The price X(t) of a risky security at time t is with a negative drift parameter µ. At time a speculator acquires an American t = 0 call option with strike price on this risky security. The option has...

The price X(t) of a risky security at time t is

with a negative drift parameter µ. At time a speculator acquires an American t = 0 call option with strike price on this risky security. The option has no finite expira- xs tion date. 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 > x_s
≥ x₀
for the first time. Otherwise, i.e. if the price of the risky security never reaches level x, the speculator will never exercise.

Determine the level x = x∗ at which the speculator should schedule to exercise this option to achieve

1) maximal mean payoff without discounting and

2) maximal mean discounted payoff (constant discount rate ).