A wide variety of conditional and stochastic variance 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. In this
paper, we first show the relationship among conditional, stochastic, integrated and
realized volatilities. Then we develop a new asymmetric volatility model, which takes
account of small and large, and positive and negative, shocks. Using the new
specification, we examine alternative volatility models that have recently been
developed and estimated in order to understand the differences and similarities in the
definitions of asymmetry and leverage. We extend the new specification to realized
volatility by taking account of measurement errors. As an empirical example, we apply
the new model to the realized volatility of Standard and Poor's 500 Composite Index
using Efficient Importance Sampling to show that the new specification of asymmetry
significantly improves the goodness of fit, and that the out-of-sample forecasts and VaR
thresholds are satisfactory.
|