Estimation of Jump Tails

We propose a new and flexible non-parametric framework for estimating the jump tails of semi martingale 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, nor 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.