To capture volatility risk, we use factors from VIX, VIX futures, and their basis. We find that portfolios with lower (higher) factor loadings on the market and volatility risk from in-sample time-series regressions, have persistent out-of-sample lower (higher) factor loadings. More importantly, by separating cases based on the sign of volatility changes, this study documents the existence of an asymmetric effect due to volatility shocks on asset returns. When volatility is shocked positively, there is a significantly negative relationship between factors associated with uncertainty and asset returns. Furthermore, after incorporating this asymmetric effect, volatility factors have significant risk premia in Fama-MacBeth crosssectional regressions.

This paper presents the first comparison of the accuracy of density forecasts for stock prices. Six sets of forecasts are evaluated for DJIA stocks, across four forecast horizons. Two forecasts are risk‐neutral densities implied by the Black–Scholes and Heston models. The third set are historical lognormal densities with dispersion determined by forecasts of realized variances obtained from 5‐min returns. Three further sets are defined by transforming risk‐neutral and historical densities into real‐world densities. The most accurate method applies the risk transformation to the Black–Scholes densities. This method outperforms all others for 87% of the comparisons made using the likelihood criterion.

This paper presents the first comparison of the accuracy of density forecasts for stock prices. Six sets of forecasts are evaluated for DJIA stocks, across four forecast horizons. Two forecasts are risk-neutral densities implied by the Black-Scholes and Heston models. The third set are historical lognormal densities with dispersion determined by forecasts of realized variances obtained from 5-min returns. Three further sets are defined by transforming risk-neutral and historical densities into realworld densities. The most accurate method applies the risk transformation to the Black-Scholes densities. This method outperforms all others for 87% of the comparisons made using the likelihood criterion.

First, to separate different market conditions, this study focuses on how VIX spot (VIX), VIX futures (VXF), and their basis (VIX VXF ) perform different roles in asset pricing. Secondly, this study decomposes the VIX index into two parts, volatility calculated from out-of-themoney call options and volatility calculated from out-of-the-money put options. The analysis shows that out-of-the-money put options capture more useful information in predicting future stock returns.

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