Mathematical Finance

2. [10 marks] Single-period multi-state model. Consider a single-period market model M= (B, S) on 2 = {W1,W2, W3}. Assume that the savings account B equals Bo = B1 = 1 and the stock price S = (S0, S1) satisfies So = 7 and Si = (Si(wi), Si(w2), Si(w3)) = (9,6,4). The real-world probability P is such that Pwi) = pi > O for i = 1,2,3. (a) Check directly that the model Mis arbitrage-free (use Definition 2.2.3). (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) Check that the contingent claim X = (X(wi), X(w2), X(W3)) = (-5, -2,0) is attain- able and compute its arbitrage price ro(X) using the replicating strategy for X. (e) Consider again the contingent claim X = (-5, -2,0). Show that the expected value does not depend on the choice of a risk-neutral probability Q E M and coincides with the arbitrage price of X. 2. [10 marks] Single-period multi-state model. Consider a single-period market model M= (B, S) on 2 = {W1,W2, W3}. Assume that the savings account B equals Bo = B1 = 1 and the stock price S = (S0, S1) satisfies So = 7 and Si = (Si(wi), Si(w2), Si(w3)) = (9,6,4). The real-world probability P is such that Pwi) = pi > O for i = 1,2,3. (a) Check directly that the model Mis arbitrage-free (use Definition 2.2.3). (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) Check that the contingent claim X = (X(wi), X(w2), X(W3)) = (-5, -2,0) is attain- able and compute its arbitrage price ro(X) using the replicating strategy for X. (e) Consider again the contingent claim X = (-5, -2,0). Show that the expected value does not depend on the choice of a risk-neutral probability Q E M and coincides with the arbitrage price of X