$($a$)$ What is the price of a European call option on a non$-$dividend$-$paying stock when the stock price is $$52,$ the strike price is $$50,$ the risk$-$free interest rate is $12\%$ per annum, the volatility is

$30\%$ per annum, and the time to maturity is three months? $(12$ points$)$

$($b$)$ What is the risk$-$neutral probability that this European call will be exercised? $(6$ points$)($c$)$ Given that this call option is exercised at the expiration date, what is the expected stock price at the expiration date in the risk$-$neutral world? $(9$ points$)$

$($d$)$ If this call is an American call, is it possible for the call to be exercised early? $(3$ points$)($e$)$ Continued from parts a and d$,$ what is the price of this American call option? $(4$ points$)$