Single period multi state model

1. [12 marks] Single-period multi-state model. Consider a single-period market model M = (B, S) on a finite sample space Q = {WI, w2, w3}. We assume that the money market account B equals Bo = 1 and Bl = 4 and the stock price S = (So, Sl) satisfies So = 2.5 and Sl = (18, 10, 2). The real-world probability P is such that P(Wi) = > 0 for i (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 (A, p) for the contingent claim X = (5, 1, 3) and compute the arbitrage price To (X) at time 0 through replication. (c) Compute the arbitrage price To (X) using the risk-neutral valuation formula with an arbitrary martingale measure Q from M. (d) Show directly that the contingent claim Y = (Y (WI), Y(W2), Y(W3)) = (10, 8, 2) is not attainable, that is, no replicating strategy for Y exists in M. (e) Find the range of arbitrage prices for Y using the class M of all martingale measures for the model M. (f) Suppose that you have sold the claim Y for the price of 3 units of cash. Show that you may find a portfolio @, p) with the initial wealth 3 such that Vl@, p) > Y, that is, Vl@, p) (wc) > Y (Wi) for i =