Now consider pricing call and put options on a $2-$step binomial tree. Each step on the binomial tree is one year. The current stock price is $$50,u=1\mathrm{.}20,d=0\mathrm{.}80,$ and the continuously compounded interest rate is $5\%.$ What is the replicating portfolio at the initial node of the binomial tree of a $2-$year American put option with a strike price of $$52?$

A$.$ Short sell $0\mathrm{.}4162$ shares and invest $$26\mathrm{.}19.$

B$.$ Short sell $0\mathrm{.}4162$ shares and invest $$28\mathrm{.}33.$

C$.$ Short sell $0\mathrm{.}5293$ shares and invest $$31\mathrm{.}55.$

D$.$ Short sell $0\mathrm{.}5293$ shares and invest $$33\mathrm{.}48.$

E$.$ Short sell $0\mathrm{.}6013$ shares and invest $$39\mathrm{.}17.$

What is the price of the above American put option?

A$.$ $$5\mathrm{.}29$

B$.$ $$4\mathrm{.}99$

C$.$ $$4\mathrm{.}89$

D$.$ $$5\mathrm{.}19$

E$.$ $$5\mathrm{.}09$