1. [12 marks] Single-period multi-state model. Consider a single-period market model M = (B. S) on a finite sample space ? = (w1, waug}. We assume that the money market account B equals Bo = 1 and B1 = 4 and the stock price S = (So, S,) satisfies So = 2.5 and S1 = (18, 10, 2). The real-world probability Pis such that P(cy) = p; > 0 for i = 1, 2,3. (a) Find the class M of all martingale measures for the model M. Is the market model M arbitrage-free? Is this market model complete? (b) Find the replicating strategy (y", y ) for the claim X = (5,1, -3) and compute the arbitrage price To( X) at time 0 through replication. (c) Find the arbitrage price wo( X) using the risk-neutral valuation formula with an arbi- trary martingale measure @ from M. (d) Show directly that the contingent claim Y = (Y(w|), Y(wz), Y(@3)) = (10,8, -2) is not attainable and find the range of arbitrage prices for Y using the class M of all martin- gale measures. (e) For the contingent claim Z = (20, 16, -4), find the minimal initial endowment a for which there exists a portfolio (7, () with Vo(r, y) = + and such that the inequality Vi(r, )(w;) > Z(w;) holds for i = 1, 2, 3. (f) Can we interpret the number I as an arbitrage price for Z? Can we complete the market by assuming that Z is an additional primary asset traded at time 0 at the initial price equal to 3