Answer:

## Explanation This question appears to be incomplete as it references "the exhibit below" which is not provided in the text. However, I can provide the general methodology for calculating the American put option value using the given parameters: ### Given Parameters: - Current stock price (S₀) = €65 - Strike price (K) = €63 - Up factor (u) = 1.20 - Down factor (d) = 0.85 - Risk-free rate (r) = 5% per period ### Step 1: Build Stock Price Tree - Time 0: S₀ = 65 - Time 1 (up): S_u = 65 × 1.20 = 78 - Time 1 (down): S_d = 65 × 0.85 = 55.25 - Time 2 (up-up): S_uu = 78 × 1.20 = 93.60 - Time 2 (up-down): S_ud = 78 × 0.85 = 66.30 - Time 2 (down-down): S_dd = 55.25 × 0.85 = 46.96 ### Step 2: Calculate Risk-Neutral Probability q = (1 + r - d) / (u - d) = (1.05 - 0.85) / (1.20 - 0.85) = 0.20 / 0.35 = 0.5714 ### Step 3: Calculate Option Values Backward **At Time 2 (European put payoffs):** - P_uu = max(63 - 93.60, 0) = 0 - P_ud = max(63 - 66.30, 0) = 0 - P_dd = max(63 - 46.96, 0) = 16.04 **At Time 1 (European values):** - P_u = [q × P_uu + (1-q) × P_ud] / (1+r) = [0.5714 × 0 + 0.4286 × 0] / 1.05 = 0 - P_d = [q × P_ud + (1-q) × P_dd] / (1+r) = [0.5714 × 0 + 0.4286 × 16.04] / 1.05 = 6.87 / 1.05 = 6.54 **At Time 0 (European value):** P_European = [q × P_u + (1-q) × P_d] / (1+r) = [0.5714 × 0 + 0.4286 × 6.54] / 1.05 = 2.80 / 1.05 = 2.67 **For American put, check early exercise at each node:** - At Time 1 (down): Intrinsic value = max(63 - 55.25, 0) = 7.75 Since 7.75 > 6.54, early exercise is optimal **American put value at Time 0:** P_American = [q × P_u + (1-q) × P_d_early] / (1+r) = [0.5714 × 0 + 0.4286 × 7.75] / 1.05 = 3.32 / 1.05 = 3.16 **Early exercise premium:** 3.16 - 2.67 = 0.49 Without the complete exhibit, I cannot provide the exact answer, but this demonstrates the calculation methodology.

Author: LeetQuiz Editorial Team