We are interested in pricing an at-the-money put option on a stock that is currently trading for $60. Suppose the annual risk-free rate is 0.02, the stock's annualized volatility is \(\sigma = .35\) , and the time to maturity is 5 months. Use continuous compounding for this problem.

A. For a one-step binomial tree, calculate the following: the risk-neutral probability of an upward stock movement, p, the amount of the upward movement, u, and the amount of the downward move, d.

B. Find the value of the option today. Draw the one-step binomial tree and clearly label the stock price and the option value at each node.

C. Now, suppose you divide the maturity into subperiods of equal length. Calculate the following: the risk-neutral probability of an upward stock movement, p, the amount of the upward movement, u, and the amount of the downward move, d.

D. Find the current price of the European put option today. Draw out the binomial tree and indicate the stock and option price at each node.

E. Again, assuming 5 subperiods, calculate the price of an American put option. Draw the binomial tree and indicate the stock and option price at each node. Also, highlight the nodes (if any) where it is optimal to exercise the option early.