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