6) Consider a binomial tree model for the stock price St. Let S0 = 70 and...

 

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6) Consider a binomial tree model for the stock price St. Let S0 = 70 and let the price rise by 11% or fall by 6% each month for the next three months. Assume also that the risk-free rate is 5.5% per annum continuously compounded. (i) State the conditions under which the market is arbitrage free. (2 marks) (ii) Calculate the price at time t = (0 of a European call option on this stock, which expires in three months and is struck at-the-money (i.e. strike price K = 70). (4 marks) A special option, called a knock-out barrier option, goes out of existence (i.e. expires without any payoff or value) if the underlying asset reaches a pre-specified barrier b>0 either from above (down-and-out) or from below (up-and-out). The down-and-out call has the following payoff at time T: !3! max (S, -k:0) if min S, 2b, 0 otherwise Assume this special option is written on the given stock, has the same strike price and maturity as the European call option described in part (ii) and the barrier b is fixed at 50. (iii) Calculate the price of this contract using the binomial tree model and risk neutral valuation. (3 marks) (iv) Determine the price of the down-and-out contract when b = 60, without performing any further calculations. (2 marks) Q7) (10 marks) A non-dividend paying stock currently trades at $75. Every two years the stock price either increases by a multiplicative factor 1.4, or decreases by a multiplicative factor 0.85. The effective risk-free rate is 4% p.a. Calculate the price of an American put option written on the stock with strike price $70 and maturity four years, using a two-period binomial model. 6) Consider a binomial tree model for the stock price St. Let S0 = 70 and let the price rise by 11% or fall by 6% each month for the next three months. Assume also that the risk-free rate is 5.5% per annum continuously compounded. (i) State the conditions under which the market is arbitrage free. (2 marks) (ii) Calculate the price at time t = (0 of a European call option on this stock, which expires in three months and is struck at-the-money (i.e. strike price K = 70). (4 marks) A special option, called a knock-out barrier option, goes out of existence (i.e. expires without any payoff or value) if the underlying asset reaches a pre-specified barrier b>0 either from above (down-and-out) or from below (up-and-out). The down-and-out call has the following payoff at time T: !3! max (S, -k:0) if min S, 2b, 0 otherwise Assume this special option is written on the given stock, has the same strike price and maturity as the European call option described in part (ii) and the barrier b is fixed at 50. (iii) Calculate the price of this contract using the binomial tree model and risk neutral valuation. (3 marks) (iv) Determine the price of the down-and-out contract when b = 60, without performing any further calculations. (2 marks) Q7) (10 marks) A non-dividend paying stock currently trades at $75. Every two years the stock price either increases by a multiplicative factor 1.4, or decreases by a multiplicative factor 0.85. The effective risk-free rate is 4% p.a. Calculate the price of an American put option written on the stock with strike price $70 and maturity four years, using a two-period binomial model.

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