$6\mathrm{.}11$ A stock index is currently $990,$ the risk$-$free rate is $5\%,$ and the dividend yield on the index is $2\%.$ 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. How much does the option holder gain by being able to exercise early? When is the gain made?

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