A stock index is currently 990, the risk-free rate is 5%, and the dividend yield on the index is 2%.

(a) Use a three-step tree to value an 18-month American put option with a strike price of 1,000 when the volatility is 20% per annum.

(b) How much does the option holder gain by being able to exercise early? When is the gain made?

(c) What position in the stock is initially necessary to hedge the risk of the put option?