Lie Symmetry Analysis of a Nonlinear Black–Scholes Equation in Illiquid Markets

Abstract

We have conducted comprehensive Lie symmetry analysis of a nonlinear Black–Scholes equation that arises in illiquid markets. The equation incorporates nonlinearities arising from market constraints, such as transaction costs and liquidity effects. We have identified the admitted Lie point symmetries, derived the optimal system of one-dimensional subalgebras, and constructed invariant solutions. For the first time, we have determined nonclassical symmetries of this equation, resulting in new exact solutions that are inaccessible through the classical symmetries of the equation. These results extend and clarify earlier work on the nonlinear Black–Scholes equation and provide a broader understanding of the solution space for the equation. The work offers new perspectives on nonlinearities in option pricing under realistic market conditions and also highlights the role of Lie symmetry methods in financial modeling.