Efficient Evaluation of Expectations of Functions of a Lévy Process and Its Extremum

We prove simple general formulas for expectations of functions a L\'evy process and its running extremum. Under additional conditions, we derive analytical using the Fourier/Laplace inversion Wiener-Hopf factorization, discuss efficient numerical methods realization these formulas. As applications, cumulative probability distribution function maximum price option to exchange power stock are calculated. The most use sinh-acceleration technique simplified trapezoid rule. program in Matlab on Mac with moderate characteristics achieves precision E-7 better several milliseconds, E-14 - fraction second.