Answer: USD 5.86

The risk-neutral probability of an up move is $57.61\%$ (calculated in the previous question). <details> <summary><strong>View Binomial Tree Diagram Details</strong></summary> * **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: $\max(0, 52-32) = 20$ </details> <br> 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: <details> <summary><strong>Step-by-Step Node Calculations</strong></summary> * **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. </details> <br> **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.