Consider a one$-$step binomial tree on stock with a current price of $$100$ that can go either up to $$120$ or down to $$80$ in $1$ year. The stock does not pay dividend. The continuously compounding interest rate is $5\%($per year$).$ We want to price the $1-$year $$100-$strike European put option on this tree. $($i$)$ What's the put option payoff $1$ year later if the stock price ends up at $$120?$

$($ii$)$ What's the payoff if the stock price ends at $$80?$

$($iii$)$ Use the tree to compute the value and delta of the put option. $($iv$)$ What's the risk$-$neutral probability of the stock price going up to the $$120$ node of the tree?

$($v$)$ What's the risk$-$neutral probability of going down to the $$80$ node?

$($Round answers to $2$ decimals$)$