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).