Stochastic volatility jump

In mathematical finance, the stochastic volatility jump (SVJ) model is suggested by Bates. This model fits the observed implied volatility surface well. The model is a Heston process for stochastic volatility with an added Merton log-normal jump. It assumes the following correlated processes:

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

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

$\mathrm{corr}(d{Z}_1,d{Z}_2)=\rho$

$\mathrm{prob}(dq=1)=\lambda dt$

where S is the price of security, μ is the constant drift (i.e. expected return), t represents time, Z₁ is a standard Brownian motion, q is a Poisson counter with density λ.