2. Let T = 2, 12 = {w1, ...,W4}, P({wi}) > 0 for i = 1,...,4, Fo = {0,12}, F1 = {{w1,w2}, {w3, w4},0,12}, and F2 be the collection of all subsets of 12. Consider a riskless asset with price process S = {S,t = 0,1, 2} where So = 1 for all t, and a risky asset with price process Sl = {St, t = 0,1, 2} such that S(wi) = 5, S](wi) = 8, S? (wi) = 9 S(w2) = 5, Si(wa) = 8, s} (wa) = 6 S(w3) = 5, Si(w3) = 4, S2(W3) = 5 s (wa) = 5, s}(wy) = 4, s(wa) = 2. Then X = max(0, S) 6, S1 - 6, S2 - 6 is the value at time T of a so-called look-back option, where this value de- pends on the maximum of the prices of the underlying asset sl over the time interval from 0 to T. (a) Draw a tree to indicate the possible paths followed by the risky asset price process S. (b) Find an equivalent martingale measure for the model. (c) Find a replicating strategy for the option whose value at time T is given by X (d) What is the arbitrage free price for the option at time zero? 2. Let T = 2, 12 = {w1, ...,W4}, P({wi}) > 0 for i = 1,...,4, Fo = {0,12}, F1 = {{w1,w2}, {w3, w4},0,12}, and F2 be the collection of all subsets of 12. Consider a riskless asset with price process S = {S,t = 0,1, 2} where So = 1 for all t, and a risky asset with price process Sl = {St, t = 0,1, 2} such that S(wi) = 5, S](wi) = 8, S? (wi) = 9 S(w2) = 5, Si(wa) = 8, s} (wa) = 6 S(w3) = 5, Si(w3) = 4, S2(W3) = 5 s (wa) = 5, s}(wy) = 4, s(wa) = 2. Then X = max(0, S) 6, S1 - 6, S2 - 6 is the value at time T of a so-called look-back option, where this value de- pends on the maximum of the prices of the underlying asset sl over the time interval from 0 to T. (a) Draw a tree to indicate the possible paths followed by the risky asset price process S. (b) Find an equivalent martingale measure for the model. (c) Find a replicating strategy for the option whose value at time T is given by X (d) What is the arbitrage free price for the option at time zero