Construct a portfolio consisting of a long position in the European put option and a long

position in the underlying stock. Assume the put option will expire in one period, and $K=$

The stock only has two outcomes at the end of the period, namely an $\text{'}$up$\text{'}$ move with

$U=1\mathrm{.}2$ and a 'down' move with $D=0\mathrm{.}8.$ The current stock price $S_0=50$ and the stock

pays no dividends. Assume a one$-$period risk$-$free interest rate $r=0\mathrm{.}05$

a$.$ At the end of the period, what are the two possibilities for stock price? What are the

corresponding payoffs of the put?

b$.$ What is the hedge ratio or delta of the put option?

c$.$ What is the payoff to the portfolio at the end of the period if the portfolio is risk$-$neutral?

d$.$ What is the price of the put option?