$1.$ The price of a stock, which pays no dividends, is $$55$ and the strike price of a one$-$year European call option on the stock is $$50.$ The risk$-$free rate is $5\%($continuously compounded$).$

a$.$ What is the lower bound for the option such that there are arbitrage opportunities if the price is below the lower bound and no arbitrage opportunities if it is above the lower bound?