The models we discussed so far in our blog posts did not include the possibility of jumps, i.e., discontinuous asset price processes. In real markets, however, such jumps do occur. While small jumps can be satisfactorily explained by diffusion, this is no longer possible for the more pronounced breaks in market behaviour that have happened several times in history (Black Monday, 9/11, Lehman Brothers,...). There are two different categories of financial models with jumps: The first are so-called jump diffusion models, where a diffusion process responsible for the "normal" evolution of prices is augmented by additional jumps added at random intervals. In these models, jumps represent rare events, such as crashes and large drawdowns.
This kind of evolution of the asset price can be represented by modelling the (log)-price as a Levy process (A Levy process is a stochastic process with independent, stationary increments.) with a non-zero Gaussian component and a jump part. The jump part is a compound Poisson process with finitely many jumps in every time interval. Depending on the distribution used for jump sizes, different jump diffusion models exist.
The second category of jump models uses infinite activity Levy processes for
propagating the state variable. Examples for models in this category are the Variance
Gamma (VG) model and the Normal Inverse Gaussian (NIG) model.