Spectral Method for the Black-Scholes Model of American Options Valuation

Abstract

In this paper, we devote ourselves to the research of numerical methods for American option pricing problems under the Black-Scholes model. The optimal exercise boundary which satisfies a nonlinear Volterra integral equation is resolved by a high-order collocation method based on graded meshes. For the other spatial domain boundary, an artificial boundary condition is applied to the pricing problem for the effective truncation of the semi-infinite domain. Then, the front-fixing and stretching transformations are employed to change the truncated problem in an irregular domain into a one-dimensional parabolic problem in [−1,1]. The Chebyshev spectral method coupled with fourth-order Runge-Kutta method is proposed for the resulting parabolic problem related to the options. The stability of the semi-discrete numerical method is established for the parabolic problem transformed from the original model. Numerical experiments are conducted to verify the performance of the proposed methods and compare them with some existing methods.

Projection and Contraction Method for the Valuation of American Options

“…We compared results from our PCFEM with results from two other methods -viz. the Chebyshev spectral method (CPSM) [23] and the front-fixing finite element method (FFEM) [10].…”

“…Numerical methods for American options have attracted increasing interest, and are mainly of two types -viz. the Monte Carlo method [6,19,22] and the partial differential equation (PDE) method [1,9,10,14,20,23,27]. The Monte Carlo method has a high computational cost due to its slow convergence, and in this article we pursue the famous Black-Scholes PDE approach, which is widely regarded as one of most effective [7,11,15].…”

“…[14,15] discuss relevant convergence analysis. To improve the solution precision, finite element and spectral methods have received more attention recently [1,10,23,27]). One may refer to Ref.…”

An efficient numerical method for the valuation of American multi-asset options

“…Therefore, efficient numerical algorithms are essential for pricing American multi-asset options accurately. Recently, PDE discretization methods such as the finite difference method (Company et al 2016a, b;Zhou and Gao 2016), the finite element method , the finite volume method , the spectral method (Song et al 2014) and so on attracted the attention of practitioners and academics. Therefore, there emerged a large number of numerical algorithms, which could be divided into two categories-the penalty method (Kovalov et al 2007;Lei et al 2017;Zvan et al 1998) and the variational inequality method (Huang and Pang 2003;Hager et al 2010).…”