Khadijeh Nedaiasl

Urs Vögeli, Khadijeh Nedaiasl, Stefan A. Sauter

In this paper, we present a Galerkin method for Abel-type integral equation with a general class of kernel. Stability and quasi-optimal convergence estimates are derived in ractional-order Sobolev norms. The fully-discrete Galerkin method is defined by employing simple tensor-Gauss quadrature. We develop a corresponding perturbation analysis which allows to keep the number of quadrature points small. Numerical experiments have been performed which illustrate the sharpness of the theoretical estimates and the sensitivity of the solution with respect to some parameters in the equation.

Khadijeh Nedaiasl, Ali Foroush Bastani, Aysan Rafiee

In this paper, an integral equation representation for the early exercise boundary of an American option contract is considered. Thus far, a number of different techniques have been proposed in the literature to obtain a variety of integral equation forms for the early exercise boundary, all starting from the Black-Scholes partial differential equation. We first present a coherent categorization of exiting integral equation methodologies in the American option pricing literature. In the reminder and based on the fact that the early exercise boundary satisfies a fully nonlinear weakly singular non-standard Volterra integral equation, we propose a product integration approach based on linear barycentric rational interpolation to solve the problem. The price of the option will then be computed using the obtained approximation of the early exercise boundary and a barycentric rational quadrature. The convergence of the approximation scheme will also be analyzed. Finally, some numerical experiments based on the introduced method are presented and compared to some exiting approaches.