Consider a European put option where the stock price is $28,$ the strike price is $$30,$ and the risk free rate is $5\%$ per annum.

$($a$)(18$ points$)$ If the life of the option is $6$ months, and there are two time steps $(3$ months each$),$ what is the value of the put if the volatility is $35\%?$

$($b$)(12$ points$)$ What is the value of the put if it is an American option and has a dividend yield of $2\mathrm{.}8\%$ per annum? $($Note: the only effect the dividend has is on calculating the value of $p,$ it is not used for discounting purposes$).$