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.

make sure to convert the annualized discrete interest rate to a $3-$month period rate using $(1+$r$)^0\mathrm{.}25.$ You need to use this both to calculate the risk$-$neutral probability and option values at each node of the option tree. Also, to avoid missing the point due to rounding errors, make sure to keep intermediate calculations, such as risk$-$neutral probability and others, to at least $5$ digits