Tails, Fears, and Risk Premia




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    • Bollerslev is with Duke University and Todorov is with Kellogg School of Management, Northwestern University. We thank Yacine Aït-Sahalia, Torben G. Andersen, Snehal Banerjee, Robert Barro, Geert Bekaert, Jules van Binsbergen, Oleg Bondarenko, Mikhail Chernov, Rama Cont, Dobrislav Dobrev, Darrell Duffie (Acting Editor), Eric Engstrom, Jeremy Graveline, Friedrich Hubalek, Jean Jacod, Claudia Kluppelberg, Kai Li, Sydney Ludvigson, Lasse Pedersen, Mark Podolskij, Petra Posadel, Walter Schachermayer, George Tauchen, and Amir Yaron, our discussants Alexandra Dias, Xavier Gabaix, and Yuhang Xing, two anonymous referees, as well as seminar participants at EDHEC, the Federal Reserve Board, UIC, the June 2009 CREATES-Stevanovic Center conference in Skagen, the October 2009 SoFiE conference on “Liquidity, Credit Risk and Extreme Events” in Chicago, the January 2010 conference on the “Interplay between Financial and Insurance Mathematics, Statistics and Econometrics” at Wolfgang Pauli Institute, Vienna University, the 2010 Warwick Business School Conference on Derivatives, Volatility and Correlation, the June 2010 FERM symposium in Taipei, the 2010 WFA meetings, and the 2010 NBER Fall Asset Pricing meeting for their comments and helpful discussions related to the paper. We also thank OptionMetrics for providing us with the options data, Lai Xu for help with extracting and organizing the data, and Nicola Fusari for help with the option pricing codes used in the Monte Carlo simulations. The research was partly funded by NSF grant SES-0957330 to the NBER. Bollerslev also acknowledges the support of CREATES, funded by the Danish National Research Foundation.


We show that the compensation for rare events accounts for a large fraction of the average equity and variance risk premia. Exploiting the special structure of the jump tails and the pricing thereof, we identify and estimate a new Investor Fears index. The index reveals large time-varying compensation for fears of disasters. Our empirical investigations involve new extreme value theory approximations and high-frequency intraday data for estimating the expected jump tails under the statistical probability measure, and short maturity out-of-the-money options and new model-free implied variation measures for estimating the corresponding risk-neutral expectations.