Numerical Solution of Fokker-Planck Equation by Variational Approach � an Application to Pricing Barrier Options

The Fokker-Planck equation is a partial differential equation (PDE) that defines the time evolution of the probability density function of a stochastic process. By applying appropriate boundary conditions to the PDE and obtaining the corresponding probability density function, one can price a variety of derivatives, assuming different stochastic processes. The current work presents a numerical procedure for solving the Fokker-Planck equation. It is based on a variational approach for discretizing the PDE and converting it into a system of ordinary differential equations. Space discretization is achieved by using non-uniform B-Splines, which are shown to be appropriate for representing initial conditions like the Dirac delta function. The proposed numerical methods are presented for pricing barrier options under the Heston stochastic volatility process.