  We present new numerical schemes for pricing perpetual Bermudan and American
options as well as $\alpha$-quantile options. This includes a new direct
calculation of the optimal exercise barrier for early-exercise options. Our
approach is based on the Spitzer identities for general L\'evy processes and on
the Wiener-Hopf method. Our direct calculation of the price of
$\alpha$-quantile options combines for the first time the Dassios-Port-Wendel
identity and the Spitzer identities for the extrema of processes. Our results
show that the new pricing methods provide excellent error convergence with
respect to computational time when implemented with a range of L\'evy
processes.
