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.So what are policymakers to do? First and foremost, reduce uncertainty. Do so by removing tail risks, and the perception of tail risks.-Olivier Blanchard, chief economist, IMF, The Economist, January 31, 2009.LACK OF INVESTOR CONFIDENCE is frequently singled out as one of the main culprits behind the massive losses in market values in the advent of the Fall 2008 financial crises. At the time, the portrayal in the financial news media of different doomsday scenarios in which the stock market would have declined even further were also quite commonplace. Motivated by these observations, we provide a new theoretical framework for better understanding the way in * Bollerslev is with Duke University and Todorov is with Kellogg School of Management, Northwestern University. We thank Yacine Aït-Sahalia, Torben

JEL classification: C13 C14 G10 G12 Keywords:Variance risk premium Time-varying jump tails Market sentiment and fears Return predictability a b s t r a c tThe variance risk premium, defined as the difference between the actual and risk-neutral expectations of the forward aggregate market variation, helps predict future market returns. Relying on a new essentially model-free estimation procedure, we show that much of this predictability may be attributed to time variation in the part of the variance risk premium associated with the special compensation demanded by investors for bearing jump tail risk, consistent with the idea that market fears play an important role in understanding the return predictability.

We consider a bivariate process Xt = (X 1 t , X 2 t ), which is observed on a finite time interval [0, T ] at discrete times 0, ∆n, 2∆n, . . . . Assuming that its two components X 1 and X 2 have jumps on [0, T ], we derive tests to decide whether they have at least one jump occurring at the same time ("common jumps") or not ("disjoint jumps"). There are two different tests for the two possible null hypotheses (common jumps or disjoint jumps). Those tests have a prescribed asymptotic level, as the mesh ∆n goes to 0. We show on some simulations that these tests perform reasonably well even in the finite sample case, and we also put them in use for some exchange rates data.

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We study the dynamic relation between aggregate stock market risks and risk premia via an exploration of the time series of equity-index option surfaces. The analysis is based on estimating a general parametric asset pricing model for the risk-neutral equity market dynamics using a panel of options on the S&P 500 index, while remaining fully nonparametric about the actual evolution of market risks. We find that the risk-neutral jump intensity, which controls the pricing of left tail risk, cannot be spanned by the market volatility (and its components), so an additional factor is required to account for its dynamics. This tail factor has no incremental predictive power for future equity return volatility or jumps beyond what is captured by the current and past level of volatility. In contrast, the novel factor is critical in predicting the future market excess returns over horizons up to one year, and it explains a large fraction of the future variance risk premium. We contrast our findings with those implied by structural asset pricing models that seek to rationalize the predictive power of option data. Relative to those studies, our findings suggest a wider wedge between the dynamics of equity market risks and the corresponding risk premia with the latter typically displaying a far more persistent reaction following market crises.

We propose new nonparametric estimators of the integrated volatility of an Itô semimartingale observed at discrete times on a fixed time interval with mesh of the observation grid shrinking to zero. The proposed estimators achieve the optimal rate and variance of estimating integrated volatility even in the presence of infinite variation jumps when the latter are stochastic integrals with respect to locally "stable" Lévy processes, that is, processes whose Lévy measure around zero behaves like that of a stable process. On a first step, we estimate locally volatility from the empirical characteristic function of the increments of the process over blocks of shrinking length and then we sum these estimates to form initial estimators of the integrated volatility. The estimators contain bias when jumps of infinite variation are present, and on a second step we estimate and remove this bias by using integrated volatility estimators formed from the empirical characteristic function of the high-frequency increments for different values of its argument. The second step debiased estimators achieve efficiency and we derive a feasible central limit theorem for them. t 0 γ s− dY s and − t 0 γ s− dY s have the same law,

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