Problem 1 (15 points). Consider the Binomial Asset Pricing Model, with u = 2, d = 1/2, r = 1/4 and S, = 4. Consider a European Call Option with strike price K = 9, maturing at time N = 3. (1) State the non arbitrage condition and compute the Risk-neutral probabilities. (2) Using a backward recursion, find the arbitrage-free price of the option at time 0, i.e. V, under the risk-neutral probability. Problem 2 (20 points). Consider the Binomial Asset Pricing Model, with u = 2,d = 1/2,r = 1/4 and Sh = 4. Consider an American Put Option with strike price K =9, maturing at time N = 3. Using a backward recursion, find the arbitrage-free price of the option at time 0, i.e. V, under the risk-neutral probability. Problem 3 (30 points). Consider a 3-step Binomial Asset Pricing Model with probability p of going up by a factor u, and probability q = 1 -pof going down by a factor d. We denote by S,, the stock price at step n. (1) Compute E, (Sa) the conditional expectation of S, given the information available up to time 1. (2) Compute E, (S,) and use it to compute E, [E, (S,)]. How does it compare with E, (S,)? (3) Use E, (S,) to compute En(S;). (4) Now assume that u = 2,d = 1/2, r = 1/4 and S, = 4. Generalize the result obtained from (3) to compute directly (in a single step using a Martingale property) the value of a European Call option with maturity T = 3 and Strike price K = 9. Problem 4 (15 points). Consider the Binomial Asset Pricing Model with probability p of going up by a factor u, and probability q of going down by a factor d. Prove that the stock price process is a Markov process. Under which condition the stock price process is also a Martingale? Problem 5 (20 points). Given a sample space of events ! = {#, 7), we consider the sequence of random i.id. variables (X,.),.> defined on ! = {#, T ] by Xn = 0 and X,(4;)= +1, if w; = H -1, if wj =T for j > 1. We define now a simple random walk (M,), >, by M, = 0 and M. = ) X; for any n > 1. (1) We assume that P(H ) = P(T) = 1/2. (a) Show that (M.), 2 is a Martingale. (b) Show that the sequence (2,),>, defined by Z,, = M2 -n is a Martingale. (2) We assume now that P(H ) = 2/3. (a) Is (M.)..> a Supermartingale or a Submartingale? (b) Is (2.).> a Supermartingale or a Submartingale