A stock price is currently $$50.$ Over each of the next two three$-$month periods it is expected to go up by $10\%$ or down by $10\%.$ The

annualized risk$-$free interest rate is $6\%$ per year. $($Note: this is a discrete $($not cc$)$ rate. Also, for a given three$-$month period, you need to

convert it to a $3-$month return, using $(1+r{)}^{0\mathrm{.}25}.)$ We will use a two$-$period binomial model to solve for the option prices.

Required:

a$.$ What is the risk$-$neutral probability for the binomial model in this setting?

b$.$ What is the value of a six$-$month European call option with a strike price of $51?$

c$.$ What is the value of a six$-$month American call option with a strike price of $51?$

d$.$ What is the value of a six$-$month European put option with a strike price of $51?$

e$.$ What is the value of a six$-$month American put option with a strike price of $51?$

Note: For all requirements, do not round intermediate calculations. Round your answers to $3$ decimal place.