Consider an American put option with 8 months to expiry, and a strike price of 50. The current price of the underlying is 47. Divide the time to expiry into two 4-months intervals. Assume that in each 4-month interval, the price can either rise by 5, or fall by 5, with unknown probability. The risk-free (continuously compounding) interest rate is 0.03.

1. a)Using a binomial tree, identify the circumstances under which early exercise would be rational for the holder of this option. [7]
2. b)What is the value of the option? [2]
3. c)What would the value of this option be if it were European and not American? Explain briefly why your answer should be no greater than that obtained in (b).
4. d)How would your answers in (b) and (c) change if the underlying asset paid a dividendof 5 in 7 months' time, and the stock price increase and decrease happen with equalprobability? [7]