This article examines the leverage effect, or the generally negative covariation between asset returns and their changes in volatility, under a general setup that allows the log-price and volatility processes to be Itô semimartingales. We decompose the leverage effect into continuous and discontinuous parts and develop statistical methods to estimate them. We establish the asymptotic properties of these estimators. We also extend our methods and results (for the continuous leverage) to the situation where there is market microstructure noise in the observed returns. We show in Monte Carlo simulations that our estimators have good finite sample performance. When applying our methods to real data, our empirical results provide convincing evidence of the presence of the two leverage effects, especially the discontinuous one. Supplementary materials for this article are available online.
The online supplementary materials contain the appendices for the article.
Yacine Aït-Sahalia’s work was supported in part by the NSF under grant SES-0850533. Jianqing Fan’s work was supported in part by the NSF under grants DMS-1206464 and DMS-1406266. 3Roger J. A. Laeven’s work was supported in part by the NWO under grant VIDI-2009. 4Christina Dan Wang’s work was supported in part by the NIH under grant R01-GM100474 and by a grant from the Bendheim Center for Finance.