Question 1
(25 marks)
A stock price is currently £70. Over each of the next two three-month periods, the stock price can go up by 7% or down by 3%. The stock does not pay any dividend. The riskfree interest rate is 2% per annum with continuous compounding.
(a) What is the value of a six-month European call option with a strike price of £73? [5 marks]
(b) What is the value of a six-month European put option with a strike price of £73? [5 marks]
(c) Verify that put-call parity holds. [5 marks]
(d) If the call option was an American option, explain whether it would be optimal to exercise it early. [5 marks]
(e) If the put option was an American option, explain whether it would be optimal to exercise it early. [5 marks]
Question 2
(25 marks)
Portfolio A consists of a one-year zero-coupon bond with a face value of £1,000 and an 11-year zero-coupon bond with a face value of £3,000. Portfolio B consists of a 7- year zero-coupon bond with a face value of £5,000. The current yield on all bonds is 7% per annum. (Answer the question with two-decimal precision in your calculations).
(a) Define duration, and explain why Portfolio B has a duration of 7 years. [5 marks]
(b) Show that Portfolios A has the same duration as Portfolio B. Interpret your result. [5 marks]
(c) Calculate the actual percentage change in the values of the two portfolios for a 0.07% per annum increase in yields. Interpret your results. [5 marks]
(d) Calculate the actual percentage changes in the values of the two portfolios for a 6% per annum increase in yields. Interpret your results. [5 marks]
(e) What do you conclude about the usefulness of duration when comparing bond portfolios, and what do you infer about the convexity of the two portfolios? [5 marks]
Question 3
(25 marks)
A one-month European put option on a non-dividend-paying stock is currently selling for £2. The stock price is £96, the strike price is £100, and the risk-free interest rate is 2% per annum with continuous compounding.
(a) Evaluate the price of the option against its theoretical upper and lower bounds. [5 marks]
(b) Explain what combination of trades would generate arbitrage profits.[5 marks]
(c) Explain what happens at maturity of the arbitrage trades (calculate the arbitrage profits and show that it is positive irrespective of what the stock price happens to be at the maturity of the arbitrage trades). [5 marks]
(d) You are informed that the stock has a volatility of 30% per annum. Determine the price of the one-month European put option based on the Black-Scholes Merton model. [5 marks]
(e) Relate your answer in (d) to your answer in (a). [5 marks]
Question 4
(25 marks)
A US-based investor is considering a two-year forward contract on the dollar price of pounds. The current USD/GBP exchange rate is $1.4. The continuously compounded USD interest rate is 1%, and the continuously compounded GBP interest rate is 2%. (Answer the question with four-decimal precision in your calculations).
(a) Calculate the forward price on the two-year forward contract. Explain the intuition behind your calculation. [5 marks]
(b) If the investor receives a market quote on the two-year forward of $1.51, what can she do to lock in an arbitrage profit? The quote is offered on the condition that she enters into an integer number of forward contracts (i.e. she cannot enter into parts of a forward contract). [5 marks]
(c) Calculate the exact arbitrage profit that she can make by taking the steps you specify in (b). [5 marks]
(d) If the investor receives a market quote for the two-year forward of $1.29, what can she do to lock in an arbitrage profit? The quote is offered on the condition that she enters into an integer number of forward contracts (i.e. she cannot enter into parts of a forward contract). [5 marks]
(e) Calculate the exact arbitrage profit that she can make by taking the steps you specify in (d). [5 marks]
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