A Stochastic Volatility Model With a General Leverage Specification

Abstract

We introduce a new stochastic volatility model that postulates a general correlation structure between the shocks of the measurement and log volatility equations at different temporal lags. The resulting specification is able to better characterize the leverage effect and propagation in financial time series. Furthermore, it nests other asymmetric volatility models and can be used for testing and diagnostics. We derive the simulated maximum likelihood and quasi maximum likelihood estimators and investigate their finite sample performance in a simulation study. An empirical illustration shows that the postulated correlation structure improves the fit of the leverage propagation and leads to more precise volatility predictions.

Keywords:

• Asymmetric stochastic volatility
• Leverage effect
• Volatility prediction

Supplementary Materials

The supplementary material accompayining this paper reports additional results for the analysis of Section 5.2.1.

Notes

Acknowledgments

I would like to thank Jun Yu, the associate editor, and two anonymous reviewers for their comments and suggestions during the preparation of this article. I would also like to thank Domenica Muri and Beatrice Malorni for their kind hospitality during the COVID-19 pandemic when this article was written.

Notes

1 The exact formulation of c_k is $c_k={\kappa }^2{\rho }_0^2{\beta }^2\hspace{0.17em}\text{exp}\hspace{0.17em}\left\{\frac{{\sigma }_h^2}{4}\right\}[2\hspace{0.17em}\text{exp}\hspace{0.17em}\{\frac{{\sigma }_h^2}{4}\}\Phi (-g_k)-\Phi {(-\frac{-g_k}{2})}^2]$, and $g_k=\kappa {\sum }_{j=0}^k{\varphi }^{k-j}{\rho }_j$, if $0<k\le m$ and $g_k=\kappa {\varphi }^k{\sum }_{j=0}^m{\varphi }^{m-j}{\rho }_j$ if k > m.

2 As pointed out by a referee, even if $\sqrt{T}/N\to 0$, the consistency and asymptotic normality properties of SML follow from those of ML, which are, however, difficult to derive even in the plain SV model. In the latter case, consistency is discussed in Douc et al. (2011, p. 492).

3 Allowing for ${\rho }_0\ne 0$ induces a nonlinear system for $\text{log}\hspace{0.17em}(y_t^2)$ which can be subsequently linearized using a Taylor expansion. This approach would result in an estimator based on the extended Kalman filter. Such estimator might be used to estimate the specification of Jacquier, Polson, and Rossi (2004) without resorting to MCMC. We leave this for future research.

4 Similar simulation results for the plain SV model ($m=0,{\rho }_0=0$) are reported by Andersen and Sørensen (1996) for GMM, Ruiz (1994) for QML, and Danielsson (1994) for SML. See Broto and Ruiz (2004) for a review of different estimation methods for the plain SV model.

5 SML estimates obtained imposing the constraint ${\rho }_0=0$, leads to the same results but for N225, where m = 5 is selected.

6 Note that our sample size is more than two times that of Yu (2005).

7 Note that the specification by Jacquier, Polson, and Rossi (2004) coincides with m = 0.