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:

$dS=\mu S\,dt+{\sqrt {\nu }}S\,dZ_{1}+(e^{\alpha +\delta \epsilon }-1)Sdq$

$d\nu =-\lambda (\nu -{\overline {\nu }})dt+\eta {\sqrt {\nu }}dZ_{2}$

$\operatorname {corr} (dZ_{1},dZ_{2})=\rho$