$3.$ Consider an option on a stock $\backslash ($ A B C $\backslash )$ that pays no dividend. The option has the following characteristics: strike price $\backslash ($ X$=22\backslash ),$ maturity $\backslash ($ T$=2\backslash )$ years. Assume that the term structure of interest rate is flat at a rate $($annually compounded$)$ of $\backslash (1\backslash \%\backslash )$ and that the stock prices can take the following path over the next two years $($each period is one year$)$:

$($a$)$ Using a dynamic replicating porfolio, compute the price $\backslash ($ P $\backslash )$ of an European Put option on stock ABC $.$

$($b$)$ Using a dynamic replicating porfolio, compute the price $\backslash ($ P $\backslash )$ of an American Put option on stock ABC $.$

$($c$)$ Which option is more valuable and why?