Explanation

To calculate the European call option value using a two-period binomial model:

Step 1: Calculate stock prices at each node

Given:

• S₀ = 50.00
• u = 1.373 (annual up factor)
• d = 0.727 (annual down factor)
• K = 55.00
• r = 5.00% annual
• T = 2 years

Stock price tree:

• Time 0: S₀ = 50.00
• Time 1 (up): S_u = 50 × 1.373 = 68.65
• Time 1 (down): S_d = 50 × 0.727 = 36.35
• Time 2 (up-up): S_uu = 68.65 × 1.373 = 94.25
• Time 2 (up-down): S_ud = 68.65 × 0.727 = 49.90
• Time 2 (down-down): S_dd = 36.35 × 0.727 = 26.42

Step 2: Calculate option payoffs at Time 2

• C_uu = max(94.25 - 55, 0) = 39.25
• C_ud = max(49.90 - 55, 0) = 0
• C_dd = max(26.42 - 55, 0) = 0

Step 3: Calculate risk-neutral probabilities

Risk-neutral probability (q): q = (1 + r - d) / (u - d) = (1.05 - 0.727) / (1.373 - 0.727) = 0.323 / 0.646 = 0.50

Step 4: Calculate option values backward

At Time 1 (up node): C_u = [q × C_uu + (1-q) × C_ud] / (1+r) = [0.50 × 39.25 + 0.50 × 0] / 1.05 = [19.625 + 0] / 1.05 = 18.69

At Time 1 (down node): C_d = [q × C_ud + (1-q) × C_dd] / (1+r) = [0.50 × 0 + 0.50 × 0] / 1.05 = 0

At Time 0: C₀ = [q × C_u + (1-q) × C_d] / (1+r) = [0.50 × 18.69 + 0.50 × 0] / 1.05 = [9.345 + 0] / 1.05 = 8.90

Therefore, the current value of the European-style call option is closest to 8.90.