A stock is selling today for $$85.$ The stock has an annual volatility of $42$ percent, and the annual nominal risk$-$free interest rate is $7$ percent.

a$.$ Calculate the fair price for a $25$ month European put option with an exercise price of $$80.$

b$.$ Calculate how much the current stock price would need to change for the purchaser of the put option to break even in $25$ months.

c$.$ Calculate the level of volatility that would make the $$80$ put option sell for $$15.($Use Goal Seek or Solver$).$