Optimal system, symmetry reductions and new closed form solutions for the geometric average Asian options

In this paper, the Lie group analysis method is applied to the geometric average Asian option pricing equation in financial problems. Firstly, the complete Lie symmetry group and infinitesimal generators of this equation are derived. Then the optimal system with one parameter for the Lie symmetry algebra are obtained, which gives the possibility to describe a complete set of invariant solutions to the pricing equation. Finally, based on the optimal system the symmetry reductions and corresponding closed form solutions for the pricing equation are proposed.