Question (Options Arbitrage - 30 marks)

The price of a non-dividend paying stock is currently $140. In one year's time the stock price will be either $160 or $120. The risk free rate with continuous compounding is 5% per annum. A European put option on the stock has a strike price of $150 and a maturity of one year. The theoretically correct value of the option is $9.1475.

Required

(a) Present the arbitrage calculations verifying that the correct value of the option is $9.1475. (5 marks)

(b) Suppose the option is trading in the market at $14. Calculate the profit available to an arbitrageur. Show all calculations. Give brief explanatory comments to show you understand the logic behind the arbitrage technique. (5 marks)

(c) Suppose the option is trading in the market at $8. Calculate the profit available to an arbitrageur. Show all calculations. Give brief explanatory comments to show you understand the logic behind the arbitrage technique. (5 marks)

(d) Use put-call parity to find the value of a European call option on the same stock with the same strike price and the same maturity. (5 marks)

(e) Explain how the actions of arbitrageurs will move the prices of the options, the shares, and the risk free asset to enforce theoretically correct pricing. (10 marks) (Total 30 marks)

Guidance for this question: In your calculations round your calculations to four decimal places. In parts (b) and (c) it is good presentation to give the tables giving the arbitrage position and cash flows at the time t = 0 (today) and at time t = 1 (in one year's time).