In this study we empirically study the variance term structure using volatility index (VIX) futures market. We first derive a new pricing framework for VIX futures, which is convenient to study variance term structure dynamics. We construct five models and use Kalman filter and maximum likelihood method for model estimations and comparisons. We provide evidence that a third factor is statistically significant for variance term structure dynamics. We find that our parameter estimates are robust and helpful to shed light on economic significance of variance factor model.

This research was supported by a grant from the Network for Study on Pensions, Aging, and Retirement to the Columbia Business School. We especially thank Pierre Collin-Dufresne for many substantive early discussions that were fundamental to the development of the paper. We also thank Elessar Chen for his research assistance. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research. NBER working papers are circulated for discussion and comment purposes. They have not been peerreviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications.

Studies have shown that markets may underprice sub-national governments' risk on the implicit assumption that these entities would be bailed out by their central government in case of financial difficulties. However, the question of whether sovereigns pay a premium on their own borrowing as a result of (implicitly or explicitly) guaranteeing sub-entities' debt has been explored only little. We use an event study approach with separate equations for two levels of government to test for a simultaneous increase in sovereign risk premia and decrease in sub-national risk premia-or a de facto transfer of risk from the latter to the former-on the day a sovereign bailout is announced. Using daily financial market data for Spain and its autonomous regions from January 2010 to June 2013, we find support for our risk transfer hypothesis. We estimate that the Spanish sovereign's spread may have increased by around 70 basis points as a result of the central government's support for fiscally distressed comunidades autónomas.

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a b s t r a c tWe test whether bear market risk, time variation in the probability of future bear market states, is priced. We construct an Arrow-Debreu security that pays off in bear market states (AD Bear) from traded Standard & Poor's (S&P) 500 index options and use its returns to measure bear market risk. We find that bear beta (exposure to bear market risk) has a strong relation with expected stock returns that is robust, persistent, and remains strong among liquid and large stocks. Historical bear beta also predicts future bear market risk exposure. We conclude that bear market risk is priced in the cross section of stock returns.