Sufficient dimension reduction

SVJ or the Stochastic Volatility Jump model is a model suggested by Jim Gatheral. This is a model that fits the observed implied volatility surface well. The model is a Heston process with an added Merton log-normal jump.

Model

The model assumes the following correlated processes:

$dS=\mu Sdt+\sqrt{\nu }Sd{Z}_1+(e^{\alpha +\delta ϵ}-1)Sdq$

$d\nu =-\lambda (\nu -\bar{\nu })dt+\eta \sqrt{\nu }d{Z}_2$

$\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}(d{Z}_1,d{Z}_2)=\rho$