Ali Foroush Bastani

Ali Foroush Bastani, Davood Damircheli

This paper presents an adaptive multiple-shooting method to solve stochastic multi-point boundary value problems. The heuristic to choose the shooting points is based on separating the effects of drift and diffusion terms and comparing the corresponding solution components with a pre-specified initial approximation. Having obtained the mesh points, we solve the underlying stochastic differential equation on each shooting interval with a first-order strongly-convergent stochastic Runge-Kutta method. We illustrate the effectiveness of this approach on 1-dimentional and 2-dimentional test problems and compare our results with other non-adaptive alternative techniques proposed in the literature.

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.