An unconditionally stable, positivity-preserving splitting scheme for nonlinear Black-Scholes equation with transaction costs.

Jianqiang Guo,Wansheng Wang

doi:10.1155/2014/525207

Abstract

This paper deals with the numerical analysis of nonlinear Black-Scholes equation with transaction costs. An unconditionally stable and monotone splitting method, ensuring positive numerical solution and avoiding unstable oscillations, is proposed. This numerical method is based on the LOD-Backward Euler method which allows us to solve the discrete equation explicitly. The numerical results for vanilla call option and for European butterfly spread are provided. It turns out that the proposed scheme is efficient and reliable.

Highlights

One of the modern financial theory’s biggest successes in terms of both approach and applicability has been the BlackScholes option pricing model developed by Black and Scholes in 1973 [1] and previously by Merton [2]:
4.115806 4.105581 where UiN,Sp and UiN,Ns denote the numerical solutions computed by our splitting scheme (45) and the nonstandard scheme proposed in [16] at the maturity date T = 0.5, respectively, and ŨiN denotes the numerical reference solution computed by the Backward Euler method with the standard second-order finite differences on a finer grid with N = 2560 and M = 320
From the theoretical analysis given in this paper and the numerical results shown we come to the following remark: the proposed scheme is efficient and reliable

Summary

Introduction

One of the modern financial theory’s biggest successes in terms of both approach and applicability has been the BlackScholes option pricing model developed by Black and Scholes in 1973 [1] and previously by Merton [2]:. Recent studies of their influence reveal that they result in a nonnegligible increase in the option price, they are generally small for institutional investors. To relax the restrictive conditions, in [16], Zhou et al proposed an unconditionally stable explicit finite difference scheme based on a nonstandard approximation of the second partial derivative. This scheme is conditionally consistent, and the truncation error depends on the ratio of the time stepsize and the square of the space stepsize.

Transformation and Spatial Semidiscretization

Splitting Time-Stepping Method

Properties of the Numerical Scheme

Numerical Experiments

Concluding Remarks