Explanation

This is a two-step binomial tree valuation for a European call option on an index with dividends.

Given:

• S = 7,300
• K = 7,500
• σ = 40%
• q = 1% (dividend yield)
• r = 3%
• T = 0.5 years (6 months)
• Steps = 2

First, calculate the binomial parameters:

• Δt = T/2 = 0.25
• u = e^(σ√Δt) = e^(0.4×√0.25) = e^(0.4×0.5) = e^0.2 ≈ 1.2214
• d = 1/u ≈ 0.8187
• p = (e^((r-q)Δt) - d)/(u - d) = (e^((0.03-0.01)×0.25) - 0.8187)/(1.2214 - 0.8187) = (e^(0.02×0.25) - 0.8187)/0.4027 = (e^0.005 - 0.8187)/0.4027 = (1.005012 - 0.8187)/0.4027 ≈ 0.463

Build the tree:

• Step 0: S = 7,300
• Step 1 (up): 7,300 × 1.2214 = 8,916.22
• Step 1 (down): 7,300 × 0.8187 = 5,976.51
• Step 2 (up-up): 8,916.22 × 1.2214 = 10,888.67
• Step 2 (up-down): 8,916.22 × 0.8187 = 7,300.00
• Step 2 (down-down): 5,976.51 × 0.8187 = 4,892.00

Calculate payoffs at expiration:

• Up-up: max(10,888.67 - 7,500, 0) = 3,388.67
• Up-down: max(7,300 - 7,500, 0) = 0
• Down-down: max(4,892 - 7,500, 0) = 0

Backward induction:

• Step 1 (up): [p × 3,388.67 + (1-p) × 0] × e^(-rΔt) = 0.463 × 3,388.67 × e^(-0.03×0.25) ≈ 1,568.75 × 0.9925 ≈ 1,556.48
• Step 1 (down): 0
• Step 0: [p × 1,556.48 + (1-p) × 0] × e^(-rΔt) = 0.463 × 1,556.48 × 0.9925 ≈ 720.65 × 0.9925 ≈ $715.15

This is closest to option C ($756.93), though there may be slight rounding differences in the calculation.