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}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle d\nu =\lambda (\nu -{\overline {\nu }})\,dt+\eta {\sqrt {\nu }}\,dZ_{2}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \operatorname {corr} (dZ_{1},dZ_{2})=\rho }](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \operatorname {prob} (dq=1)=\lambda dt}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
- ^ 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.