Lie Symmetry Net: Preserving Conservation Laws in Modelling Financial Market Dynamics via Differential Equations

TL;DR

This paper introduces Lie Symmetry Net (LSN), a novel framework that models financial market dynamics by preserving the intrinsic symmetries of differential equations, leading to significant accuracy improvements.

Contribution

The paper presents LSN, a new symmetry-aware neural network that incorporates conservation laws from Lie symmetries to improve modeling of financial differential equations.

Findings

LSN effectively captures Lie symmetries in financial models.

Achieves over tenfold error reduction compared to existing methods.

Demonstrates robustness across various financial differential equations.

Abstract

This paper employs a novel Lie symmetries-based framework to model the intrinsic symmetries within financial market. Specifically, we introduce Lie symmetry net (LSN), which characterises the Lie symmetries of the differential equations (DE) estimating financial market dynamics, such as the Black-Scholes equation. To simulate these differential equations in a symmetry-aware manner, LSN incorporates a Lie symmetry risk derived from the conservation laws associated with the Lie symmetry operators of the target differential equations. This risk measures how well the Lie symmetries are realised and guides the training of LSN under the structural risk minimisation framework. Extensive numerical experiments demonstrate that LSN effectively realises the Lie symmetries and achieves an error reduction of more than one order of magnitude compared to state-of-the-art methods. The code is available at https://github.com/Jxl163/LSN_code.