Optimal system of Lie group invariant solutions for the Asian option PDE

Asian options are useful financial products as they guard against large price manipulations near the termination date of the contract. In addition, they are often cheaper than their vanilla European counterparts. Previous analyses of the Asian option partial differential equation (PDE) have obtained analytical solutions for the fixed strike (arithmetically averaged) Asian option (and then only with certain assumptions on the boundary conditions). Using Lie symmetry analysis we obtain an optimal system of Lie point symmetries and demonstrate that many (usually ad hoc) reductions of the Asian option PDE are contained in this minimal set. We analyse each reduction member and the feasibility of its resulting invariant solution with the boundary conditions. We show that the numerical simulations on a reduced equation are more efficient than on the original specified problem. In addition, we have found new analytical solutions in terms of Fourier transforms for the floating strike Asian option as well as the fixed strike Asian option without the simplification of the domain.