Exponential Time Integration and Second-Order Difference Scheme for a Generalized Black-Scholes Equation

We apply an exponential time integration scheme combined with a central difference scheme on a piecewise uniform mesh with respect to the spatial variable to evaluate a generalized Black-Scholes equation. We show that the scheme is second-order convergent for both time and spatial variables. It is proved that the scheme is unconditionally stable. Numerical results support the theoretical results

A posteriori mesh method for a system of singularly perturbed initial value problems

A system of singularly perturbed initial value problems with weak constrained conditions on the coefficients is considered. First the system of second-order singularly perturbed problems is transformed into a system of first-order singularly perturbed problems with integral terms, which facilitates the subsequent stability and a posteriori error analyses. Then a hybrid difference method with the use of interpolating quadrature rules is utilized to approximate the transformed system. Next a posteriori error analysis for the discretization scheme on an arbitrary mesh is presented. A solution-adaptive algorithm based on a posteriori error estimation is devised to generate a posteriori mesh and obtain approximation solution. Finally numerical experiments show a uniform convergence behavior of second-order for the scheme, which improves the previous results and achieves the optimal convergence order under the given discrete scheme

Asymptotic Behaviors of Intermediate Points in the Remainder of the Euler-Maclaurin Formula

The Euler-Maclaurin formula is a very useful tool in calculus and numerical analysis. This paper is devoted to asymptotic expansion of the intermediate points in the remainder of the generalized Euler-Maclaurin formula when the length of the integral interval tends to be zero. In the special case we also obtain asymptotic behavior of the intermediate point in the remainder of the composite trapezoidal rule

A HODIE finite difference scheme for pricing American options

Abstract In this paper, we introduce a new numerical method for pricing American-style options, which has long been considered as a very challenging problem in financial engineering. Based on the HODIE (high order via differential identity expansion) finite difference scheme, we discretize the spatial variable on a piecewise uniform mesh, and meanwhile, use the implicit Euler method to discretize the time variable. Under such a discretization, we show that the resulting matrix is an M-matrix, which ensures the stability of the current scheme in the maximum-norm sense. By applying the discrete maximum principle, an error estimate of the current scheme is theoretically obtained first and then tested numerically. It is shown that our method is first order and second order convergent in the time and spatial directions, respectively. The results of various numerical experiments show that this new approach is quite accurate, and can be easily extended to price other kinds of American-style options

Cubic Spline Method for a Generalized Black-Scholes Equation

We develop a numerical method based on cubic polynomial spline approximations to solve a a generalized Black-Scholes equation. We apply the implicit Euler method for the time discretization and a cubic polynomial spline method for the spatial discretization. We show that the matrix associated with the discrete operator is an M-matrix, which ensures that the scheme is maximum-norm stable. It is proved that the scheme is second-order convergent with respect to the spatial variable. Numerical examples demonstrate the stability, convergence, and robustness of the scheme

Analysis of a Hybrid Finite Difference Scheme for the Black-Scholes Equation Governing Option Pricing

Abstract. In this paper we present a hybrid finite difference scheme on a piecewise uniform mesh for a class of Black-Scholes equations governing option pricing which is path-dependent. In spatial discretization a hybrid finite difference scheme combining a central difference method with an upwind difference method on a piecewise uniform mesh is used. For the time discretization, we use an implicit difference method on a uniform mesh. Applying the discrete maximum principle and barrier function technique we prove that our scheme is second-order convergent in space for the arbitrary volatility and the arbitrary asset price. Numerical results support the theoretical results

A Robust Spline Collocation Method for Pricing American Put Options

In this paper a robust numerical method is proposed for pricing American put options. The Black-Scholes differential operator in the original form is discretized by using a quadratic spline collocation method on a piecewise uniform mesh for the spatial discretization and the implicit Euler scheme for the time discretization. The position of collocation points is chosen so that the spline difference operator satisfies the discrete maximum principle, which guarantees that the scheme is maximum-norm stable. The error estimation is derived by applying the maximum principle to the discrete linear complementarity problem in two mesh sets. It is proved that the scheme is second-order convergent with respect to the spatial variable and first-order convergent with respect to the time variable. Numerical results demonstrate that the scheme is stable and accurate