Transitional densities of diffusion processes: a new approach to solving the Fokker-Planck equation

Many common option pricing problems require numerical solution techniques. One standard tool is to solve a finite difference approximation to the instrument's fundamental partial differential equation (PDE), with appropriate boundary conditions. The approximation will converge to the true solution of the PDE asymptotically as the time and price steps go to zero. In practice, of course, finite step sizes must be selected as a compromise between accuracy and computation time. But this can lead to numerical difficulties in computation. Often these problems are difficult to spot a priori because they only occur for certain parameter values that seem improbable economically. Unfortunately, in searching for correct estimates, a calibration routine can still stumble onto such problem parameter values, and then break down or deliver highly inaccurate results. One such numerical problem occurs at time 0, when the probability density at the initial asset price is a Dirac delta function. In this short article, Hurn, Jeisman and Lindsay offer a simple solution to the problem by reformulating the calculation into one based on the transitional cumulative distribution function (CDF) instead of the transitional density. Simulation analysis suggests that the new approach may not greatly reduce calculation times, although performance can be enhanced with a minor modification. But a plain vanilla real world example of fitting a simple interest rate model on 1-month LIBOR demonstrates the fragility of the standard methodology and the improvement in robustness provided by the new technique.