1. [10 marks] Path-dependent American claim. Let M = (B, S) be the CRR model with r = 0 and the stock price S satisfying So = 8, SY = 11, ST = 7. Consider a path- dependent American claim with maturity 7 = 2 and the reward process g defined as fol- lows: go = 11, 91 = 12 and the random variable 92 is given by 92(S", 57") = 16, 92($1, 5,") 8, 92($1, S,") = 10 and 92 ($1, S,") = 18. (a) Let P be the probability measure under which the process S/B is a martingale. Com- pute the arbitrage price process (m(X"), t = 0, 1) for the American claim using the recursive relationship m(X" ) = max gt, BE( It+1(X" ) Bi+1 with the terminal condition *2(X") = 92. Find the rational exercise time To of this claim by its holder. (b) Find the replicating strategy y for the claim up to the random time ro and check that the equality Vi(y) = m(X") is valid for all t s 78. (c) Determine whether the arbitrage price process (m(X"); t = 0, 1, 2) is either a martin- gale or a supermartingale under P with respect to the filtration F. (d) Find a probability measure Q on the space (2, F2) such that the arbitrage price process (m (X"); t = 0, 1, 2) is a martingale under Q with respect to the filtration F and compute the Radon-Nikodym density of Q with respect to P on (2, F2). (e) Let P be a probability measure under which the process B/S is a martingale. Define the process (7,(X"), t = 0, 1) through the recursive relationship mi(X" ) = max gt, St Ep ( Fit+1 (X" ) St+1 Ft with 72(X") = 92. Is it true that the equality ,(X") = m(X") holds for all t = 0, 1, 2? Justify your answer but do not perform any computations with numbers