A fast high-order finite difference algorithm for pricing American options

Abstract

We describe an improvement of Han and Wu's algorithm [H. Han, X.Wu, A fast numerical method for the Black-Scholes equation of American options, SIAM J. Numer. Anal. 41 (6) (2003) 2081-2095] for American options. A high-order optimal compact scheme is used to discretise the transformed Black-Scholes PDE under a singularity separating framework. A more accurate free boundary location based on the smooth pasting condition and the use of a non-uniform grid with a modified tridiagonal solver lead to an efficient implementation of the free boundary value problem. Extensive numerical experiments show that the new finite difference algorithm converges rapidly and numerical solutions with good accuracy are obtained. Comparisons with some recently proposed methods for the American options problem are carried out to show the advantage of our numerical method.

A Radial Point Interpolation Method for Pricing Options on a Dividend Paying Asset

“…They defined an algorithm which is computationally efficient and guarantees to generate prices that exclude arbitrage possibilities. Some other relevant works that can be worth mentioning here are those of Khaliq et al [7] who developed adaptive θ-methods for solving the Black-Scholes PDE for American options; Zhao et al [17] who discussed some compact finite difference methods for pricing American options on a single asset with methods for dealing with optimal exercise boundary, and Tangman and Bhuruth [14] who described an improvement of Han and Wu's algorithm [4] for American options. The RPIM has the following advantages ( [9]): The shape function has the Kronecker delta property, which facilitates easy treatment of the essential boundary conditions; the moment matrix used in constructing shape functions is always invertible for irregular nodes; and the polynomials can be exactly reproduced up to desired order by polynomial augmentation.…”

An Integral Equation Method with High-Order Collocation Implementations for Pricing American Put Options

“…Wong and Zhao (2008) derive artificial boundary conditions for a CEV model with Black-Scholes model as a special case. Tangman et al (2008) provide a survey to the numerical methods for pricing American options and develop a new finite difference method to deal with the singularity existing at the strike price in the payoff function which deceases the accuracy of the solution. Fusai et al (2007) investigate quadrature method for solving the free boundary problem.…”

A high-order front-tracking finite difference method for pricing American options under jump-diffusion models

“…Methods keeping track of the boundary and discretizing the problem in changing domain are called front-tracking methods. This approach has been considered in [15], [30], [32]. An alternative approach is to employ a time-dependent change of variable to map the changing domain into a fixed domain.…”

“…An alternative approach is to employ a time-dependent change of variable to map the changing domain into a fixed domain. Such methods are called front-fixing methods and they have been considered in [17], [28], [32], [36], [38]. Most of these methods use spatially secondorder accurate finite differences.…”

“…Most of these methods use spatially secondorder accurate finite differences. In [17], finite element method was used, and in [32], fourth-order accurate finite differences were employed. The implicit discretization lead to a nonlinear system of equations at each time step.…”