The value of a share per unit develops according to a geometric Brownian motion with drift given by

\[X(t)=10 e^{0.2 t+0.1 S(t)}, t \geq 0\]

where \(\{S(t), t \geq 0\}\) is the standardized Brownian motion. An investor owns a European call option with running time \(\tau=1\) [year] and with strike price

\[x_{s}=\$ 12\]

on a unit of this share.

(1) Given a discount rate of \(\alpha=0.04\), determine the mean discounted profit of the holder of the option.

(2) For what value of the drift parameter \(\mu\) do you get the fair price of the option?

(3) Determine this fair price.