Problem 4. (20 Points) Assume that the price S of a risky asset follows a binomial model with S(0) = 50, u = 10%, d = 10%. In this market, the risk-free return is 5%. The time horizon will be N = 3.

1. (a)(3 Points) Determine the tree corresponding to the prices of an American put option with strike price X = 50 and expiry date N = 3.
2. (b)(10 Points) Hedge the aforementioned option, i.e., provide the positions in the risky asset and the risk-free asset at every node of the tree.
3. (c)(7 Points) Assume that you are the writer of the option and that the holder decides to exercise at the node (2, 2). Do you expect to have any (strictly positive) profit? If yes, determine it. Do you think this is an arbitrage opportunity? Explain your conclusion.