The S&P 500, which covers the 500 largest companies on the U.S. stock market, is currently trading at 3,915.65 (index points). The dividend yield of the index is 4.2% per annum and the risk-free interest rate in U.S. is 3.6% per annum (both with continuous compounding for all maturities). The volatility of the index over the next 8 months is also estimated to be 28.5% 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 9-month European put option on the index with a strike of 4,350. Calculate also the value of the option by using the Black-Scholes formula. Compare the values and comment.

(b) A 9-month American put option with a strike of 4,350. 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 9 months at a price of 4,350. Calculate also the theoretical value of this forward position. Compare these values and comment. (Hint: the fair forward price that would be agreed for new contracts today is different to 4,500 and hence the forward contract in question has a non-zero value.)

(d) An American up-and-out barrier put option with a strike of 4,350 and knockout barrier of 4,750 maturing in 9 months. An American up-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 rise above a pre-determined barrier during the options lifetime. When the price of the underlying asset is above 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 increased from 4,750 to 4,900? Comment on the reason why.