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:

d S = μ S d t + ν S d Z 1 + ( e α + δ ε − 1 ) S d q {\displaystyle dS=\mu S\,dt+{\sqrt {\nu }}S\,dZ_{1}+(e^{\alpha +\delta \varepsilon }-1)S\,dq}
d ν = λ ( ν − ν ¯ ) d t + η ν d Z 2 {\displaystyle d\nu =\lambda (\nu -{\overline {\nu }})\,dt+\eta {\sqrt {\nu }}\,dZ_{2}}
corr ⁡ ( d Z 1 , d Z 2 ) = ρ {\displaystyle \operatorname {corr} (dZ_{1},dZ_{2})=\rho }
prob ⁡ ( d q = 1 ) = λ d t {\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.
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