Invariant solutions of a European option pricing equation under a continuous-time capital asset pricing model

Obaidullah, Usaamah; Gwaxa, B.

doi:10.47974/JIM-2159

Open Access Research Article

Invariant solutions of a European option pricing equation under a continuous-time capital asset pricing model

Usaamah Obaidullahusaamah.obaidullah@gmail.comSchool of Mathematics University of the WitwatersrandJohannesburg, 2001, South AfricaView full profile → , ^*B. GwaxaCorresponding authorSameerah.Jamal@wits.ac.zaSchool of Mathematics University of the Witwatersrand; Department of Science and Innovation (DSI) National Research Foundation (NRF) Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS)School of Mathematics University of the Witwatersrand, JohannesburgJohannesburg, Wits, 2001, South Africa0000-0003-3702-8368View full profile →

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pp. 1–12Online FirstMarch 2026DOI: 10.47974/JIM-2159XML

Received:: 01 May 2025
Published Online:: 27 Mar 2026
Article type:: Research Article
Language:: EN
Article no.:: JIM-2159
Pages:: 1–12

Abstract

This work investigates a nonlinear European option pricing equation under a continuous-time Capital Asset Pricing Model. In particular, we determine its point symmetries and thereby find analytical solutions using the Lie symmetry approach. In our investigation, we vary the beta coefficient, a measure of an asset’s systematic risk. We examine the model for both a non-zero and a zero market beta in our study. We show that in the first instance, this model is invariant under a three-dimensional Lie algebra, and in the latter situation, it is invariant under a six-dimensional Lie algebra. For both the non-zero and zero market beta models, we get reductions and with various parameter settings, interesting graphical solutions are investigated.

Keywords

Option pricing Lie symmetries Exact solutions

Subject Classifications

35K0535K1537C79

References

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