A stock has a current price of $$47.$ An option on this stock that expires in eight months has an exercise price of $$45.$ The stock will pay a dividend of $$1\mathrm{.}50$ in six months. Assume an annualized volatility of $20\%$ and a continuously compounded risk$-$free rate of $4\%$ per annum. Use the Black$-$Sholes$-$Merton model to price this option$\mathrm{.}1)$ Suppose the option is a European put. Calculate the value of the put. $2)$ Suppose this option is an American call. Use Black$$s approximation to calculate the value of this call.