Finite Volume Method for Pricing European and American Options under Jump-Diffusion Models

Abstract

A class of finite volume methods is developed for pricing either European or American options under jump-diffusion models based on a linear finite element space. An easy to implement linear interpolation technique is derived to evaluate the integral term involved, and numerical analyses show that the full discrete system matrices are M-matrices. For European option pricing, the resulting dense linear systems are solved by the generalised minimal residual (GMRES) method; while for American options the resulting linear complementarity problems (LCP) are solved using the modulus-based successive overrelaxation (MSOR) method, where the H+-matrix property of the system matrix guarantees convergence. Numerical results are presented to demonstrate the accuracy, efficiency and robustness of these methods.

Modulus-based Successive Overrelaxation Iteration Method for Pricing American Options with the Two-asset Black–Scholes and Heston's Models Based on Finite Volume Discretization

“…Furthermore, the linear system of the equation in line 9 in Algorithm 5.2 may be solved exactly, and the coefficient matrix D + Ω − ωL can be more diagonal dominant than D−ωL in the PSOR method (5.2) for the positive diagonal matrix Ω. Hence, MSOR method may be more effective in practice [18,56].…”

An RBF–FD method for pricing American options under jump–diffusion models

“…For the spatial discretization of the PIDE typically a traditional finite difference method (FDM) is applied as in [6,26,27,39,40] or a finite element method (FEM) as in [20,29,49] for the PDE case or a finite volume method as in [48,18]. There are several other numerical methods available in the literature to solve the governing equation.…”