Problem $4\mathrm{.}2($Evaluating the Black$-$Scholes formula$)$ Consider an European option on a non$-$dividend$-$paying stock. Currently the stock price is $$50,$ the option exercise price is $$51,$ the risk$-$free interest rate is $5\mathrm{.}0\%,$ the volatility is $45\%$ per annum, and the time to maturity of the option is $6$ months. Using the Black$-$Scholes European option price formula, evaluate the price of this option, assuming that: $($a$)$ it is an European call option $($b$)$ it is an European put option $($c$)$ Verify that put$-$call parity holds