Stochastic volatility and stochastic leverage

Abstract

This paper proposes the new concept of stochastic leverage in stochastic volatility models. Stochastic leverage refers to a stochastic process which replaces the classical constant correlation parameter between the asset return and the stochastic volatility process. We provide a systematic treatment of stochastic leverage and propose to model the stochastic leverage effect explicitly, e.g. by means of a linear transformation of a Jacobi process. Such models are both analytically tractable and allow for a direct economic interpretation. In particular, we propose two new stochastic volatility models which allow for a stochastic leverage effect: the generalised Heston model and the generalised Barndorff-Nielsen & Shephard model. We investigate the impact of a stochastic leverage effect in the risk neutral world by focusing on implied volatilities generated by option prices derived from our new models. Furthermore, we give a detailed account on statistical properties of the new models.

Estimation of the Continuous and Discontinuous Leverage Effects

“…Kalnina and Xiu (2016), Curato (2015), Curato and Sanfelici (2015), Veraart and Veraart (2012)) define CLE as the quadratic covariation between the Brownian motion parts of X and σ , i.e. ρ t dt = d [ W X , W σ ] t .…”

“…3 In view of the differences in definitions, the econometric and asymptotic properties of our CLE estimator are very different from those of the two CLE estimators in Kalnina and Xiu (2016). Finally, among others, Veraart and Veraart (2012) extend the Heston model to allow for a time-varying feature, and Curato (2015) and Curato and Sanfelici (2015) employ a Fourier transform based method to estimate a stochastic continuous leverage effect.…”

Estimation of Correlation for Continuous Semimartingales

“…A test for a time‐varying correlation for example could then be based on statistics such as . See Veraart & Veraart (2011) for some details on models incorporating stochastic leverage. Another typical extension is to allow for additional jumps in the model, such that we need to apply methods for jump detection first in order to be able to focus on the volatility part only.…”

How precise is the finite sample approximation of the asymptotic distribution of realised variation measures in the presence of jumps?

“…Note that the parameter ρ in the BNS model is not a correlation parameter as such (since it is not restricted to be in [−1, 1]), but it plays its role, see e.g. Veraart and Veraart (2010) for more details.…”