A numerical scheme for pricing American options with transaction costs under a jump diffusion process

Abstract

In this paper we develop a numerical method for a nonlinear partial integro-differential complementarity problem arising from pricing American options with transaction costs when the underlying assets follow a jump diffusion process. We first approximate the complementarity problem by a nonlinear partial integro-differential equation (PIDE) using a penalty approach. The PIDE is then discretized by a combination of a spatial upwind finite differencing and a fully implicit time stepping scheme. We prove that the coefficient matrix of the system from this scheme is an M-matrix and that the approximate solution converges to the viscosity solution to the PIDE by showing that the scheme is consistent, monotone, and unconditionally stable. We also propose a Newton's iterative method coupled with a Fast Fourier Transform for the computation of the discretized integral term for solving the fully discretized system. Numerical results will be presented to demonstrate the convergence rates and usefulness of this method.

Pricing options on investment project expansions under commodity price uncertainty

“…Remark 3.2. The upwind finite difference method has been used for solving various types of differential equations (see, for example, [14,28,19,13,16,17]). In particular it is used for a nonlinear Black-Scholes equation arising in pricing a conventional option in [16] in which the authors also prove the convergence of the method by showing that it is consistent, monotone, stable.…”

Numerical solution of an obstacle problem with interval coefficients

“…scheme is normally used to approximate (1)-(2) by a finite-dimensional obstacle or constrained optimization problem (see, for example, [11,12,21,42]). Numerical solution of the resulting large-scale discretied and finite-dimensional optimization problems has been discussed in the open literature such as [11,12,25,26,42,27,23]. While these existing methods provide some effective and efficient tools for solving obstacle problems without uncertainties, how to find a solution to a large-scale finite-dimensional obstacle problem effectively in the presence of parameter uncertainties still remains a challenge.…”

“…As mentioned before, A(p) in 3is usually a positive-definite symmetric matrix if an appropriate numerical scheme such as one of those in [22,42,27,23] is used for the discretization of (1). Thus, there exists a positive constant α such that…”

“…Adding up both sides of (25) and (26) and dividing both sides of the resulting inequality by 2 give λ x µ − x * 2 2 ≤ µn. Taking the square-root on both sides of the above estimate we have (23).…”

European option valuation under the Bates PIDE in finance: A numerical implementation of the Gaussian scheme

“…Authors in [8] proposed a mathematical model in finance to calculate the option price of a European contract. After that, empirical evidences indicated that their model assumptions on the log-normality of the return of the underlying asset and constant volatility are usually inconsistent with market prices, for more refer to the discussions in [26].…”