Lie Symmetry Analysis Of An Option Pricing Equation In The Non-arbitrage Liquid Market

With the continuous expansion of the economic market,many financial derivatives emerge as the times require,its pricing problem has been widely concerned.This paper considers the option pricing problem of a non-arbitrage flow market(B-H)model with observable parameters without risk-free arbitrage opportunities in the market.This article first uses equivalent changes to transform the B-H model into a nonlinear partial differential equation.Based on the Lie symmetry analysis method,based on the continuation of the vector field,the symmetry of the partial differential equation is analyzed according to the situation,so as to obtain the single parameter variation group,the Lie bracket and the Lie adjoint representation corresponding to the equation.Then,according to the different symmetry,the one-dimensional sub-algebraic optimal system theory and the single parameter change group are used to further reduce the equation,reduce the dimensionality of the nonlinear partial differential equation to make it into an ordinary differential equation,and then solve the ordinary differential equation.Get the special solution of the reduced equation.Finally,discuss the economic significance of the solution,and study the impact of elasticity factors,volatility,time,etc,on asset prices in the model.