Explanation

To identify the arbitrage opportunity, we need to check if the call option is mispriced relative to its theoretical value using the binomial model.

Given data:

• Strike price (K) = 55.00
• Current stock price (S₀) = 50.00
• Call price (C₀) = 13.50
• Up move probability (π) = 0.60
• S⁺ = 67.80, S⁻ = 27.05
• C⁺ = 21.52, C⁻ = 0.00
• Risk-free rate (r) = 3.00%

Calculate theoretical call value:

First, calculate the risk-neutral probability: $p = \frac{(1+r) - d}{u - d}$ Where: $u = \frac{S⁺}{S₀} = \frac{67.80}{50.00} = 1.356$ $d = \frac{S⁻}{S₀} = \frac{27.05}{50.00} = 0.541$

$p = \frac{1.03 - 0.541}{1.356 - 0.541} = \frac{0.489}{0.815} = 0.60$

Theoretical call value: $C₀ = \frac{p \times C⁺ + (1-p) \times C⁻}{1+r} = \frac{0.60 \times 21.52 + 0.40 \times 0}{1.03} = \frac{12.912}{1.03} = 12.54$

Arbitrage analysis:

• Theoretical value = 12.54
• Market price = 13.50
• The call is overpriced by 0.96

Arbitrage strategy: Since the call is overpriced, we should:

1. Sell the call option (receive 13.50)
2. Buy the underlying stock (hedge the short call position)
3. Lend cash (invest the remaining proceeds at risk-free rate)

This creates a risk-free profit because we're selling the overpriced call and creating a synthetic long position that replicates the call's payoff at a lower cost.