Use DerivaGem to calculate the value of:

a. A regular European call option on a non-dividend-paying stock where the stock price is

$50, the strike price is $50, the risk-free rate is 5% per annum, the volatility is 30%, and the time to maturity is one year

b. A down-and-out European call which is as in

(a) with the barrier at $45

c. A down-and-in European call which is as in

(a) with the barrier at $45 Show that the option in

(a) is worth the sum of the values of the options in

(b) and (c).