The common stock of the P$.$U$.$T$.$T$.$ Corporation has been trading in a narrow price range for the past

month, and you are convinced it is going to break far out of that range in the next $6$ months. You do not

know whether it will go up or down, however. The current price of the stock is $$125$ per share, and the

price of a $6-$month call option at an exercise price of $$125$ is $$6\mathrm{.}93.$

a$.$ If the risk$-$free interest rate is $5\mathrm{.}00\%$ per year, what must be the price of a $6-$month put option on

P$.$U$.$T$.$T$.$ stock at an exercise price of $$125?($The stock pays no dividends.$)$

b$.$ How far would it have to move in either direction for you to make a profit on your initial

investment?