Consider a binomial world in which the current stock price of $80$ can either go up by $25$ percent or down by $20$ percent. The risk$-$free rate is $7$ percent. Assume a two$-$period world and an exercise price of $80.$

Compute the following

a$)$ Determine the two possible stock prices for the Time Period $1$ and $2$

b$)$ What would be the Put's price at Time Period $1$ and $2,$ if the Put is American.

c$)$ What would be the theoretical value of Put at Time $0,$ if the Put is American.

d$)$ Now consider the Put to be European and compute the Hedge Ratio, construction of Hedge Portfolio, and the value of Hedge Portfolio at Time $0.$

e$)$ If the Put is European, compute the value of Hedge Portfolio and updated hedge ratio and portfolio at Time $1.$