Explanation

This is a binomial option pricing problem. Given:

• Stock price (S) = $75
• Strike price (K) = $90
• Risk-free rate (r) = 5%
• Time to expiration (T) = 3 years
• Risk-neutral probability of up move (p) = 60%

Since the stock price rises or falls by a proportional amount each year, we need to calculate the up and down factors. The volatility is given as 18.25%, which suggests we can use the standard binomial model parameters:

• u = e^(σ√Δt) = e^(0.1825×√1) = e^0.1825 ≈ 1.200
• d = 1/u ≈ 0.833

However, we're told the risk-neutral probability is approximately 60%, which is consistent with: p = (e^(rΔt) - d)/(u - d)

With 60% probability, the option will only have value if the stock price goes up significantly over 3 years. The call option pays max(S-K, 0).

Using the binomial tree:

• After 3 up moves: S = 75 × 1.200^3 ≈ 75 × 1.728 = $129.60 Payoff = max(129.60 - 90, 0) = $39.60
• Probability of 3 up moves = 0.6^3 = 0.216
• Present value = 39.60 × 0.216 × e^(-0.05×3) ≈ 39.60 × 0.216 × 0.8607 ≈ $7.36

This matches option D.