# Stochastic volatility jump

In mathematical finance, the stochastic volatility jump (SVJ) model is suggested by Bates.[1] This model 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:

${\displaystyle dS=\mu S\,dt+{\sqrt {\nu }}S\,dZ_{1}+(e^{\alpha +\delta \varepsilon }-1)S\,dq}$
${\displaystyle d\nu =\lambda (\nu -{\overline {\nu }})\,dt+\eta {\sqrt {\nu }}\,dZ_{2}}$
${\displaystyle \operatorname {corr} (dZ_{1},dZ_{2})=\rho }$

[where S = Price of Security, μ = constant drift (i.e. expected return), t = time, Z1 = Standard Brownian Motion etc.]