Inverse Realized Laplace Transforms for Nonparametric Volatility Density Estimation in Jump-Diffusions

Abstract

This article develops a nonparametric estimator of the stochastic volatility density of a discretely observed Itô semimartingale in the setting of an increasing time span and finer mesh of the observation grid. There are two basic steps involved. The first step is aggregating the high-frequency increments into the realized Laplace transform, which is a robust nonparametric estimate of the underlying volatility Laplace transform. The second step is using a regularized kernel to invert the realized Laplace transform. These two steps are relatively quick and easy to compute, so the nonparametric estimator is practicable. The article also derives bounds for the mean squared error of the estimator. The regularity conditions are sufficiently general to cover empirically important cases such as level jumps and possible dependencies between volatility moves and either diffusive or jump moves in the semimartingale. The Monte Carlo analysis in this study indicates that the nonparametric estimator is reliable and reasonably accurate in realistic estimation contexts. An empirical application to 5-min data for three large-cap stocks, 1997–2010, reveals the importance of big short-term volatility spikes in generating high levels of stock price variability over and above those induced by price jumps. The application also shows how to trace out the dynamic response of the volatility density to both positive and negative jumps in the stock price.