Pricing European and American Options under Heston Model using Discontinuous Galerkin Finite Elements

This paper deals with pricing of European and American options, when the underlying asset price follows Heston model, via the interior penalty discontinuous Galerkin finite element method (dGFEM). The advantages of dGFEM space discretization with Rannacher smoothing as time integrator with nonsmooth initial and boundary conditions are illustrated for European vanilla options, digital call and American put options. The convection dominated Heston model for vanishing volatility is efficiently solved utilizing the adaptive dGFEM. For fast solution of the linear complementary problem of the American options, a projected successive over relaxation (PSOR) method is developed with the norm preconditioned dGFEM. We show the efficiency and accuracy of dGFEM for option pricing by conducting comparison analysis with other methods and numerical experiments

Mixed hp‐DGFEM for incompressible flows II: Geometric edge meshes

We consider the Stokes problem of incompressible fluid flow in three‐dimensional polyhedral domains discretized on hexahedral meshes with hp‐discontinuous Galerkin finite elements of type Qk for the velocity and Qk−1 for the pressure. We prove that these elements are inf‐sup stable on geometric edge meshes that are refined anisotropically and non‐quasiuniformly towards edges and corners. The discrete inf‐sup constant is shown to be independent of the aspect ratio of the anisotropic elements and is of O(k−3/2) in the polynomial degree k, as in the case of conforming Qk−Qk−2 approximations on the same meshe

High-Order Maximum Principle Preserving (MPP) Techniques for Solving Conservation Laws with Applications on Multiphase Flow

We develop numerical methods to solve the linear scalar conservation law fulfilling the maximum principle. To do this we use continuous and discontinuous Galerkin finite elements and achieve the preservation on the maximum principle via the Flux Corrected Transport (FCT) method. We use high-order polynomial spaces with Bernstein basis functions and obtain the optimal convergence rates with spaces of up to third order for smooth solutions that are monotone. This methodology produces good quality results for spaces up to (around) third order. However, when higher-order spaces are used non-physical oscillations are introduced, which is true nevertheless the methods are maximum principle preserving. These oscillations can be highly reduced by defining tighter bounds. Using discontinuous Galerkin finite elements we present a new FCT-like methodology based on single cell flux corrections. This method combines a mass conservative low-order Maximum Principle Preserving (MPP) solution with a non-mass conservative high-order MPP solution. The process is designed to recover mass conservation locally (with respect to degrees of freedom). Using this scheme we obtain the optimal convergence rates with spaces of up to third order for smooth solutions that are monotone. The method is designed to overcome problems when high-order spaces are used and, under this context, we obtained better results than with the standard FCT method. We present two methods to transport a smoothed Heaviside level set function using a one-stage reinitialization based on artificial compression. The first method allows arbitrarily large compression which might lead to non-physical behavior. To overcome this difficulty the second method self-balances the artificial dissipation and compression. Finally, we use the level set solver with a Navier-Stokes solver to simulate incompressible two-phase flow