(40 points). In this question we will price a European put and an American put. Assume that the underlying stock has a current price of $25 and pays no dividends. The risk-free rate is 5% per year, and the stock price follows the following binomial tree: $25 $33 $30 $27 $25 $20 $15 We will price puts with a strike price of $25 and maturity of 2 years. a) (10 points). We want to figure out the price of a European put with K=$25. Write down the payoff of the European put at time 2, and obtain the risk-neutral probabilities of the up and down movements in the binomial tree in all nodes. b) (10 points). What is the price of the European put? c) (10 points). Now, we want to replicate the European put with a portfolio of the stock and the risk-free bond. How many shares of the stock do you need to have in your portfolio in period o (at the beginning)? d) (10 points). Now, we want to price the American put, still with K=$25. Compare (at each node in times o and 1) the payoff from exercising the option to the payoff you get from not exercising it, and show whether you want to exercise the option early. What is the price of the American put?