A Flexible Galerkin Scheme for Option Pricing in Lévy Models

Abstract

One popular approach to option pricing in Lévy models is through solving the related partial integro differential equation (PIDE). For the numerical solution of such equations powerful Galerkin methods have been put forward e.g. by Hilber et al. (2013). As in practice large classes of models are maintained simultaneously, flexibility in the driving Lévy model is crucial for the implementation of these powerful tools. In this article we provide such a flexible finite element Galerkin method. To this end we exploit the Fourier representation of the infinitesimal generator, i.e. the related symbol, which is explicitly available for the most relevant Lévy models. Empirical studies for the Merton, NIG and CGMY model confirm the numerical feasibility of the method.

A Finite Elements Strategy for Spread Contract Valuation Via Associated PIDE

“…We adopt the theoretical framework in [14,13,1,28] and consider the Sobolev-Slobodeckii spaces V = H α η (R 2 ) and H = L 2 η (R 2 ). For concrete definitions of weighted Sobolev-Slobodeckii spaces see [13,14].…”

“…The solution of the ( 21) is defined in the whole R 2 plane, in order to implement a numerical method we need first to localize to a bounded domain. A standard procedure is to find suitable function ψ asymptotically similar to the exact solution when x → ∞ (see [1,29]). By subtracting ψ : I × R 2 → R from u we obtain a new equation equivalent to (19) to be solved for v = u − ψ…”

“…For this notion of solution, Fast Fourier Transform (FFT) based methods have been proposed in [5,6] which are quite easy to implement, flexible and generic although they do not handle boundary conditions efficiently and are only applicable to linear problems with non space-dependent coefficients. Other possible notions of solution are viscosity solutions [7,8,9,10] to be approximated using finite difference methods and variational solutions [1,11,12] to be approximated using variational methods. In [13,14], important theoretical advances are presented to establish the connection between variational solutions and the corresponding expectations that European option prices represent.…”

“…Recently, Gass and Glau presented in [1] a flexible Galerkin scheme for option pricing in multidimensional Levy models, the authors developed an approach to set up a FEM solver for option prices in Levy models that they call it the symbol method. They show a way to implement a Galerkin discretization of the integrodifferential operator whenever its symbol is known and the approximated solution is approximated as the linear combination of the smooth mother basis function's displacements.…”

“…Inspired by the use of the Preconditioned Conjugated gradient (PCG) method for symmetric Toeplitz and BTTB systems covered in [25] we propose the use of the Preconditioned BICGSATB (Krilov) method introduced in [26], which works for general (non-symmetric) matrices, along with a circular preconditioner. As mentioned in [1], the pricing error is sensitive to the accuracy of the stiffness matrix entries and the authors emphasized the necessity of accurately compute them. Pointing in that direction, we discretize the terms in the integro-differential operator not related to the pure jump using exact Gaussian quadrature (as it is suggested in [27] for the case of classical FEM) and suppress their corresponding terms in the symbol.…”