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

Xuelian Jiang, Tongtian Zhu, Yingxiang Xu, Can Wang, Yeyu Zhang, Fengxiang He

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.

Submission Length: Regular submission (no more than 12 pages of main content)

Changes Since Last Submission:

1. We added experiments for Maxwellian Tails model, in addition to the presented Black-Scholes equation, Vašiček equation, and KdV equation; please kindly referred to Section 5.2. The results suggest that our approach significantly outperforms all existing PINN methods.
2. We added an explanation below on the motivation of using the conservation loss instead of the established Lie symmetry loss term, in page 13, and as follows,

“Our conservation law-based approach can capture a broader range of symmetries, compared with using a pre-fixed symmetry operator. More specifically, our approach preserves Lie symmetry operators G₂, l(t), and g(t) (see Equation (7)), while exsiting method can only cover G₂. We experimentally validated the advances in Figure 9.”

We also conducted additional experiments on real financial data, the OptionMetrics dataset based on the Nasdaq 100 index, to verify our approach, which suggests that our method outperformes existing methods.

Code: https://github.com/Jxl163/LSN_code

Assigned Action Editor: ~Bamdev_Mishra1

Submission Number: 4052