$($PCP$)$ The current market price of a $2-$month European put option on a nondividend$-$paying stock with strike price of $$50$ is $$4.$ The stock price is $$47$ and the risk$-$free interest rate is $3\%.$

a$.$ If a $2-$month call option with the same strike price is currently selling for $$1,$ what opportunities are there for an arbitrageur? How can you exploit arbitrage?

b$.$ Would the above market prices still provide an arbitrage opportunity if the option has a $1-$month maturity and the stock price is $$46\mathrm{.}877?$