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

Norm Preconditioners for discontinuous Galerkin hp-Finite Element methods.

We consider a norm-preconditioning approach for the solution of discontinuous Galerkin finite element discretizations of second order PDE with non-negative characteristic form. In particular, we perform an analysis for the general case of discontinuous hp-finite element discretizations. Our solution method is a norm-preconditioned three-term GMRES routine. We find that for symmetric positive-definite diffusivity tensors the convergence of our solver is independent of discretization, while for the semidefinite case both theory and experiment indicate dependence on both h and p. Numerical results are included to illustrate performance on several test cases

On Discontinuous Galerkin Methods for Singularly Perturbed and Incompressible Miscible Displacement Problems

This thesis is concerned with the numerical approximation of problems of fluid flow, in particular the stationary advection diffusion reaction equations and the time dependent, coupled equations of incompressible miscible displacement in a porous medium. We begin by introducing the continuous discontinuous Galerkin method for the singularly perturbed advection diffusion reaction problem. This is a method which coincides with the continuous Galerkin method away from internal and boundary layers and with a discontinuous Galerkin method in the vicinity of layers. We prove that this consistent method is stable in the streamline diffusion norm if the convection field flows non-characteristically from the region of the continuous Galerkin to the region of the discontinuous Galerkin method. We then turn our attention to the equations of incompressible miscible displacement for the concentration, pressure and velocity of one fluid in a porous medium being displaced by another. We show a reliable a posteriori error estimator for the time dependent, coupled equations in the case where the solution has sufficient regularity and the velocity is bounded. We remark that these conditions may not be attained in physically realistic geometries. We therefore present an abstract approach to the stationary problem of miscible displacement and investigate an a posteriori error estimator using weighted spaces that relies on lower regularity requirements for the true solution. We then return to the continuous discontinuous Galerkin method. We prove in an abstract setting that standard (continuous) Galerkin finite element approximations are the limit of interior penalty discontinuous Galerkin approximations as the penalty parameter tends to infinity. We then show that by varying the penalization parameter on only a subset of the domain we reach the continuous discontinuous method in the limit. We present numerical experiments illustrating this approach both for equations of non-negative characteristic form (closely related to advection diffusion reaction equations) and to the problem of incompressible miscible displacement. We show that we may practically determine appropriate discontinuous and continuous regions, resulting in a significant reduction of the number of degrees of freedom required to approximate a solution, by using the properties of the discontinuous Galerkin approximation to the advection diffusion reaction equation. We finally present novel code for implementing the continuous discontinuous Galerkin method in C++