We will derive a two$-$state put option value in this problem. Data: S$0=$ $$180$; X $=$ $$190$; $1+$ r $=1\mathrm{.}10.$ The two possibilities for ST are $$210$ and $$110.$

Required:

a$.$ The range of S is $$100$ while that of P is $$80$ across the two states. What is the hedge ratio of the put? $($Negative value should be indicated by a minus sign. Round your answer to $2$ decimal places.$)$

b$.$ Form a portfolio of four shares of stock and five puts. What is the $($nonrandom$)$ payoff to this portfolio? $($Round your answer to $2$ decimal places.$)$

c$.$ What is the present value of the portfolio? $($Round your answer to $2$ decimal places.$)$

d$.$ Given that the stock currently is selling at $$180,$ calculate the put value. $($Do not round intermediate calculations and round your answer to $2$ decimal places.$)$