On the construction of infinitely many nonclassical symmetries of the Black–Scholes equation

Abstract The Black–Scholes equation plays an important role in financial mathematics and continues to attract interest due to its theoretical significance and pedagogical value. While numerous studies have been conducted on the Black–Scholes equation using its Lie point symmetries, comparatively little attention has been given to its nonclassical symmetries. In this paper, we present a method for constructing infinitely many nonclassical symmetries of the Black–Scholes equation. Our approach employs the expanded Lie group method, wherein transformation groups act on the equation’s variables as well as its parameters. This enables the construction of an invertible point transformation that maps the original equation to a simplified version containing two fewer terms. We show that a class of nonclassical symmetries of the simplified equation are governed by an auxiliary (1+1)-dimensional partial differential equation involving an arbitrary function. Solutions of this equation for each admissible specification of the arbitrary function yields distinct nonclassical symmetries, resulting in an infinite class of such symmetries. These nonclassical symmetries are then transformed back to produce corresponding nonclassical symmetries of the original Black–Scholes equation. We derive four representative nonclassical symmetries for the reduced equation, which are mapped to the original Black–Scholes equation. The proposed framework offers a systematic approach for finding nonclassical symmetries of the Black–Scholes equation and related equations, enabling the construction of exact solutions that cannot be obtained using classical symmetry methods.