Pricing Barrier and Bermudan Style Options Under Time-Changed Lévy Processes: Fast Hilbert Transform Approach

We construct efficient and accurate numerical algorithms for pricing discretely monitored barrier and Bermudan style options under time-changed Lévy processes by applying the fast Hilbert transform method to the log-asset return dimension and quadrature rule to the dimension of log-activity rate of stochastic time change. Some popular stochastic volatility models, like the Heston model, can be nested in the class of time-changed Lévy processes. The computational advantages of the fast Hilbert transform approach over the usual fast Fourier transform method, like exponential decay of errors in terms of the step size in the transform and avoidance of recovering option prices at the monitoring time instants, can be extended to pricing barrier and Bermudan style options under time-changed Lévy processes. We manage to compute the fair value of a dividend-ruin model with both embedded reflecting (dividend) barrier and absorbing (ruined) barrier. We also consider pricing of Bermudan options in conjunction with the determination of the associated critical asset prices. Our numerical tests demonstrate high level of accuracy, efficiency and reliability of the fast Hilbert transform approach when compared to other numerical schemes in the literature.