TD - Payoff diagrams, a power option and the Black-Scholes equation

Question
(a) Draw payoff diagrams (from the holder’s point of view) for a European
Vanilla Call and a European Vanilla Put, both with strike price E and
expiry T. On financial grounds, would you expect American Vanilla
Calls and Puts to be more or less expensive than their European equivalents?
(b) A POWER OPTION is an option whose payoff is given by
V (S, T) = ASn
where A and n are constants. Show that the Black-Scholes equation
admits solutions of the form
V (S,t) = g(t)S
n
provided that g(t) satisfies the ordinary differential equation
dg
dt +
µ
1
2
σ
2n + r
¶
(n − 1)g = 0.
Hence or otherwise find the fair value for a power option. What financial product does the option become in the special cases n = 0 and
n = 1?