At time \(t=0\) a speculator acquires an American call option with infinite expiration time and strike price \(x_{s}\). The price [in \$] of the underlying risky security at time \(t\) is given by \(X(t)=x_{0} e^{B(t)}\). The speculator makes up his mind to exercise this option at that time point, when the price of the risky security hits for the first time level \(x\) with \(x>x_{S} \geq x_{0}\).

(1) What is the speculator's mean discounted payoff \(G_{\alpha}(x)\) under a constant discount rate \(\alpha\) ?

(2) What is the speculator's payoff \(G(x)\) without discounting?

In both cases, the cost of acquiring the option is not included in the speculator's payoff.