f L U' I. 1. [12 marks] Single-period multi-state model. Consider a single-period market model M = (H, S} on a nite sample space 51 = {whwmwg}. We assume that the money market account H equals B\". = 1 and B, = 4 and the stock price 8 = {30,31} satises Sn 2 5 and S; = [36,21], 4]. The real-world probability IP\" is such that IF'[w,) = p,- 3} [1 fort = 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 for the contingent claim X : [5, 1, 3} and com- pute the arbitrage price MOE) at time {1 through replication. {c} Compute the arbitrage price emit} using the risk-neutral valuation formula with an arbitrary martingale measure Q from the class ML {d} Show directly that the contingent claim 1\" = [Y[w1),Y[w2}, Y{w3]) : (10,8, 2} is not attainable, that is, no replicating strategy for lr\" exists in M. {e} Find the range of arbitrage prices for 1" using the class MI of all martingale measures for the model M. {f} Suppose that you have sold at time 1'] the claim Y at the price of 3 units of cash. Show that you may nd a portfolio {3, {p} with the initial wealth a: = 3 such that H{s,c,9} :3 Y, that is, V1[c,