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? $($You are required to draw a $3-$step Binomial tree with correct values at each node.$)$