[25points] Question 1 [default,02] (a) Consider the following one period trinomial model: {W, W2, W3}, P(w;) = 1/3 for i = 1, 2, 3, a bank account B with interest rate r = 0, and one stock S with So = 6 and S(w) = { 2, if w=w, if w = W2, 12, if w = W3. = 6, We denote with C(K) the European call option (on the stock) with strike price K 0; this has payoff C(K) := (S K) at time 1. Answer the following questions and justify carefully with either proofs or counterexamples. (b) [2 points] Is the market (B, S) arbitrage free? (c) [3 points] Is the call option C(K) with strike K 4 replicable? (d) [6 points] Find arbitrage free prices Co(K) (at time 0, in the given market (B, S)) of a call with strike K = 4. (e) [6 points] Consider the enlarged market (B, S, C(K)) made of: bank account, stock, call option with strike K 4 sold at time 0 at an arbitrage-free price Co(4). Is this market complete? Does the answer depend on which arbitrage-free price the option C(K) is sold at? = (f) [8 points] Enlarge the market (B, S, C(4)) considered in the previous item with call options with strike K = 5, sold at time 0 at price Co(K). We do not assume that Co (5) is necessarily an arbitrage free price; instead we assume that Co(K) satisfies the inequalities Co(K) Co(K) Co(K) + K K (So Kz)* < Co(K2) S. (A) (B) It can be shown that in any market model where at least one of these inequalities fails there is an arbitrage. Does the converse hold, i.e. do our assumptions imply that the enlarged market (B, S, C(K), C(K)) is arbitrage free? If yes, prove it; if not, explicitly find values of Co(K), Co(K) which satisfy (A), (B) and for which the market admits an arbitrage.