Nonparametric Estimation of the Leverage Effect: A Trade-Off Between Robustness and Efficiency

ABSTRACT

We consider two new approaches to nonparametric estimation of the leverage effect. The first approach uses stock prices alone. The second approach uses the data on stock prices as well as a certain volatility instrument, such as the Chicago Board Options Exchange (CBOE) volatility index (VIX) or the Black–Scholes implied volatility. The theoretical justification for the instrument-based estimator relies on a certain invariance property, which can be exploited when high-frequency data are available. The price-only estimator is more robust since it is valid under weaker assumptions. However, in the presence of a valid volatility instrument, the price-only estimator is inefficient as the instrument-based estimator has a faster rate of convergence.We consider an empirical application, in which we study the relationship between the leverage effect and the debt-to-equity ratio, credit risk, and illiquidity. Supplementary materials for this article are available online.

KEYWORDS:

• Derivatives
• High-frequency data
• Implied volatility
• Spot correlation
• VIX

Supplementary Materials

Appendix A contains the proof of Theorem 1 of the main text. Appendix B contains the proof of Theorem 2 of the main text. Appendix C contains the proof of Lemma A2, which is stated in Appendix A. Appendix D contains the proof of Lemma A3, which is stated in Appendix A.

Acknowledgment

We benefited much from discussions with Torben Andersen, Tim Bollerslev, Oleg Bondarenko, Stéphane Bonhomme, Francis Diebold, Kirill Evdokimov, Christian Hansen, Zhiguo He, Jean Jacod, Bryan Kelly, Jia Li, Yingying Li, Per Mykland, Benoit Perron, Mark Podolskij, Jeffrey Russell, Olivier Scaillet, Kevin Sheppard, George Tauchen, and Dan Wang. We thank seminar participants at Boston University, Duke University, Purdue University, Princeton University, Queen's University, University of Chicago, University of Pennsylvania, and University of Rochester, as well as participants of the 11th International Symposium on Econometric Theory and Applications, Frontiers in Financial Econometrics at Hitotsubashi University, Financial Econometrics Conference at the Toulouse School of Economics, Econometric Society European Meeting 2014, the SOFIE 2014 conference, the CIREQ Econometrics Conference 2014, the High-Frequency Data and High-Frequency Trading Conference at Chicago, the Econometric Study Group 2013 conference at Bristol, the 9th GNYEC, the First Conference in Econometric Theory at Universidad de San Andrés, the 5th Annual Modeling High-Frequency Data in Finance Conference, and the 11th World Congress of the Econometric Society.