Q3) (30 marks) Suppose that a stock price is modelled by a two period recombining binomial model, where each period is six months, with the following parameters: risk free interest rate r = 4% p.a.; volatility = 10% p.a.; initial share price S = 150. The up step is given by ure Consider a European call option, expiring in one year's time with exercise price 110. (i) Construct a two period recombining tree, showing the share price at each node, and derive the risk neutral probabilities assuming that no dividends are payable. [5] (ii) Using the binomial tree in (i), determine the value of the option. [5] Assume now that a dividend of 45 is payable immediately before the expiry of the option. (iii) (a) Write down the share prices at the final three nodes of the binomial tree in (ii) assuming the same total return as before. [5] (b) Write down the payoff at each of these three nodes of the option. [5] 2 (iv) Using the risk neutral probabilities from (1) determine the value of the payoff under the option at the end of the first time period, assuming the share price has jumped up in the first time period . [2] (v) Suppose now that the option is American, not European. (a) Determine the payoff at the node described in (iv) (i.e. at the end of the first time period, assuming the share price has moved up in the first time period). [3] (b) Using (a), determine whether the holder of the American option should exercise early at the node described in (iv). [5] Q3) (30 marks) Suppose that a stock price is modelled by a two period recombining binomial model, where each period is six months, with the following parameters: risk free interest rate r = 4% p.a.; volatility = 10% p.a.; initial share price S = 150. The up step is given by ure Consider a European call option, expiring in one year's time with exercise price 110. (i) Construct a two period recombining tree, showing the share price at each node, and derive the risk neutral probabilities assuming that no dividends are payable. [5] (ii) Using the binomial tree in (i), determine the value of the option. [5] Assume now that a dividend of 45 is payable immediately before the expiry of the option. (iii) (a) Write down the share prices at the final three nodes of the binomial tree in (ii) assuming the same total return as before. [5] (b) Write down the payoff at each of these three nodes of the option. [5] 2 (iv) Using the risk neutral probabilities from (1) determine the value of the payoff under the option at the end of the first time period, assuming the share price has jumped up in the first time period . [2] (v) Suppose now that the option is American, not European. (a) Determine the payoff at the node described in (iv) (i.e. at the end of the first time period, assuming the share price has moved up in the first time period). [3] (b) Using (a), determine whether the holder of the American option should exercise early at the node described in (iv). [5]
