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Research articles

ScienceAsia 50 (2024):ID 2024035 1-7 |doi: 10.2306/scienceasia1513-1874.2024.035


Numerical simulation of a nonlinear model in finance by Broyden?s method


Xianfu Zenga, Hongwei Liub, Haiyan Songc,*

 
ABSTRACT:     In this paper, we study the stationary Black-Scholes model arising in finance with transaction costs. This model becomes interesting when the time does not play a role such as, for instance, in perpetual options. The equation describing this model is a nonlinear second-order boundary value problem and there is no analytic solutions in closed form for such a nonlinear equation. After discretization via the centered finite difference formula we have to solve a nonlinear algebraic system which would be a serious problem when we use a small discretization mesh. We solve this nonlinear system by the residual-based Broyden?s method, which is an efficient quasi-Newton method and is convenient to implement by a desk computer. We give a convergence analysis of the Broyden?s method by assuming a lower and upper bound of the converged solution of the Black-Scholes model. Numerical results are given to show that the convergence rate of the method is robust with respect to the discretization mesh and the problem parameters.

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a College of International Economics and Trade, Ningbo University of Finance and Economics, Ningbo 315175, Zhejiang, China
b School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, Guangxi, China
c School of Computer and Data Engineering, NingboTech University, Ningbo 315100, Zhejiang, China

* Corresponding author, E-mail: haiyansong@nbt.edu.cn

Received 12 Jan 2023, Accepted 21 Jan 2024