A stock is currently trading at $14.56 and has a volatility of 32% per annum and a continuous dividend yield of 1.30% per annum with continuous compounding. The riskfree interest rate is 2.51% per annum with continuous compounding for all maturities. Given your expertise in stock option valuation you decide to use a fourstep binomial tree to calculate the following derivative prices (to four decimal places):

1. (a) A oneyear European put option with a strike of $15.00. Calculate also the value of the option by using the BlackScholes formula. Compare and comment.
2. (b) A oneyear American put option with a strike of $15.00. Is the answer different to the answer from (a)? Explain.
3. (c) A short position in a forward contract on the stock for delivery in one year at a price of $15.00. Calculate also the theoretical value of the forward contract. Compare and comment. (Hint: the fair forward price that would be agreed for new contracts today is around $16.71 = $15.00, hence the forward contract in question has a nonzero value.)
4. (d) A European upandout barrier put option with a strike of $15.00 and knockout barrier of $17.00 maturing in one year. An upandout put option gives the holder the right to sell the underlying asset at the strike price on the expiration date so long as the price of that asset did not go above a predetermined barrier during the option's lifetime. When the price of the underlying asset rises above the barrier, the option is "knockedout" and no longer carries any value. Comment on the price of this option relative to the option in (a) and explain any differences.