We propose a new and flexible nonparametric framework for estimating the jump tails of Itô semimartingale processes. The approach is based on a relatively simple-to-implement set of estimating equations associated with the compensator for the jump measure, or its intensity, that only utilizes the weak assumption of regular variation in the jump tails, along with in-fill asymptotic arguments for directly estimating the “large” jumps. The procedure assumes that the large-sized jumps are identically distributed, but otherwise allows for very general dynamic dependencies in jump occurrences, and, importantly, does not restrict the behavior of the “small” jumps or the continuous part of the process and the temporal variation in the stochastic volatility. On implementing the new estimation procedure with actual high-frequency data for the S&P 500 aggregate market portfolio, we find strong evidence for richer and more complex dynamic dependencies in the jump tails than hitherto entertained in the literature.

ABSTRACT 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.