Pricing Options under Levy models using Spectral methods

Abstract

Spectral methods have been actively developed in the last decades. The main advantage of these methods is to yield exponential order of accuracy when the function is smooth. However, for discontinuous functions, their accuracy deteriorates due to the Gibbs phenomenon. When functions are contaminated with the Gibbs phenomenon, proper workarounds can be applied to recover their accuracy. In this dissertation, we review the spectral methods and their convergence remedies such as grid stretching, discontinuity inclusion and domain decomposition methods in pricing options. The basic functions of L´evy processes models are also reviewed. The main purpose of this dissertation is to show that high order of accuracy can be recovered from spectral approximations. We explored and designed numerical methods for solving PDEs and PIDEs that arise in finance. It is known that most standard numerical methods for solving financial PDEs and PIDEs are reduced to low order accurate results due to the discontinuity at strike prices in the initial condition. Firstly the Black Scholes (BS) PDE was solved numerically. The computation of the PDE is done by using barycentric spectral methods. Three different payoffs call options are used as initial and boundaries conditions. It appears that the grid stretching, the discontinuity inclusion and the domain decomposition methods provide efficient ways to remove Gibbs phenomenon. On the other hand, these methods restore the high accuracy of spectral methods in pricing financial options. The spectral domain decomposition method appears to be the most accurate workaround when we solve a BS PDE in this dissertation. Secondly, a financial PIDE was discretized and solved by using a barycen tric spectral domain decomposition method algorithm. The method is applied to two different options pricing problems under a class of infinite activity L´evy models. The use of barycentric spectral domain decomposition methods allows the computation of ODEs obtained from the discretization of the PIDE. The ODEs are solved by exponential time integration scheme. Several numerical tests for the pricing of European and butterfly options are given to illustrate the efficiency and accuracy of this algorithm. We also show that the option Greeks such as the Delta and Gamma sensitivity measures are computed with no spurious oscillation. The methods produce accurate results.