \leq K \ n \cdot [(K - S_0) + P] & \text{if } S_T > K \end{cases} ]

Interpretation

• If the stock price $S_T$ is less than or equal to the strike price $K$, you keep the stock and the premium. Your profit is the difference in stock price plus the premium.
• If the stock price $S_T$ exceeds the strike price $K$, the option is exercised, and you sell the stock at $K$. Your profit is the difference between the strike price and the initial stock price plus the premium.

Example Calculation

Let's use the provided example values:

• Initial stock price $S_0 = 50$
• Strike price $K = 55$
• Premium received $P = 2$
• Number of shares $n = 100$

If $S_T \leq 55$: $\text{Total Profit} = 100 \cdot [(S_T - 50) + 2]$

If $S_T > 55$: [ \text{Total Profit} = 100 \cdot [(55 - 50) + 2] = 100 \cdot [5 + 2] = 100 \cdot 7 = 700 ]

Do you want more details or have any questions?

Relative Questions

1. What is the impact on profit if the stock price falls below the initial stock price?
2. How does increasing the premium $P$ affect the total profit?
3. What happens to the total profit if the number of shares $n$ is increased?
4. Can the total profit be negative in a covered call strategy? Under what conditions?
5. How does the strike price $K$ influence the exercise loss?
6. What are the risks associated with selling call options?
7. How would dividends affect the covered call strategy?
8. What are alternative strategies to covered calls for generating income from stocks?

Tip

Always consider the potential for the stock price to exceed the strike price when selling call options, as this can significantly impact your total profit.