Fitting jump diffusion processes using the EM algorithm

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 process, 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 leptokurtic, 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. New recursive methods that use the EM algorithm are proposed and benchmarked against direct maximum likelihood using simulation. In addition to being relatively simple to implement, the new method appears to have better numerical properties and is less sensitive to the choice of starting values. Finally, it yields flexible conditional mean and volatility estimates which may be used in an exploratory data analysis.