Newton-Fréchet Iterative Solution of Nonlinear Local-Stochastic Volatility Models

Articles in Press

Document Type : Research article

Authors

1 Department ‎of Applied ‎Mathematics, ‎Faculty ‎of‎ ‎Mathematics‎, Statistics and ‎Computer ‎Science‎, ‎University of Tabriz‎, ‎Tabriz‎, ‎Iran

2 Graduate of Biomedical Engineering from Tabriz University of Technology (Sahand), Tabriz, Iran

10.22128/ansne.2026.3409.1231

Abstract

Financial models frequently exhibit nonlinear behavior, necessitating robust numerical techniques for precise analysis. This study investigates the numerical implementation of a generalized Newton method within a Local-Stochastic Volatility framework. The nonlinear partial differential equation (PDE) governing the model is linearized via the generalized Newton procedure, employing the Fr'echet derivative to construct the iterative scheme. By imposing appropriate boundary conditions and utilizing representative financial data, we examine the convergence properties of the resulting sequence. Numerical results demonstrate that the iterative method achieves stable and rapid convergence, thereby validating the existence of a unique solution within this framework and highlighting the efficacy of Newton-type approaches for solving nonlinear PDEs in financial modeling. Furthermore, this approach significantly reduces computational costs compared to traditional methods, offering a practical and efficient tool for real-time pricing in dynamic market environments. The proposed method also ensures high accuracy in handling complex derivative structures, making it a valuable asset for financial institutions seeking to optimize their risk management strategies.

Keywords

Main Subjects

Analytical and Numerical Solutions for Nonlinear Equations

Articles in Press, Accepted Manuscript
Available Online from 08 September 2026

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History

  • Receive Date: 28 July 2026
  • Revise Date: 02 September 2026
  • Accept Date: 05 September 2026
  • Publish Date: 08 September 2026

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