Fitting jump diffusion processes using the EM algorithm
by
Peter Thomson
Statistics Research Associates Ltd
Coauthors: Jodie Duncan (CSIRO Mathematical and Information Sciences)

Jump diffusion processes are often used as an alternative to geometric Brownian motion within continuous-time dynamic financial time series models. The advantages of the jump diffusion process are that it can not only account for discrete jumps in the path of the processs, but it also provides a simple way of replacing the Gaussian return distributions that arise in geometric Brownian motion models by Gaussian mixture distributions. The latter leads to more appropriate and meaningful models for the heavy-tailed distributions typically met in finance. An added attraction is that many of the analytical pricing formulae established for models based on geometric Brownian motion can be generalised to include the jump diffusion process.

However maximum likelihood estimation for jump diffusion models is not straightforward and careful numerical optimisation is typically required to identify the appropriate maximum likelihood estimates. Existing methods for fitting jump diffusion models are reviewed and their limitations discussed. A new recursive method that uses the EM algorithm is proposed and benchmarked against existing techniques using simulation.

Date received: April 30, 2002