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:

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

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

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

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

References

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