Math Problem Statement
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.
Solution
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
- 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:
Step 2: Analyze the portfolio's payoff at
Case 1: Stock price at maturity
- Both call options expire worthless since the stock price is below both strikes.
- The short-sold stock is repurchased at a price .
- The portfolio payoff is simply the repurchase of the stock at , which leads to a net payoff of .
In this case, the payoff is:
Case 2:
- The $40-strike call will be exercised (because ) and have a payoff of .
- The $50-strike call expires worthless.
- The short-sold stock is repurchased at .
Thus, the total payoff is:
Case 3:
- Both calls are exercised.
- The $40-strike call has a payoff of .
- The $50-strike call has a payoff of .
- The short-sold stock is repurchased at .
Thus, the total payoff is:
Step 3: Determine the minimum gain
- From Case 2 and Case 3, we observe that when , the minimum gain is $8.
- In Case 1, , the payoff is , which is maximized at , yielding a payoff of .
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:
- How is arbitrage determined in financial markets?
- What is the significance of continuously compounded interest in options pricing?
- How would the presence of dividends change the analysis of this portfolio?
- What are the key differences between European and American options?
- 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.
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Math Problem Analysis
Mathematical Concepts
Options Pricing
Arbitrage
Financial Markets
Continuously Compounded Interest
Formulas
Net cash inflow = Stock price - Option 1 price - Option 2 price
Call payoff = max(S_T - Strike price, 0)
Portfolio payoff = 48 - S_T + (S_T - 40) + (S_T - 50)
Theorems
Put-Call Parity
Arbitrage Pricing Theory
Black-Scholes Model (implicitly)
Suitable Grade Level
Undergraduate Finance or Economics
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