A stock price evolves in a standard binomial tree. Each period it can either go up to u = 1.2 times its previous price or down to d = 0.8 times its previous price. Consider a two period model (t = 0, 1, and 2), as depicted below, where each period corresponds to one year. The risk-free (net) return between t = 0 and t = 1 is r1 = 5%, while between t = 1 and t = 2 it is r2 = 10%. The initial stock price is S = 100 and the stock pays no dividends.

(a) Price an exotic Put option with maturity at t = 2 (and no early exercise allowed) and with a strike price equal to the maximum stock price up to that point, that is, K = max{S, S1, S2}, where S1 is the stock price at t = 1, and S2 is the stock price at t = 2.

(b) Price a derivative with maturity at t = 2, that obliges you to sell the stock at t = 2 (and not earlier) for price max{S, S1, S2}, where S1 is the stock price at t = 1, and S2 is the stock price at t = 2.