Convergence of Price and Sensitivities in Carr's Randomization Approximation Globally and Near Barrier

Barrier options under wide classes of L\'evy processes with exponential jump densities, including Variance Gamma model, KoBoL (a.k.a. CGMY) model and Normal Inverse Gaussian processes, are studied. The leading term of asymptotics of the option price and the leading term of asymptotics in Carr's randomization approximation to the price are calculated, as the price of the underlying approaches the barrier. We prove that the order of asymptotics is the same in both cases, and the asymptotic coefficient in the asymptotic formula for Carr's randomization approximation converges to the asymptotic coefficient for the price, as the number of time steps N → ∞, and justify Richardson extrapolation of arbitrary order. Similar results are derived for sensitivities and the leading terms of their asymptotics in Carr's randomization approximation. The convergence of prices and sensitivities is proved in appropriate weighted H¨older spaces