Lie symmetry analysis on pricing power options under the Heston dynamic and some fractional financial models

The rise of computational mathematics in financial markets has accelerated the bloom of various financial models. For instance, the Black-Scholes-Merton model, the Vasicek model, the Cox-Ingersoll-Ross model, the Heston model, etc. Each of these models often produces challenging partial differential equations. The Lie symmetry method appears to be a powerful tool to solve these types of equations. In this study, we apply Lie's method to the power options model under the Heston dynamic. The infinitesimal operators, the optimal systems, the invariant solutions, and the conservation laws are presented. Lie analysis is also an efficient tool to solve the fractional differential equations which involve the differentiation of a function with respect to its independent variable(s) to a non-integer order. Fractional differential equations are well known for their ability to describe the memory effect in various natural phenomena. We apply the Lie symmetry analysis to a time-fractional Black-Scholes-Merton model, as well as an arbitrage-free stock price model. The results of the analysis which include the infinitesimal operators or generators, the optimal systems, and the invariant solutions of the above models are presented. The visual representations of the invariant solutions are provided alongside discussions and comparisons with the solutions from their corresponding non-fractional models.