1. [20 marks] Single-period market model. Consider a single-period security market model M = (B, S) on a finite state space 12 {W1,W2, W3}. Assume that the savings account B equals Bo =1, B1 = 2 and the stock price S satisfies So = 15 and = Si = (Si(wi), Si(wa), S(W3)) = (36, 27, 24). The real-world probability P on N is such that P(wi) = pi > 0 for i = 1, 2, 3. (a) Check directly using only the definition of an arbitrage opportunity that the model M (B, S) is arbitrage-free. = (b) Find the class M of all risk-neutral probability measures for the model M. Is the market model M complete? (c) Describe the class A of all attainable contingent claims. (d) Show that the contingent claim X = (X(wi), X(w2), X(W3)) = (10, 7, 6) is attainable and compute its arbitrage price To(X) using the repli- cating strategy for X. (e) Consider the contingent claim X = (30, 21, 18). Show that the ex- pected value X EO ( Bi does not depend on the choice of a risk-neutral probability measure QE M. Is this claim attainable? 1. [20 marks] Single-period market model. Consider a single-period security market model M = (B, S) on a finite state space 12 {W1,W2, W3}. Assume that the savings account B equals Bo =1, B1 = 2 and the stock price S satisfies So = 15 and = Si = (Si(wi), Si(wa), S(W3)) = (36, 27, 24). The real-world probability P on N is such that P(wi) = pi > 0 for i = 1, 2, 3. (a) Check directly using only the definition of an arbitrage opportunity that the model M (B, S) is arbitrage-free. = (b) Find the class M of all risk-neutral probability measures for the model M. Is the market model M complete? (c) Describe the class A of all attainable contingent claims. (d) Show that the contingent claim X = (X(wi), X(w2), X(W3)) = (10, 7, 6) is attainable and compute its arbitrage price To(X) using the repli- cating strategy for X. (e) Consider the contingent claim X = (30, 21, 18). Show that the ex- pected value X EO ( Bi does not depend on the choice of a risk-neutral probability measure QE M. Is this claim attainable