Hilbert transform, spectral filters and option pricing

We show how spectral filters can improve the convergence of numerical schemes which use discrete Hilbert transforms based on a sinc function expansion, and thus ultimately on the fast Fourier transform. This is relevant, for example, for the computation of fluctuation identities, which give the distribution of the maximum or the minimum of a random path, or the joint distribution at maturity with the extrema staying below or above barriers. We use as examples the methods by Feng and Linetsky (Math Finance 18(3):337–384, 2008) and Fusai et al. (Eur J Oper Res 251(4):124–134, 2016) to price discretely monitored barrier options where the underlying asset price is modelled by an exponential Lévy process. Both methods show exponential convergence with respect to the number of grid points in most cases, but are limited to polynomial convergence under certain conditions. We relate these rates of convergence to the Gibbs phenomenon for Fourier transforms and achieve improved results with spectral filtering

Fourier transform methods for the pricing of barrier options and other exotic derivatives

This thesis focuses on the numerical calculation of fluctuation identities with both dis- crete and continuous monitoring and the wider application of finding a general numerical solution to the Wiener-Hopf equation on a semi-infinite or finite interval. The motivating application is pricing path-dependent options. It is demonstrated that, with the use of spectral filters, exponential convergence can be achieved for the pricing of discretely monitored double-barrier options. We thus describe the first exponentially convergent pricing method for this type of option with general L ́evy processes and a CPU time which is independent of the number of monitoring dates. Using a numerical implementation of the inverse Laplace transform, the numerical method to calculate fluctuation identities is extended to continuous monitoring. This pro- vides the first method for calculating continuously monitored fluctuation identities which can be used for general L ́evy processes. Furthermore a detailed error bound is obtained, providing additional insight into the pricing methods based on fluctuation identities and the numerical solution of the Wiener-Hopf equation in general. Pricing algorithms for other exotic options such as α-quantile, perpetual Bermudan and perpetual American options are also devised and a new method to compute the optimal exercise boundary for the latter two types of contract is presented. These methods show excellent error performance with computational time. Finally, an application of these new numerical methods to the general solution of the Wiener-Hopf equation is presented. The methods are applied to three new test cases which we derived analytically and the results are presented to show that this new method has an error convergence with grid size which has twice the speed of the current state of the art