#### Authors:

Zahid Hussain,Sadaqat Hussain,Suhail Abbas,Shams-ur-Rehman,Shahid Hussain,#### DOI NO:

https://doi.org/10.26782/jmcms.2021.09.00005#### Keywords:

Black-Scholes Equation,Generators,LE,OS,Invariant solution,#### Abstract

*In this manuscript, the Lie group technique is applied to construct a new OS and invariant solutions of a one-dimensional LA, which describes the symmetries properties of a nonlinear Black-Scholes model. The structure of LA depends on one parameter. We have shown a novel way to construct the so-called OS of subalgebras of the Black-Scholes equation by utilizing the given symmetries. We transform the symmetries of the Black-Scholes equation into a simple ordinary differential equation called the Lie equation, which provides us a way through which to construct a new optimal scheme of subalgebras of the Black-Scholes through applying the concept of LE. The OS which consists of minimal representatives is utilized to develop the invariant solution for the Black-Scholes equation. The fundamental use of the Lie group analysis to the differential equation is the categorization of group invariant solutions of differential equations via OS. Finally, we have utilized the OS to construct the invariant solution of the Black-Scholes equation*.

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