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:


 * $$dS=\mu S\,dt+\sqrt{\nu} S\,dZ_1+(e^{\alpha +\delta \varepsilon} -1)S \, dq$$


 * $$d\nu =\lambda (\nu - \overline{\nu}) \, dt+\eta \sqrt{\nu} \, dZ_2$$


 * $$\operatorname{corr}(dZ_1, dZ_2) =\rho$$


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

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