Problem 2. Consider a three-period model of Example 1.2.1 on p. 9 with So = 4, u = 2, d = 1 and interest rate r = 1, so that = q = . For n = 0, 1, 2, 3, define Mn = max Sk, the maximum of the stock k=0,..., price between times zero and n. Consider a floating strike lookback option, whose payoff at time T = 3 is given by MT - ST > 0. Let un(s, m) be price of the option at time n if Sn =s and Mn = m. In particular, V3(s,m) = m - s. (i) Develop an algorithm for computing un recursively. In particular, write a formula for Un in terms of Un+1. (ii) Apply the algorithm developed in (i) to compute vo(4,4). (iii) Provide a formula for ons,m), the number of shares of stock that should be held by the replicating portfolio at time n if Sn = s and Mn = m. You may assume that the functions un(:, :), n=0,..., 3 are already known. Problem 3. Repeat problem 2 for the European up-and-out call option with the strike 4 and up-and-out barrier 10, i.e. let us consider the settings of a three-period model of Example 1.2.1 on p. 9 with So = 4, u = 2, d = 1 and interest rate r = 1, so that p = q = 3. For n = 0,1,2,3, define Mn = max Sk, the k=0,...,n maximum of the stock price between times zero and n. Consider an up-and-out call option with the strike K = 4 and barrier B= 10, whose payoff at time T = 3 is given by max(ST K,0), if M B. Let un(s, m) be price of the option at time n if Sn =s and Mn = m. In particular, 03 (s, m) = max(s K0Im 0. Let un(s, m) be price of the option at time n if Sn =s and Mn = m. In particular, V3(s,m) = m - s. (i) Develop an algorithm for computing un recursively. In particular, write a formula for Un in terms of Un+1. (ii) Apply the algorithm developed in (i) to compute vo(4,4). (iii) Provide a formula for ons,m), the number of shares of stock that should be held by the replicating portfolio at time n if Sn = s and Mn = m. You may assume that the functions un(:, :), n=0,..., 3 are already known. Problem 3. Repeat problem 2 for the European up-and-out call option with the strike 4 and up-and-out barrier 10, i.e. let us consider the settings of a three-period model of Example 1.2.1 on p. 9 with So = 4, u = 2, d = 1 and interest rate r = 1, so that p = q = 3. For n = 0,1,2,3, define Mn = max Sk, the k=0,...,n maximum of the stock price between times zero and n. Consider an up-and-out call option with the strike K = 4 and barrier B= 10, whose payoff at time T = 3 is given by max(ST K,0), if M B. Let un(s, m) be price of the option at time n if Sn =s and Mn = m. In particular, 03 (s, m) = max(s K0Im