The risk-neutral probability of an up move is $57.61%$ (calculated in the previous question).

• Node A (Initial):
  • Stock: $50$
  • Formula: $\max(0, 52-50) = 2$
  • Value: $5.86$
  • Branches to Node B and Node C
• Node B (Up from A):
  • Stock: $50 \times 1.2 = 60$
  • Formula: $\max(0, 52-60) = 0$
  • Condition: $1.65 > 0$
  • Branches to Node D and Node E
• Node C (Down from A):
  • Stock: $50 \times 0.8 = 40$
  • Formula: $\max(0, 52-40) = 12$
  • Condition: $10.46 < 12$
  • Branches to Node E and Node F
• Node D (Up from B):
  • Stock: $50 \times 1.2 \times 1.2 = 72$
  • Formula: $\max(0, 52-72) = 0$
• Node E (Down from B / Up from C):
  • Stock: $50 \times 1.2 \times 0.8 = 48$
  • Formula: $\max(0, 52-48) = 4$
• Node F (Down from C):
  • Stock: $50 \times 0.8 \times 0.8 = 32$
  • Formula: $\max(0, 52-32) = 20$

The figure shows the stock price and the respective option value at each node. At the final nodes, the value is calculated as $\max(0, K-S)$ and the following payoffs are obtained:

• Node [D]: Intrinsic value of the put option = $\max(52-72, 0) = 0$
• Node [E]: Intrinsic value of the put option = $\max(52-48, 0) = 4$
• Node [F]: Intrinsic value of the put option = $\max(52-32, 0) = 20$

Next, assess the option values at each of the other nodes as follows:

• Node [B]: $(0.5761 \times 0 + 0.4239 \times 4) \times \exp(-0.12 \times 3/12) = 1.65$, which is greater than the intrinsic value of the option at this node equal to $\max(0, 52-60) = 0$, so the option should not be exercised early at this node.

• Node [C]: $(0.5761 \times 4 + 0.4239 \times 20) \times \exp(-0.12 \times 3/12) = 10.46$, which is lower than the intrinsic value of the option at this node equal to $\max(0, 52-40) = 12$, so the option should be exercised early at node C with the value of the option at node C being $12$.

• Node [A]: $(0.5761 \times 1.65 + 0.4239 \times 12) \times \exp(-0.12 \times 3/12) = 5.86$, which is greater than the intrinsic value of the option at this node equal to $\max(0, 52-50) = 2$, so the option should not be exercised early at this node.

Therefore, the no-arbitrage price of the option at node A = USD $5.86$.

A is incorrect. USD $2.00$ is the intrinsic value of the option at the initial date, node A.

B is incorrect. USD $5.23$is the value of the option if it is a European put option, and thus only exercised at expiration in `$6`$ months.