Consider a European put option on an non-dividend paying stock with a maturity of six months

Consider a European put option on an non-dividend paying stock with a maturity of six months. The current stock price is $80 per share, and the continuously compounded risk-free interest rate is 5% per year. The strike price of the put option is $85. Assume that the stock price follows a log-normal distribution with a volatility of 30% per year. The risk neutral probability is 0.46257

Set up a two-period binomial model and use McDonald's formula to compute stock prices in the binomial tree.

Questions:

1. Use the risk-neutral probability method to compute the value of the put option at node u: $__________.

2. The value of the put option at node d is: $______________.

3. Based on your answer for questions 1 and 2, use the risk-neutral probability to compute the price of the put option at node 0: $_________.

4. Now, let's compute discounted cash flow in one step.

We use π to denote the risk-neutral probability. Starting from node 0, the probability of reaching node uu (the stock price at this node is 110.7225) is π²; the probability of reaching node ud (S_ud=82.0252) is 2×π(1-π); and the probability of reaching node dd (S_dd=60.7658) is (1-π)² . Denote the payoff of the put option at these nodes to be P_uu, P_ud, and P_dd, respectively.

The expected payoff of the put option under the risk-neutral probability can be computed as:

EP=π² P_uu+2 π(1-π) P_ud +(1-π)² P_dd.

Now, let's compute the discounted expected cash flow: e^{-0.05 * 0.5} EP=$____________. Compare your solution with the one obtained in question 3.