Consider a stock that is currently trading at $$50$ and has an annualized volatility of $30\%.$ The stock

is not expected to pay any dividends in the next $3$ months. The annualized risk$-$free rate is $4\%.$

A$.$ Calculate the Black$-$Scholes price of a cash$-$or$-$nothing binary call option that matures in

$3$ months. The option pays nothing if the stock price at maturity is less than $$40,$ and pays

a fixed amount of $$40$ if the stock price at maturity is higher than $$40.$

B$.$ Calculate the Black$-$Scholes price of an asset$-$or$-$nothing binary call option that pays

nothing if the stock price at maturity $(3$ months$)$ is less than $$40,$ and pays the value of the

stock price at maturity $($or delivers the stock$)$ if the stock price at maturity is higher than

$$40.$

C$.$ Use your answers from parts $($A$)$ and $($B$),$ and only your answers from parts $($A$)$ and

$($B$),$ to find the Black$-$Scholes price of a European call option on the same underlying stock

that matures in $3$ months and has a strike price of $$40.$ Please explain your answer carefully.