We consider a two$-$period binomial model with the following properties: each period lasts one year and the current stock price is $\backslash ($ S$\_\{0\}=4\backslash ).$ On each period, the stock price doubles when it moves up and is reduced by half when it moves down. The annual interest rate on the money market is $\backslash (25\backslash \%\backslash ).($Notice that this interest rate is not realistic, but allows for easier calculations$).$

We consider two options on this market:

$-$ A call option with maturity $\backslash ($ T$=2\backslash )$ years and strike price $\backslash ($ K$=6\backslash )$;

$-$ A put option with maturity $\backslash ($ T$=2\backslash )$ years and strike price $\backslash ($ K$=6\backslash ).$

$($a$)$ For both options above, find the price at time $0.$

$($b$)$ For both options above, find the replicating portfolio composition at each time and position.

$($c$)$ Assume now that the interest rate is $\backslash (0\backslash \%\backslash ).$ Find the new prices of both options. Comment on the change in prices.

$($d$)$ We denote the call price at time $0$ by $\backslash ($ C$\_\{0\}\backslash )$ and the put price at time $0$ by $\backslash ($ P$\_\{0\}\backslash ).$ For both $\backslash ($ r$=25\backslash \%\backslash )$ and $\backslash ($ r$=0\backslash \%\backslash ),$ find the value of $\backslash ($ C$\_\{0\}-$P$\_\{0\}\backslash ).($See Problem $2$ for the impact of this question.$)$