$(*)($FFOM $12\mathrm{.}27$ modified$)$ 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?