Answer: USD 10.00

## Explanation To value this American put option using a one-step binomial tree, we need to consider the possibility of early exercise since it's an American option. **Given:** - Strike price (K) = $50 - Spot price (S) = $40 - Up movement = +$8, Down movement = -$8 - Time period = 6 months (0.5 years) - Risk-free rate (r) = 6.2% **Step 1: Calculate stock prices at expiration** - Up state: S_u = $40 + $8 = $48 - Down state: S_d = $40 - $8 = $32 **Step 2: Calculate option payoffs at expiration** - Put payoff in up state: max(K - S_u, 0) = max(50 - 48, 0) = $2 - Put payoff in down state: max(K - S_d, 0) = max(50 - 32, 0) = $18 **Step 3: Calculate risk-neutral probabilities** - u = 48/40 = 1.2 - d = 32/40 = 0.8 - Risk-neutral probability: p = (e^(rΔt) - d)/(u - d) - p = (e^(0.062×0.5) - 0.8)/(1.2 - 0.8) = (1.0315 - 0.8)/0.4 = 0.2315/0.4 = 0.57875 **Step 4: Calculate European put value (no early exercise)** - European put = e^(-rΔt) × [p × payoff_up + (1-p) × payoff_down] - = e^(-0.062×0.5) × [0.57875 × 2 + 0.42125 × 18] - = 0.9695 × [1.1575 + 7.5825] - = 0.9695 × 8.74 = $8.47 **Step 5: Check for early exercise** - Immediate exercise value = max(K - S, 0) = max(50 - 40, 0) = $10 - Since $10 > $8.47, early exercise is optimal **Therefore, the American put value is $10.00** The key insight is that for American put options, when the immediate exercise value exceeds the risk-neutral expected value, early exercise is optimal, making the American put worth more than its European counterpart.

Author: LeetQuiz .