Bootstrapping Laplace Transforms of Volatility

Abstract

This paper studies inference for the realized Laplace transform (RLT) of volatility in a fixed-span setting using bootstrap methods. Specifically, since standard wild bootstraps provide inconsistent inference, we propose local Gaussian (LG) and modified wild (MW) bootstrap procedures, and establish their first-order asymptotic validity. Moreover, motivated by its superior finite sample performance in simulations, we use Edgeworth expansions to show that the LG inference achieves second-order asymptotic refinements. To further broaden the scope of the bootstraps, we provide new Laplace transform-based estimators of the spot variance as well as the covariance, correlation and beta between two semimartingales, and adapt our inference procedures to the requisite scenario. We establish central limit theory for our estimators and show first-order asymptotic validity of their associated bootstraps. Not surprisingly, a Monte Carlo study shows that the LG bootstrap outperforms the MW bootstrap and existing (first-order) feasible inference theory in finite samples. Moreover, it demonstrates that our new spot measure estimators and inference procedures are very accurate. Finally, we illustrate the use of the new methods by examining the volatility of, and the coherence between, stocks and bonds during the large equity sell-off in December 2018.