Stochastic volatility jump explained
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:
dS=\muSdt+\sqrt{\nu}
| \alpha+\delta\varepsilon |
| SdZ | |
| 1+(e |
-1)Sdq
d\nu=λ(\nu-\overline{\nu})dt+η\sqrt{\nu}dZ2
\operatorname{corr}(dZ1,dZ2)=\rho
\operatorname{prob}(dq=1)=λdt
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.
Notes and References
- 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.
