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We consider a problem of predicting of the ultimate maximum  of the process over a finite interval of time. Mathematically, this problem relates to a particular optimal stopping problem. In this paper we discuss exponential Levy processes. As the Levy processes, we discuss $\alpha$-stable Levy processes, $0<\alpha\leq 2$,  and generalized hyperbolic Levy processes. The method of solution uses the representations of these processes as time-changed Brownian motions with drift. Our results generalize results of papers by Toit and Peskir and by Shiryaev and Xu, and Zhou.

On predicting the ultimate maximum for exponential Lévy processes