12:29 Avg f Discrete time finance .pdf 1. Be as precise as possible and use mathematical notation where necessary (a) Is it possible in an arbitrage-free general single period market model with = {1,2a}, two stocks and a Bank Account that there are contingent claims that are not attainable? Justify your answer. 13 marks] (b) in an arbitrage free general single period market model, show that for an attainable contingent claim XE = 1, where is the price of the corresponding replicating strategy and Q is a risk-neutral measure 14 marks] (c) State a definition of an arbitrage strategy in the context of a general multi-period market model. 14 marks (d) Consider an arbitrage-free incomplete single-period market model. A new attainable asset is introduced in this model at a price that complies with the no arbitrage principle. Explain why the model will remain incomplete 15 marks (e) Show that the variance of the return of a portfolio consisting of equal proportions of wealth invested in assets whose returns to i-1..-, are identically distributed and uncorrelated with E(r.) = and Var(r) = o. i = 1,..., goes to zero as n goes to infinity 14 marks] (f) In a sample space 12 - {w.w.w.w.ws), write the o-algebra generated by the random variable X Wwws -{ = [2 marks] (5) Consider a two-period market model with a bond with Bi = 1 and risk-free rate -0.02 and one risky asset whose price dynamics are given in the following table: So 50 50 50 50 s 60 60 40 40 S. 8050 40 i. Consider a strategy in the above market that takes the following values at -0.1: 1-3, 8-20 and 14. www Calculate the values of so that the strategy is self-financing Page 2 of 6 Justify why the strategy you calculated is adapted to the filtration generated by the stock price 15 marks (h) Give an example of an investment outcome that shows that a concave utility func tion expresses risk-aversion 13 marks [Total 30 marks] 12:29 Avg f Discrete time finance .pdf 1. Be as precise as possible and use mathematical notation where necessary (a) Is it possible in an arbitrage-free general single period market model with = {1,2a}, two stocks and a Bank Account that there are contingent claims that are not attainable? Justify your answer. 13 marks] (b) in an arbitrage free general single period market model, show that for an attainable contingent claim XE = 1, where is the price of the corresponding replicating strategy and Q is a risk-neutral measure 14 marks] (c) State a definition of an arbitrage strategy in the context of a general multi-period market model. 14 marks (d) Consider an arbitrage-free incomplete single-period market model. A new attainable asset is introduced in this model at a price that complies with the no arbitrage principle. Explain why the model will remain incomplete 15 marks (e) Show that the variance of the return of a portfolio consisting of equal proportions of wealth invested in assets whose returns to i-1..-, are identically distributed and uncorrelated with E(r.) = and Var(r) = o. i = 1,..., goes to zero as n goes to infinity 14 marks] (f) In a sample space 12 - {w.w.w.w.ws), write the o-algebra generated by the random variable X Wwws -{ = [2 marks] (5) Consider a two-period market model with a bond with Bi = 1 and risk-free rate -0.02 and one risky asset whose price dynamics are given in the following table: So 50 50 50 50 s 60 60 40 40 S. 8050 40 i. Consider a strategy in the above market that takes the following values at -0.1: 1-3, 8-20 and 14. www Calculate the values of so that the strategy is self-financing Page 2 of 6 Justify why the strategy you calculated is adapted to the filtration generated by the stock price 15 marks (h) Give an example of an investment outcome that shows that a concave utility func tion expresses risk-aversion 13 marks [Total 30 marks]
