Consider a non$-$dividend$-$paying stock whose current price S$(0)=$ S is $$50.$

After each period, there is a $50\%$ chance that the stock price goes up by $15\%.$ If the stock

price does not go up$,$ then it drops by $15\%.$ A European call option and a European put

option on this stock expire on the same day in $4$ months at $$55$ strike. Current risk$-$free

interest rate is $3\mathrm{.}6\%$ per annum, compounded monthly. Count a month as one period.

$($a$)$ Construct a four$-$period binomial lattice tree to calculate the stock price after four

months.

$($b$)$ Construct a four$-$period binomial lattice tree to calculate the current $($t $=0)$ call option

price.

$($c$)$ Construct a four$-$period binomial lattice tree to calculate the current $($t $=0)$ put option

price.

$($d$)$ Calculate the put option price using the Put$-$Call Parity and comparae the result with

that of part $($c$)$

$($e$)$ Find the price of an American put option on the stock that has the same strike price

and expiration date as the European ones.