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In the paper, we develop a very fast and accurate method for pricing doublebarrier options with continuous monitoring in wide classes of L\'evy models;the calculations are in the dual space, and the Wiener-Hopf factorization isused. For wide regions in the parameter space, the precision of the order of$10^{-15}$ is achievable in seconds, and of the order of $10^{-9}-10^{-8}$ - infractions of a second. The Wiener-Hopf factors and repeated integrals in thepricing formulas are calculated using sinh-deformations of the lines ofintegration, the corresponding changes of variables and the simplifiedtrapezoid rule. If the Bromwich integral is calculated using the Gaver-Wynn Rhoacceleration instead of the sinh-acceleration, the CPU time is typicallysmaller but the precision is of the order of $10^{-9}-10^{-6}$, at best.Explicit pricing algorithms and numerical examples are for no-touch options,digitals (equivalently, for the joint distribution function of a L\'evy processand its supremum and infimum processes), and call options. Several graphs areproduced to explain fundamental difficulties for accurate pricing of barrieroptions using time discretization and interpolation-based calculations in thestate space.

Efficient evaluation of double-barrier options and joint cpdf of a L\'evy process and its two extrema