This question involves American put option valuation with early exercise considerations.

Given:

• Current stock price = $40
• Strike price = $50
• Up state price = $48
• Down state price = $32
• Risk-free rate = 6.2%
• Time period = 0.5 years

Risk-neutral probability: u = 48/40 = 1.2 d = 32/40 = 0.8 P_up = [e^{rΔt} - d] / (u - d) = [e^{0.062×0.5} - 0.8] / (1.2 - 0.8) = [e^{0.031} - 0.8] / 0.4 = [1.03148 - 0.8] / 0.4 = 0.5787

European put value calculation: Put payoff in up state = max(50 - 48, 0) = $2 Put payoff in down state = max(50 - 32, 0) = $18

European put value = [2 × 0.5787 + 18 × (1 - 0.5787)] × e^{-0.062×0.5} = [1.1574 + 18 × 0.4213] × 0.9694 = [1.1574 + 7.5834] × 0.9694 = 8.7408 × 0.9694 = $8.47

American put consideration: The immediate exercise value = max(50 - 40, 0) = $10 Since $10 > $8.47, early exercise is optimal, making the American put worth $10.

This demonstrates the key difference between American and European options - American options can be exercised early when it's beneficial, which in this case occurs because the immediate exercise value exceeds the calculated European option value.