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 $$50,u=1\mathrm{.}20,d=0\mathrm{.}80,$ and the continuously compounded interest rate is $5\%.$ What is the replicating portfolio at the initial node of the binomial tree of a $2-$year European put option with a strike price of $$52?$

A$.$ Short sell $0\mathrm{.}2773$ shares and invest $$17\mathrm{.}65.$

B$.$ Short sell $0\mathrm{.}2918$ shares and invest $$20\mathrm{.}38.$

C$.$ Short sell $0\mathrm{.}3699$ shares and invest $$18\mathrm{.}62.$

D$.$ Short sell $0\mathrm{.}3911$ shares and invest $$20\mathrm{.}46.$

E$.$ Short sell $0\mathrm{.}4025$ shares and invest $$24\mathrm{.}32.$

What is the price of the above European put option?

A$.$ $$3\mathrm{.}13$

B$.$ $$3\mathrm{.}27$

$$4\mathrm{.}19$

D$.$ $$3\mathrm{.}66$

E$.$ $$4\mathrm{.}87$