Optimal error analysis of a non-uniform IMEX-L1 finite element method for time fractional PDEs and PIDEs

Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) is studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint elliptic part having (space-time) variable coefficients. The proposed scheme is based on a combination of an IMEX-L1 method on graded mesh in the temporal direction and a finite element method in the spatial direction. With the help of a discrete fractional Gr\"{o}nwall inequality, optimal error estimates in $L^2$- and $H^1$-norms are derived for the problem with initial data $u_0 \in H_0^1(\Omega)\cap H^2(\Omega)$. Under higher regularity condition $u_0 \in \dot{H}^3(\Omega)$, a super convergence result is established and as a consequence, $L^\infty$ error estimate is obtained for 2D problems. Numerical experiments are presented to validate our theoretical findings.Comment: 33 page

Robust Spectral Methods for Solving Option Pricing Problems

Doctor Scientiae - DScRobust Spectral Methods for Solving Option Pricing Problems by Edson Pindza PhD thesis, Department of Mathematics and Applied Mathematics, Faculty of Natural Sciences, University of the Western Cape Ever since the invention of the classical Black-Scholes formula to price the financial derivatives, a number of mathematical models have been proposed by numerous researchers in this direction. Many of these models are in general very complex, thus closed form analytical solutions are rarely obtainable. In view of this, we present a class of efficient spectral methods to numerically solve several mathematical models of pricing options. We begin with solving European options. Then we move to solve their American counterparts which involve a free boundary and therefore normally difficult to price by other conventional numerical methods. We obtain very promising results for the above two types of options and therefore we extend this approach to solve some more difficult problems for pricing options, viz., jump-diffusion models and local volatility models. The numerical methods involve solving partial differential equations, partial integro-differential equations and associated complementary problems which are used to model the financial derivatives. In order to retain their exponential accuracy, we discuss the necessary modification of the spectral methods. Finally, we present several comparative numerical results showing the superiority of our spectral methods

B-Spline Based Methods: From Monotone Multigrid Schemes for American Options to Uncertain Volatility Models

In the first part of this thesis, we consider B-spline based methods for pricing American options in the Black-Scholes and Heston model. The difference between these two models is the assumption on the volatility of the underlying asset. While in the Black-Scholes model the volatility is assumed to be constant, the Heston model includes a stochastic volatility variable. The underlying problems are formulated as parabolic variational inequalities. Recall that, in finance, to determine optimal risk strategies, one is not only interested in the solution of the variational inequality, i.e., the option price, but also in its partial derivatives up to order two, the so-called Greeks. A special feature for these option price problems is that initial conditions are typically given as piecewise linear continuous functions. Consequently, we have derived a spatial discretization based on cubic B-splines with coinciding knots at the points where the initial condition is not differentiable. Together with an implicit time stepping scheme, this enables us to achieve an accurate pointwise approximation of the partial derivatives up to order two. For the efficient numerical solution of the discrete variational inequality, we propose a monotone multigrid method for (tensor product) B-splines with possible internal coinciding knots. Corresponding numerical results show that the monotone multigrid method is robust with respect to the refinement level and mesh size. In the second part of this thesis, we consider the pricing of a European option in the uncertain volatility model. In this model the volatility of the underlying asset is a priori unknown and is assumed to lie within a range of extreme values. Mathematically, this problem can be formulated as a one dimensional parabolic Hamilton-Jacobi-Bellman equation and is also called Black-Scholes-Barenblatt equation. In the resulting non-linear equation, the diffusion coefficient is given by a volatility function which depends pointwise on the second derivative. This kind of non-linear partial differential equation does not admit a weak H^1-formulation. This is due to the fact that the non-linearity depends pointwise on the second derivative of the solution and, thus, no integration by parts is possible to pass the partial derivative onto a test function. But in the discrete setting this pointwise second derivative can be approximated in H^1 by L^1-normalized B-splines. It turns out that the approximation of the volatility function leads to discontinuities in the partial derivatives. In order to improve the approximation of the solution and its partial derivatives for cubic B-splines, we develop a Newton like algorithm within a knot insertion step. Corresponding numerical results show that the convergence of the solution and its partial derivatives are nearly optimal in the L^2-norm, when the location of volatility change is approximated with desired accuracy

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described