At time t = 0 a speculator acquires an American call option with infinite expira-tion time and strike price x s . The price X(t) of the underlying risky security at time t is given by The speculator...

At time t = 0 a speculator acquires an American call option with infinite expira-tion time and strike price x_s. The price X(t) of the underlying risky security at time t is given by

The speculator makes up his mind to exercise this option at that time point, when the price of the risky security hits a level x with

x > xs ≥ x0

for the first time,.

1) What is the speculator's mean discounted payoff Gα(x) under a constant discount rate
?

2) What is the speculator's payoff without discounting?

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