Stochastic volatility jump

In mathematical finance, the stochastic volatility jump (SVJ) model is suggested by Bates.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.
/ref> 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:

: $dS=\mu S\,dt+\sqrt S\,dZ_1+(e^ -1)S \, dq$

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

: $\operatorname(dZ_1, dZ_2) =\rho$

: $\operatorname(dq=1) =\lambda dt$
where ''S'' is the price of security, ''μ'' is the constant drift (i.e. expected return), ''t'' represents time, ''Z''₁ is a standard Brownian motion, ''q'' is a Poisson counter with density ''λ''.

References

Mathematical finance
Financial models

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