At time \(t=0\) a speculator acquires a European call option with strike price \(x_{S}\) and finite expiration time \(\tau\). Thus, the option can only be exercised at time \(\tau\) at price \(x_{s}\) independently of its market value at time \(\tau\). The random price \(X(t)\) of the underlying risky security develops according to

\[X(t)=x_{0}+D(t)\]

where \(\{D(t), t \geq 0\}\) is the Brownian motion with positive drift parameter \(\mu\) and volatility \(\sigma\). If \(X(\tau)>x_{s}\), the speculator will exercise the option. Otherwise, the speculator will not exercise. Assume that

\[x_{0}+\mu t>3 \sigma \sqrt{t}, \quad 0 \leq t \leq \tau\]

(1) What will be the mean undiscounted payoff of the speculator (cost of acquiring the option not included)?

(2) Under otherwise the same assumptions, what is the investor's mean undiscounted profit if

\[X(t)=x_{0}+B(t) \text { and } x_{0}=x_{s} ?\]