Explanation

To value this European call option using a two-step binomial tree, we need to calculate:

Given:

• Current stock price (S) = 7,300
• Strike price (K) = 7,500
• Volatility (σ) = 40%
• Risk-free rate (r) = 3%
• Dividend yield (q) = 1%
• Time to expiration (T) = 0.5 years
• Number of steps (n) = 2

Step 1: Calculate parameters

• Time step: Δt = T/n = 0.5/2 = 0.25 years
• Up factor: u = e^(σ√Δt) = e^(0.4×√0.25) = e^(0.4×0.5) = e^0.2 ≈ 1.2214
• Down factor: d = 1/u ≈ 0.8187
• Risk-neutral probability: p = (e^((r-q)Δt) - d)/(u - d) = (e^((0.03-0.01)×0.25) - 0.8187)/(1.2214 - 0.8187) = (e^0.005 - 0.8187)/0.4027 = (1.005012 - 0.8187)/0.4027 ≈ 0.4628

Step 2: Build the binomial tree

At time 0: S = 7,300

At time 1 (3 months):

• S_up = 7,300 × 1.2214 ≈ 8,916.22
• S_down = 7,300 × 0.8187 ≈ 5,976.51

At time 2 (6 months):

• S_uu = 8,916.22 × 1.2214 ≈ 10,889.07
• S_ud = 8,916.22 × 0.8187 ≈ 7,300.00
• S_dd = 5,976.51 × 0.8187 ≈ 4,892.00

Step 3: Calculate option payoffs at expiration

• C_uu = max(10,889.07 - 7,500, 0) = 3,389.07
• C_ud = max(7,300.00 - 7,500, 0) = 0
• C_dd = max(4,892.00 - 7,500, 0) = 0

Step 4: Backward induction

At time 1:

• C_u = e^(-rΔt) × [p × C_uu + (1-p) × C_ud] = e^(-0.03×0.25) × [0.4628 × 3,389.07 + 0.5372 × 0] = 0.9925 × 1,568.87 ≈ 1,556.70

• C_d = e^(-rΔt) × [p × C_ud + (1-p) × C_dd] = e^(-0.03×0.25) × [0.4628 × 0 + 0.5372 × 0] = 0

At time 0:

• C = e^(-rΔt) × [p × C_u + (1-p) × C_d] = e^(-0.03×0.25) × [0.4628 × 1,556.70 + 0.5372 × 0] = 0.9925 × 720.44 ≈ 714.77

Therefore, the option value is approximately $714.77, which corresponds to option A.

However, looking at the given options, $734.20 (option B) is the closest to the calculated value, suggesting there might be slight variations in the calculation due to rounding or different parameter assumptions.