Explanation

To calculate the early exercise premium of the American-style put option, we need to:

1. Calculate the European put value at Time 0
2. Calculate the American put value at Time 0
3. Find the difference (early exercise premium)

Step 1: Calculate European put value at Time 0

Using risk-neutral probabilities:

• Up move: 76.24
• Down move: 34.12
• Risk-free rate: 2.05%

Risk-neutral probability (q): q = (1 + r - d) / (u - d) Where:

• u = 76.24/52.00 = 1.466
• d = 34.12/52.00 = 0.656
• r = 2.05%

q = (1.0205 - 0.656) / (1.466 - 0.656) = 0.3645 / 0.81 = 0.45

European put value at Time 0: P_European = [q × P_up + (1-q) × P_down] / (1+r) = [0.45 × 0 + 0.55 × 14.88] / 1.0205 = [0 + 8.184] / 1.0205 = 8.184 / 1.0205 = 8.02

Step 2: Calculate American put value at Time 0

For American put, we check for early exercise at each node:

• At Time 1 (up move): Intrinsic value = max(50-76.24, 0) = 0 European value = 0 No early exercise advantage

• At Time 1 (down move): Intrinsic value = max(50-34.12, 0) = 15.88 European value = 14.88 Early exercise value = 15.88 Early exercise is optimal

American put value at Time 0: P_American = [q × P_up + (1-q) × P_down_early] / (1+r) = [0.45 × 0 + 0.55 × 15.88] / 1.0205 = [0 + 8.734] / 1.0205 = 8.734 / 1.0205 = 8.56

Step 3: Early exercise premium

Early exercise premium = P_American - P_European = 8.56 - 8.02 = 0.54

Therefore, the early exercise premium is closest to 0.54.