Asymmetry and Leverage in Stochastic Volatility Models: An Exposition

The accurate specification and modelling of risk are integral to optimal portfolio and risk management. In this context, a wide variety of conditional and stochastic volatility models has been used to estimate latent volatility (or risk). In both the conditional and stochastic volatility literature, there has been some confusion between the definitions of asymmetry and leverage. This paper examines alternative univariate SV models that have recently been developed and estimated in to order to understand the differences and similarities between the definitions of asymmetry and leverage. Five univariate SV models, namely the basic SV model, SV model with leverage, and three different types of asymmetric SV models, are analysed in order to clarify the distinction between asymmetry and leverage. Alternative specifications of SV models are defined according to the use of standardized or unstandardized returns, with or without leverage, in order to evaluate the differential impacts of positive and negative returns on future volatility, namely symmetry, asymmetry, type I asymmetry (or leverage), type II asymmetry, and type III asymmetry.