Convergence of a fitted finite volume method for the penalized Black–Scholes equation governing European and American Option pricing

Abstract

In this paper we present an analysis of a numerical method for a degenerate partial differential equation, called the Black-Scholes equation, governing American and European option pricing. The method is based on a fitted finite volume spatial discretization and an implicit time stepping technique. The analysis is performed within the framework of the vertical method of lines, where the spatial discretization is formulated as a Petrov-Galerkin finite element method with each basis function of the trial space being determined by a set of two-point boundary value problems. We establish the stability and an error bound for the solutions of the fully discretized system. Numerical results are presented to validate the theoretical results. Mathematics Subject Classification (2000)65 M 60 · 65 K 10 · 90 C 33

American option pricing problem transformed on finite interval

“…These results are consistent with the estimate (10), showing profit in the order of convergence when computing with large values of ξ . We stress that theoretical considerations suggest first order of spatial convergence [1,19] while some of the numerical results in [17] and also Table 2 motivate investigation of the superconvergence property.…”

“…In this section we focus on the spatial discretization of the following penalized problem (7) by the FVM, developed by Wang [19] and further in [1,17].…”

A Superlinearly Convergent Method for a Class of Complementarity Problems with Non-Lipschitzian Functions

“…This condition is weaker than the ones used in [1,5]. (2) Under the condition that M is a P 0 -matrix, the reformulated system H(t, z) = 0, where H is defined in (2.10), has a unique solution and also enjoys a nonsingularity property. This is vital to apply the BiCGStab iterative solver to get an approximate solution of the resulting linear system (3.1).…”

“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php as the reaction-diffusion problems [3,4], the nonlinear parabolic complementarity problem [21], and European and American option valuation [2,22]. In order to find a numerical solution for these problems, we can use a discretization method such as the central difference, the piecewise linear finite element, or a finite volume method [21,22] to reduce these problems into (1.1) or (1.2).…”

A robust finite difference scheme for pricing American put options with Singularity-Separating method

“…The implicit Euler scheme with upwind spatial difference method does not have this disadvantage, but this difference scheme is only first-order convergent. Recently, a stable fitted finite volume method (cf., for example, [1,18]) is employed for the discretization of the Black-Scholes equation. But it is only first-order convergent.…”