  The leverage effect-- the correlation between an asset's return and its
volatility-- has played a key role in forecasting and understanding volatility
and risk. While it is a long standing consensus that leverage effects exist and
improve forecasts, empirical evidence paradoxically do not show that most
individual stocks exhibit this phenomena, mischaracterizing risk and therefore
leading to poor predictive performance. We examine this paradox, with the goal
to improve density forecasts, by relaxing the assumption of linearity in the
leverage effect. Nonlinear generalizations of the leverage effect are proposed
within the Bayesian stochastic volatility framework in order to capture
flexible leverage structures, where small fluctuations in prices have a
different effect from large shocks. Efficient Bayesian sequential computation
is developed and implemented to estimate this effect in a practical, on-line
manner. Examining 615 stocks that comprise the S\&P500 and Nikkei 225, we find
that relaxing the linear assumption to our proposed nonlinear leverage effect
function improves predictive performances for 89\% of all stocks compared to
the conventional model assumption.
