The value (in \(\$\) ) of a share per unit develops, apart from the constant factor 10 , according to a geometric Brownian motion \(\{X(t), t \geq 0\}\) given by

\[X(t)=10 e^{B(t)}, 0 \leq t \leq 120\]

where \(\{B(t), t \geq 0\}\) is the Brownian motion process with volatility \(\sigma=0.1\).

At time \(t=0\) a speculator pays \(\$ 17\) for becoming owner of a unit of the share after 120 [days], irrespective of the then current market value of the share.

(1) What will be the mean undiscounted profit of the speculator at time point \(t=120\) ?

(2) What is the probability that the investor will lose some money when exercising at this time point?

In both cases, take into account the amount of \(\$ 17\), which the speculator had to pay in advance.