The price of a stock is currently $$40.$ Over each of the next two six$-$month periods, the stock is expected to go up by $10\%($u $=1\mathrm{.}1).$ The downward movement is d $=1/$u$.$ The risk$-$free interest rate is $10\%$ per year with discrete compounding. Using the Binomial Option Pricing Model, answer the following questions:

a$)$ What is the value of a one$-$year European put option with a strike price of $$42?$ For each state in the binomial tree $($there are two six$-$month periods$),$ show your calculations for the stock price and the put price, including at time zero $($today$).[$Hint: You need to use the effective semiannual interest rate associated with an annual interest rate of $10$ percent. Divide by the number of periods in a year.$]$

For the European put option worked in part $($a$),$ construct the risk$-$free portfolio, and show that in each period, you earn the risk$-$free rate. Assume that you use $1,000$ puts and the required proportion $($h$)$ of stocks to be hedged in each period. In period zero and period one, indicate clearly what the strategy is to maintain the portfolio risk$-$free from period zero to period one, and from period one to period two. Specify the value of the portfolio in each period.

What is the value of a one$-$year American put option with a strike price of $$42?$ Again, show your calculations. There is no need to construct the risk$-$free portfolio for part $($c$).$