An Introduction to Stochastic Volatility Jump Models

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.^[2]

Here, as an example, we present the basic case of a combination of Merton's jump diffusion model and Heston's stochastic volatility model. Merton's model is based on the classic Black-Scholes model but extended to include discontinuous asset returns. The jumps are assumed to be independent from the diffusion. The stochastic volatility in Heston's model is a mean-reverting square-root process. Bates (1996)^[1] was one of the first to describe this particular combination of models.

The stochastic differential equations (SDE) for the asset level and the variance under the risk neutral measure are given by

$$         \begin{align}             \frac{dS_t}{S_t} & = \mu_t dt + \sqrt{\eta_t} dW^1_t + (J_t-1) dq_t \label{eqn:level_sde} \\             d\eta_t       & = \kappa (\theta - \eta_t) dt + \sigma \sqrt{\eta_t} dW^2_t \label{eqn:variance_sde}           \end{align}         $$

with correlated Brownian motions

$$ E\left[ dW_1(t) dW_2(t) \right] = \rho dt $$

and where \(dq_t\) is the Poisson process given by

$$         \begin{equation*}             dq_t = \left\{             \begin{array}{l}                 0 \;\; \textrm{ with probability } 1-\lambda_t dt \\                 1 \;\; \textrm{ with probability } \lambda_t dt             \end{array}\right.         \end{equation*}         $$

The jumps, i.e. \(J_t\), in the asset level SDE, Equation \(\ref{eqn:level_sde}\), are assumed to be log-normally distributed with mean log-jump \(\alpha\) and standard deviation \(\beta\), i.e. \(E\left[J\right] = e^{\alpha+\beta\varepsilon}\) where \(\varepsilon \sim \textrm{N}(0,1)\). Thus, the choice of drift parameters that makes \(S_t e^{-rt}\) a martingale is^[3]

$$ \begin{eqnarray*} \mu_t & = & r - \lambda_t m \\ m & = & E\left[J\right] - 1 \end{eqnarray*} $$

The variance process \(\eta_t\), Equation \(\ref{eqn:variance_sde}\), is mean reverting with mean reversion level \(\theta\), mean reversion strength \(\kappa\) and constant volatility \(\sigma\).

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

2. Cont, R., Empirical Properties of Asset Returns, Quantitative Finance, Vol. 1, Institute of Physics Publishing, 2001

3. Glasserman, P., Monte Carlo Methods in Financial Engineering, Springer-Verlag New York Inc., 2004