Stochastic Volatility Jump Diffusion (SVJD) is a type of model commonly used for equity returns that includes both stochastic volatility and jumps. The advantage of the model is that it is possible to replicate stylized facts such as heavy tails and volatility clustering and mean reversion, negative correlation between returns and volatility, and sudden large movements in the price of the asset.

Equity return distributions tend to demonstrate fat tails not captured by a generalised Brownian motion model (GBM). One way of solving this is to incorporate stochastic volatility and jumps into the model by applying the SVJD model (Bates, 1996). The SVJD model exhibits outcomes with higher levels of volatility compared to a geometric Brownian motion, mainly due to the jump processes incorporated in the model. This allows the model to capture unexpected extreme, negative, market returns.

The stochastic differential equations (SDE) for the asset level and the variance under the risk neutral measure gives the SVJD model by

 $$\begin{align} \frac{dS_t}{S_t} & = (r-d) dt + \sqrt{V(t)} dW_s(t) + (J-1) dN(t) \label{eqn:level_sde} \\ dV(t) & = \kappa (\theta - V(t)) dt + \epsilon \sqrt{V(t)} dW_v(t) \label{eqn:variance_sde} \end{align} $$

 where \(dW_s(t)\) and \(dW_v(t)\) are Wiener processes and correlated with \(\rho\). The parameters r and d are the risk neutral rate and the dividend yield respectively, whereas \(\epsilon\) denotes the volatility of variance. \(N(t)\) is the number of random jumps over the interval [0,t] determined by a Poisson process with intensity \(\lambda\), and \(J\) is the random jump size defined by:

$$ J = e^{\frac{-1}{2}\sigma^2_J+\sigma_JZ}$$

where \(Z \sim \textrm{N}(0,1)\), \(\mu_J\) \(\in\) \(\mathbb{R}\) is the average jump size and \(\sigma_J\) > 0 is the jump volatility. 

A demonstration of the SVJD model can be found in the animation below, where one clearly can see the contribution from both the diffusion with stochastic volatility and from the jumps.

 

 

References

  1. Bates, D., Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options, The Review of Financial Studies, Vol. 9, No. 1, pp. 69-107, 1996