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.