Consider a non-dividend-paying stock whose current price equals $54 per share.
A pair of one-year European calls on this stock with strikes of $40 and $50 is available in the market for the
observed prices of $4 and $2, respectively.
The continuously compounded, risk-free interest rate is given to be 10%.
George suspects that there exists an arbitrage portfolio in the above market consisting of the following
components:
• short-sale of one share of stock,
• buy the $40-strike call,
• buy the $50-strike call.
What is the minimum gain from this suspected arbitrage portfolio?
(a) The above is not an arbitrage portfolio.
(b) $0.84
(c) $8.00
(d) $13.05
(e) None of the above.

To analyze George's suspected arbitrage portfolio, we must break down the components of his strategy and calculate the potential profit or loss.

### Information provided:
- **Current stock price (S₀)** = $54
- **Call option prices**:
  - $40-strike call: $4
  - $50-strike call: $2
- **Continuously compounded risk-free rate (r)** = 10% (0.10 annually)
- **Time to maturity (T)** = 1 year
- **Portfolio components**:
  - **Short-sell one share of stock**.
  - **Buy one $40-strike call**.
  - **Buy one $50-strike call**.

### Step 1: Analyze the cash flow at $ t = 0 $
- **From short-selling the stock**, George receives $54 immediately.
- **Buying the $40-strike call** costs $4.
- **Buying the $50-strike call** costs $2.

Thus, the net initial cash flow at time 0 is:

\[
\text{Net cash inflow} = 54 - 4 - 2 = 48
\]

### Step 2: Analyze the portfolio's payoff at $ t = 1 $

#### Case 1: Stock price at maturity $ S_T \leq 40 $
- Both call options expire worthless since the stock price is below both strikes.
- The short-sold stock is repurchased at a price $ S_T \leq 40 $.
- The portfolio payoff is simply the repurchase of the stock at $ S_T $, which leads to a net payoff of $ -(S_T) $.

In this case, the payoff is:

\[
\text{Payoff} = 48 - S_T
\]

#### Case 2: $ 40 < S_T \leq 50 $
- The $40-strike call will be exercised (because $ S_T > 40 $) and have a payoff of $ S_T - 40 $.
- The $50-strike call expires worthless.
- The short-sold stock is repurchased at $ S_T $.

Thus, the total payoff is:

\[
\text{Payoff} = 48 - S_T + (S_T - 40) = 48 - 40 = 8
\]

#### Case 3: $ S_T > 50 $
- Both calls are exercised.
- The $40-strike call has a payoff of $ S_T - 40 $.
- The $50-strike call has a payoff of $ S_T - 50 $.
- The short-sold stock is repurchased at $ S_T $.

Thus, the total payoff is:

\[
\text{Payoff} = 48 - S_T + (S_T - 40) + (S_T - 50) = 48 - 40 - 50 + S_T + S_T - S_T = 8
\]

### Step 3: Determine the minimum gain
- From Case 2 and Case 3, we observe that when $ S_T > 40 $, the minimum gain is $8.
- In Case 1, $ S_T \leq 40 $, the payoff is $ 48 - S_T $, which is maximized at $ S_T = 40 $, yielding a payoff of $ 48 - 40 = 8 $.

Thus, the **minimum gain** from this arbitrage strategy is $8.

### Final Answer: 
The minimum gain is **$8.00**, so the correct answer is **(c) $8.00**.

---

Would you like further details or have any questions?

Here are 5 related questions to expand on this:
1. How is arbitrage determined in financial markets?
2. What is the significance of continuously compounded interest in options pricing?
3. How would the presence of dividends change the analysis of this portfolio?
4. What are the key differences between European and American options?
5. Can you explain the concept of a call spread and how it relates to this example?

**Tip**: In arbitrage strategies, timing and market inefficiencies are critical. Always consider transaction costs and liquidity constraints when constructing real-world portfolios.

Consider a non-dividend-paying stock whose current price equals $54 per share. A pair of one-year European calls on this stock with strikes of $40 and $50 is available in the market for the observed prices of $4 and $2, respectively. The continuously compounded, risk-free interest rate is given to be 10%. George suspects that there exists an arbitrage portfolio in the above market consisting of the following components: • short-sale of one share of stock, • buy the $40-strike call, • buy the $50-strike call. What is the minimum gain from this suspected arbitrage portfolio? (a) The above is not an arbitrage portfolio. (b) $0.84 (c) $8.00 (d) $13.05 (e) None of the above.

Explore how to calculate the minimum gain from an arbitrage portfolio consisting of European call options with strikes at $40 and $50 on a non-dividend-paying stock priced at $54. Using continuously compounded interest and option payoff formulas, this finance problem demonstrates arbitrage strategies in financial markets. Suitable for undergraduate finance students.

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