3. (a) Consider a binomial model of a stock market where the price P of a stock can have only (i.e. either-or) two values at every time point. If C is the option value and y the number of stocks for each option at every time step, i. estimate the value of the share; (2 marks) ii. now assuming a riskless investment, when the stock value for a high market (i.e. market going up) is exactly balanced by the stock value for a low market (i.e. market going down), show that y can be expressed as a rate of change of the option value with respect to the stock price change. (3 marks) (b) A stock is presently trading at 200p per share at an annual interest rate of r% to get a selling price of either 100p or 300p. At time t = 0, if there are x units of stock and y units of option, the latter priced at C (pence) each, i. write down the total cost at t = 0, including both stock and option. (2 marks) ii. If at t = 0 the buyer has the right to purchase y units of stock at a price of 250p per share after 1 time period, find the relation between x and y at arbitrage. (4 marks) iii. What would be the value of the maximum profit after 1 time period, under the arbitrage condition? (3 marks) (c) A portfolio is constructed with N assets (C1, C2, C3, ....,Cn), all of which have the same unit price p at time t = 0. If the total asset value of the portfolio at time t = 0) is F, show that the return from the combined portfolio at a later time t = 1 is a weighted sum of the returns from the individual assets taken together. 3. (a) Consider a binomial model of a stock market where the price P of a stock can have only (i.e. either-or) two values at every time point. If C is the option value and y the number of stocks for each option at every time step, i. estimate the value of the share; (2 marks) ii. now assuming a riskless investment, when the stock value for a high market (i.e. market going up) is exactly balanced by the stock value for a low market (i.e. market going down), show that y can be expressed as a rate of change of the option value with respect to the stock price change. (3 marks) (b) A stock is presently trading at 200p per share at an annual interest rate of r% to get a selling price of either 100p or 300p. At time t = 0, if there are x units of stock and y units of option, the latter priced at C (pence) each, i. write down the total cost at t = 0, including both stock and option. (2 marks) ii. If at t = 0 the buyer has the right to purchase y units of stock at a price of 250p per share after 1 time period, find the relation between x and y at arbitrage. (4 marks) iii. What would be the value of the maximum profit after 1 time period, under the arbitrage condition? (3 marks) (c) A portfolio is constructed with N assets (C1, C2, C3, ....,Cn), all of which have the same unit price p at time t = 0. If the total asset value of the portfolio at time t = 0) is F, show that the return from the combined portfolio at a later time t = 1 is a weighted sum of the returns from the individual assets taken together
