Consider a call option on a stock with a strike price of $$195$ and one year to expiration. The current stock price is $$190.$ Every six months, the stock price either increases by a factor of $1\mathrm{.}5$ or decreases by a factor of $0\mathrm{.}5.$ The risk$-$free rate is $5\%$ per year, or $2\mathrm{.}5\%$ per six$-$month period.

a$.$ Draw a two$-$period binomial tree for the stock price as well as the option value.

b$.$ What is the value of the call option at the up node in six months$$ time $($i$.$e$.,$ Cu$)?$

c$.$ What is the value of the call option at the down node in six months$$ time $($i$.$e$.,$ Cd$)?$

d$.$ What is the value of the call option today?

e$.$ What is the option$$s intrinsic value? What is its time value?