Alternative models for FX, arbitrage opportunities and efficient pricing of double barrier options in regime-switching Lévy models

We analyze the qualitative differences between prices of double barrier no-touch options in the Heston model and pure jump KoBoL model calibrated to the same set of the empirical data, and discuss the potential for arbitrage opportunities if a jump process fits the joint dynamics of the underlying and exotic option of interest well but the counterparty uses a diffusion model for pricing, which fits vanilla prices equally well or even marginally better. We explain fundamental difficulties for pricing barrier options under Lévy processes of infinite activity and locally finite variation, with drift, using the numerical methods available in the literature, and develop a new efficient method based on the Gaver-Wynn-Rho acceleration applied to the Bromwich integral. The SINH-acceleration and simplified trapezoid rule are used to evaluate perpetual double barrier options for each value of the spectral parameter in the GWR-algorithm. The program in Matlab running on a Mac with moderate characteristics achieves the precision of the order of E-5 and better in several dozens of milliseconds; the precision E-07 is achievable in about 0.1 s. We extend the GWR-SINH method to regime-switching models and regime-switching models with memory. We outline approximations of models with stochastic volatility and interest rate and lifted Heston model by regime-switching models and the fractional Heston model by a regime-switching model with memory.