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 for stochastic volatility with an added Merton log-normal jump. It assumes the following correlated processes:

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

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

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

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

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.

References

1. David S. Bates, "Jumps and Stochastic volatility: Exchange Rate Processes Implicity in Deutsche Mark Options", The Review of Financial Studies, volume 9, number 1, 1996, pages 69–107.