A European call that expires in six months with a strike price of $$30$ is selling at $$2.$ The underlying stock is priced at $$29,$ and does not pay any dividends. The term structure is flat and all risk free interest rates are $10\%$ per annum, continuously compounded.

What is the price of a European put option that expires in $6$ months and has a strike price of $$30?$ Demonstrate the methodology used.

Explain carefully the arbitrage opportunities if the European put price is $$3$