$(20$ points$)$ The value of a derivative that pays off $$100$ after one year if a company

has defaulted during the year is $$5.$ The value of a derivative that pays off $$100$ after

one year if a company has not defaulted is $$97.$

$($a$)$ What is the risk$-$free rate?

$($b$)$ What is the risk neutral probability of default?