Algorithmic solution of the power option PDE by the lie group approach

From the era of Black–Scholes, European options have dwelt on pay-offs that are a linear function of the asset price. In this paper, we will look at a special case of exotic options - power options - whose payoffs are nonlinear functions of the underlying asset price. Exotic options are derivatives which have features that makes them more complex than commonly traded products - thus finding their fair value is not an always easy task. Previous analyses of the power option partial differential equation (PDE) have only obtained closed form solutions either by guessing solutions, similarity methods or the martingale approach [8, 9, 25]. Using Lie symmetry analysis we obtain an optimal system of the Lie point symmetries of the power option PDE and demonstrate an algorithmic method for finding solutions to the equation. In addition, we find a new analytical solution to the asymmetric type of the power option.