Discretization of the Dirac delta function for application in option pricing

Abstract: This paper compares two different approximations of the Dirac deltafunction used in a Fokker-Planck equation. Both methods deal with the singularity problem in the initial condition. The Dirac delta approximation, constructed in MATLAB with a method derived by Tornberg and Engquist, was compared to an already given method Aït-Sahalia. The methods were implemented as the initial condition in the Fokker-Planck equation, e.g approximating a probability density function. In most cases Aït-Sahalia and Tornberg-Engquist were interchangeable. During specific circumstances one method was significantly more accurate than the other. Increasing the amount of time/spatial steps enhanced the differences in error while having less time/spatial steps made the difference in error converge. The Aït-Sahalia method produces slightly more accurate results in more cases.