Answer: 0.54

## 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**.

Author: LeetQuiz Editorial Team