Abstarct: Options are one of the most widely used financial instruments created for risk management. Pricing of an option as a derivative instrument is very important. Because incorrect pricing creates arbitrage opportunities in the market. In this thesis, exact solutions, and conservation laws of some non-linear pricing models are presented. The solutions have been found by the method of symmetries and Lie algebra invariant. In this method, with reduction of symmetric parameters, differential equations with partial derivatives are converted into ordinary differential equations. After finding the solutions and analyzing them, the conservation law of Barles-Soner, fractional Barles-Soner , and nonlinear Bakstein-Howison equations are obtained by direct method and formal Lagrange. In the sequel, a new method for neural networks to find the best model with the lowest pricing error in the market is introduced. In this research, the option pricing of S&P500 index, from August 18, 2022 to August 18, 2023, has been done under two models, Barles-Soner and Bakstein-Howison. Then it has been checked with neural networks. The results show the Barles-Soner model has less error comparing with another one.