Option pricing: the reduced-form SDE model

We use partial differential equations (PDEs) to describe the pricing process of options in an illiquid market. These equations are derived from stochastic differential equations built on the Ito process. With the help of Lie symmetry analysis, this paper focuses on the pricing of a model that incorporates the effect of large traders in an illiquid market. The nonlinear PDE representing this model incorporates a nonzero risk-neutral interest rate. This PDE is singularly perturbed and quadratic in the highest derivative. Using the method of Lie symmetry analysis, we obtain symmetries in the mathematical package Program Lie, and these symmetries are used to analyse the equation and to reduce the PDE to ordinary differential equations. When the equations are solved, they yield group invariant solutions to the PDE. We give a graphical representation of the obtained solutions. These invariant solutions are new to the field and can be used in place of simulations.