The example to be used is as follows: The Hang Seng Index, which covers the largest companies on the Hong Kong stock market, is currently trading at 25,900 (index points). The dividend yield of the index is 2.2% per annum and the risk-free interest rate in Hong Kong is 1.7% per annum (both with continuous compounding for all maturities). The volatility of the index over the next 12 months is also estimated to be 38% per annum. Given your expertise in option valuation you decide to use a four-step binomial tree to calculate the following derivative values (in units of index points, to two decimal places):

(a) A 12-month European put option on the index with a strike of 28,000. Calculate also the value of the option by using the Black-Scholes formula. Compare the values and comment.

(b) A 12-month American put option with a strike of 28,000. Is the answer different to the answer from (a)? Explain why if so.

(c) A short position in a forward contract on the index for delivery in 12 months at a price of 28,000. Calculate also the theoretical value of the forward position. Compare these values and comment. (Hint: the fair forward price that would be agreed for new contracts today is different to 28,000 and hence the forward contract in question has a non-zero value.)

(d) An American down-and-out barrier put option with a strike of 28,000 and knockout barrier of 20,000 maturing in 12 months. An American down-and-out put option gives the holder the right to sell the underlying asset at the strike price at any time on or prior to the expiration date so long as the price of that asset did not go below a pre-determined barrier during the options lifetime. When the price of the underlying asset falls below the barrier, the option is "knocked-out" and no longer carries any value. Comment on the value of this option relative to the option in (b) and explain any differences. Would the value of this option change if the knockout barrier was decreased from 20,000 to 15,000? Comment on the reason why.