Efficient evaluation of expectations of functions of a stable Lévy process and its extremum

Integral representations for expectations of functions of a stable Lévy process $X$ and its supremum $\bar{X}$ are derived. As examples, cumulative probability distribution functions (cpdf) of $X_T,{\text{barX}}_T$, the joint cpdf of $X_T$ and ${\text{barX}}_T$, and the expectation of $(\text{be}{X}_T-{\text{barX}}_T{)}_{+}$, $\text{be}>1$, are considered, and efficient numerical procedures for cpdfs are developed. The most efficient numerical methods use the conformal acceleration technique and simplified trapezoid rule.