Problem $3($American put with continuous dividend, $9$pts$).$ Consider an American put option.

The time to maturity is $2$ years. We adopt a two$-$step binomial tree, with each step equals to one

year. Moreover, in each time step, the stock price either moves up by $20\%$ or moves down by $20\%.$

The strike price of the option is $$52.$ The current stock price is $$50.$ The risk$-$free interest rate is

$7\%$ per annum in continuous compounding. The continuous dividend rate is $2\%$ per annum.

Calculate the price of the American put option.