Consider a European put option on a non-dividend-paying stock where the stock price is $45, the strike price is $43, the risk-free rate is 3% per annum, the volatility is 30% per annum and the time to maturity is 9 months. Consider a three-step tree. (Hint: t = 3 months).

(a) Compute u and d.

(b) Compute the European put price using a three-step binomial tree.

(c) If the option in (b) is American instead of European, would it ever be optimal to exercise it earlier than maturity at any of the nodes of the tree? In which node is the option early exercised? Compute the American put price.

(d) Calculate the delta of the European put option at each time step in (b).

(e) Using the result obtained in (b) and the put-call parity, compute the price of an European call with same underlying, strike and maturity of the put option.

(f) If the call option in (e) is American instead of European, would it ever be optimal to exercise it earlier than maturity at any of the nodes of the tree? Explain in detail and give the price of this Amer- ical call option.