Stochastic volatility jump

In mathematical finance, the stochastic volatility jump (SVJ) model is suggested by Bates.[http://faculty.baruch.cuny.edu/lwu/890/Bates96.pdf 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.] 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:

: $dS=\backslash mu\; S\backslash ,dt+\backslash sqrt\{\backslash nu\}\; S\backslash ,dZ\_1+(e^\{\backslash alpha\; +\backslash delta\; \backslash varepsilon\}\; -1)S\; \backslash ,\; dq$

: $d\backslash nu\; =\backslash lambda\; (\backslash nu\; -\; \backslash overline\{\backslash nu\})\; \backslash ,\; dt+\backslash eta\; \backslash sqrt\{\backslash nu\}\; \backslash ,\; dZ\_2$

: $\backslash operatorname\{corr\}(dZ\_1,\; dZ\_2)\; =\backslash rho$

: $\backslash operatorname\{prob\}(dq=1)\; =\backslash lambda\; dt$

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.