Symmetry analysis of Black-Scholes equation for small values of volatility and rate of return

In this paper, we present approximate symmetries of the Black-Scholes equation for small values of volatility and rate of return parameters. A novel method for obtaining the approximate symmetry of a singularly perturbed partial differential equation (PDE) is introduced. Further, we compute the optimal system in the singular case. Finally, by combining two methods, a new approach that calculates the approximate generators for admitted Lie groups of asset price is provided.