On a sparse and stable solver on graded meshes for solving high-dimensional parabolic pricing PDEs

The purpose of this work is to investigate the multi-dimensional Black-Scholes partial differential equation with variable coefficients numerically. The problem is of practical importance due to option pricing at the presence of multi assets. Since by increasing the dimension, the curse of dimensionality restricts the computations, the proposed solver will be constructed based on sparse arrays. Toward this goal, fourth-order finite difference approximations on graded meshes are introduced and then employed through semi-discretization. Then a sixth order Runge-Kutta solver is employed for finding the resolution of the derived set of ordinary differential equations. The stability of the proposed scheme is furnished in detail as well. Numerical testings are given to uphold the accuracy and efficacy of the proposed procedure.