In this paper, we extend the long-run risks model of Bansal and Yaron (BY, 2004) to allow both a long- and a short-run volatility component in consumption growth, long-run risks, and dividend growth. Our two volatility model better captures macroeconomic volatility than a single volatility model, and can reconcile simultaneously the large negative market variance risk premium, di?ering predictability in excess returns, consumption, dividends, and stock market volatility, all of which are di±cult to explain previously by the BY model.

In this paper, we derive the corresponding implied VIX formulas under the locally riskneutral valuation relationship proposed by Duan (1995) when various forms of GARCH model are proposed for S&P 500 index. The empirical study shows that the GARCH implied VIX is consistently and significantly lower than the CBOE VIX for all kinds of GARCH model investigated. Moreover, the magnitude of the difference suggests that the GARCH option pricing model is not capable of capturing the variance premium, which indicates the incompleteness of the GARCH option pricing under the locally risk-neutral valuation relationship. The source of this kind of incompleteness is then theoretically analyzed. It is shown that the framework of GARCH option pricing model fails to incorporate the price of volatility risk or variance premium.