Conservation Laws and Symmetry Reductions of a Generalized Nonlinear Black-Scholes Equation

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DOI: https://doi.org/10.28924/2291-8639-23-2025-310
Published: Nov 28, 2025
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Maba Boniface Matadi, Conservation Laws and Symmetry Reductions of a Generalized Nonlinear Black-Scholes Equation, Int. J. Anal. Appl., 23 (2025), 310.

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Maba Boniface Matadi

Abstract

In this paper, we apply the generalized double reduction theory to a nonlinear extension of the Black-Scholes equation, which is a foundational model in financial mathematics for pricing European options. The Lie symmetry method is used to identify point symmetries of the nonlinear PDE, and conservation laws are derived using the multiplier approach. We demonstrate how the symmetry reductions lead to simplified invariant solutions and discuss their implications for understanding nonlinear market behaviors. Numerical simulations are used to illustrate how nonlinearity modifies traditional option pricing surfaces. Finally, we discuss potential real-world applications such as volatility smiles, hedging strategies, and robustness of risk metrics.

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