Question 1 [15 marks]

A price on a non-dividend paying stock is currently £50. Over each of the next two six-month periods the stock is expected to go up by 5% or down by 10%. The risk- free interest rate is 3% per annum with continuous compounding.

(a) What is the value of a one-year European call option with a strike price of £48?   [5 marks]

(b) What is the value of a one-year American call option with a strike price of £48?   [5 marks]

(c) Discuss how your answer to (b) would change if the stock instead actually did pay cash dividend?        [5 marks]

Question 2 [20 marks]

A non-dividend paying stock costs £100 at the end of today. You estimate that this  stock has an expected return of 20% per year and a volatility of 50% per year. Assume that the stock return follows a Geometric Brownian motion. Consider that a year is made up of 250 trading days.

(a) Calculate the expected value of the stock price and standard deviation of the

change in stock price at the end of the next trading day.                         [5 marks]

(b) Calculate the 99% confidence interval for the stock price at the end of the next  trading day. Calculate the 99% confidence interval for the stock price at the end of the next calendar day, i.e. consider that a year is made up of 365 calendar days.   [5 marks]

(c) Comment on the difference in your answer to (b) versus (c)             [5 marks]

(d) Consider the likelihood of observing a stock price below £92 the next day. [5 marks]

Question 3 [15 marks]

It is July 16. A company has a portfolio of stocks worth £30 million. The beta of the portfolio is 1.0. The company would like to use the December futures contract on a stock index to change beta of the portfolio to zero during the period July 16 to November 16. The index is currently 1,250, and each contract is on £250 times the index.

(a) What position should the company take?                             [5 marks]

(b) If the company instead wants to increase its portfolio beta from 1 to 2, what position in futures contracts should it take?   [5 marks]

Do the positions in (a) and (b) contain basis risk?          [5 marks]

Question 4 [20 marks]

A trader owns 55,000 units of a particular asset and decides to hedge the value of her position with futures contracts on another related asset. Each futures contract is on 5,000 units. The spot price of the asset that is owned is £28 and the standard deviation of the change in this price over the life of the hedge is estimated to be £0.43. The futures price of the related asset is £27 and the standard deviation of the change in this over the life of the hedge is £0.40. The coefficient of correlation between the spot price change and futures price change is 0.95.

(a) What is the minimum variance hedge ratio?     [5 marks]

(b) Should the hedger take a long or short futures position?    [5 marks]

(c) What is the optimal number of futures contracts with no tailing of the hedge? [5 marks]