The fast Fourier transform (FFT) method is now a standard calibration engine. However, in many situations, such as pricing of deep out-of-the-money (OTM) European options, the FFT produces large errors. We propose fast and accurate realizations of the integration-along-cut (IAC) method that explicitly control the error of pricing OTM options. For one strike (and OTM options), the IAC method is significantly faster than FFT-based methods. Even if prices for many strikes are needed, the IAC method, together with quadratic interpolation, successfully competes with the convolution method developed by Lord et al, the Fourier cosineseries-based method suggested by Fang and Oosterlee, the saddlepoint method suggested by Carr and Madan and the refined and enhanced versions of the FFT recently suggested by Boyarchenko and Levendorskii. For calculations of sensitivities, a relative advantage of the IAC is even greater. The method is applicable to a wide class of Lévy-driven models, Koponen-Boyarchenko-Levendorskii (KoBoL)processes (also known as the Carr-Geman-Medan-Yor (CGMY) model) of finite variation, variance gamma processes and normal inverse Gaussian processes.