Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T18:27:25.116Z Has data issue: false hasContentIssue false
  • English
  • Français

EFFICIENT ESTIMATION OF INTEGRATED VOLATILITY FUNCTIONALS UNDER GENERAL VOLATILITY DYNAMICS

Published online by Cambridge University Press:  17 July 2020

Jia Li  and
Yunxiao Liu
Jia Li*
Affiliation:
Duke University
Yunxiao Liu
Affiliation:
University of North Carolina at Chapel Hill
*
Address correspondence to Jia Li, Department of Economics, Duke University, Durham, NC 27708, USA; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

We provide an asymptotic theory for the estimation of a general class of smooth nonlinear integrated volatility functionals. Such functionals are broadly useful for measuring financial risk and estimating economic models using high-frequency transaction data. The theory is valid under general volatility dynamics, which accommodates both Itô semimartingales (e.g., jump-diffusions) and long-memory processes (e.g., fractional Brownian motions). We establish the semiparametric efficiency bound under a nonstandard nonergodic setting with infill asymptotics, and show that the proposed estimator attains this efficiency bound. These results on efficient estimation are further extended to a setting with irregularly sampled data.

Type
ARTICLES
Information
Econometric Theory , Volume 37 , Issue 4 , August 2021 , pp. 664 - 707
Copyright
© Cambridge University Press 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

We are grateful for comments from a co-editor and two anonymous referees, which have greatly improved the paper. We also thank Tim Bollerslev, Peter Hansen, Andrew Patton, Vladas Pipiras, George Tauchen, and Jean Jacod for helpful comments. J.L.’s research was partially supported by NSF Grant SES-1326819.

References

REFERENCES

Aït-Sahalia, Y. & Xiu, D. (2019) Principal component analysis of high-frequency data. Journal of the American Statistical Association 114(525), 287–303.CrossRefGoogle Scholar
Alòs, E. & Nualart, D. (2003) Stochastic integration with respect to the fractional Brownian motion. Stochastics and Stochastic Reports 75(3), 129–152.CrossRefGoogle Scholar
Andersen, T. & Bollerslev, T. (1997) Heterogeneous information arrivals and return volatility dynamics: Uncovering the long-run in high frequency returns. Journal of Finance 52(3), 975–1005.CrossRefGoogle Scholar
Andersen, T. & Bollerslev, T. (1998) Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review 39(4), 885–905.CrossRefGoogle Scholar
Andersen, T.G., Bollerslev, T., Diebold, F.X., & Labys, P. (2001) The distribution of realized exchange rate volatility. Journal of the American Statistical Association 96(453), 42–55.CrossRefGoogle Scholar
Andersen, T.G., Bollerslev, T., Diebold, F.X., & Labys, P. (2003) Modeling and forecasting realized volatility. Econometrica 71(2), 579–625.CrossRefGoogle Scholar
Baillie, R.T., Bollerslev, T., & Mikkelsen, H.O. (1996) Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 74(1), 3–30.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A., & Shephard, N. (2008) Designing realized kernels to measure the ex post variation of equity prices in the presence of noise. Econometrica 76(6), 1481–1536.Google Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2001) Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics Journal of the Royal Statistical Society, Series B 63(2), 167–241.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2002a) Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society, Series B 64(2), 253–280.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2002b) Estimating quadratic variation using realized variance. Journal of Applied Econometrics 17(5), 457–477.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2003) Realized power variation and stochastic volatility models. Bernoulli 9(2), 243–265.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2004a) Econometric analysis of realized covariation: High frequency based covariance, regression, and correlation in financial economics. Econometrica 72(3), 885–925.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2004b) Power and bipower variation with stochastic volatility and jumps. Journal of Financial Econometrics 2(1), 1–37.CrossRefGoogle Scholar
Bickel, P.J., Klaassen, C.A.J., Ritov, Y., & Wellner, J.A. (1998) Efficient and Adapative Estimation for Semiparametric Models. Springer-Verlag.Google Scholar
Bollerslev, T. & Todorov, V. (2011) Estimation of jump tails. Econometrica 79(6), 1727–1783.Google Scholar
Broadie, M., Chernov, M., & Johannes, M. (2007) Model specification and risk premia: Evidence from futures options. The Journal of Finance 62(3), 1453–1490.CrossRefGoogle Scholar
Clément, E., Delattre, S., & Gloter, A. (2013) An infinite dimensional convolution theorem with applications to the efficient estimation of the integrated volatility. Stochastic Processes and their Applications 123(7), 2500–2521.CrossRefGoogle Scholar
Comte, F. & Renault, E. (1996) Long memory continuous time models. Journal of Econometrics 73(1), 101–149.CrossRefGoogle Scholar
Comte, F. & Renault, E. (1998) Long memory in continuous-time stochastic volatility models. Mathematical Finance 8(4), 291–323.CrossRefGoogle Scholar
Delbaen, F. & Schachermayer, W. (1994) A general version of the fundamental theorem of asset pricing. Mathematische Annalen 300, 463–520.CrossRefGoogle Scholar
Ding, Z., Granger, C.W., & Engle, R.F. (1993) A long memory property of stock market returns and a new model. Journal of Empirical Finance 1(1), 83–106.CrossRefGoogle Scholar
Duffie, D., Pan, J., & Singleton, K.J. (2000) Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68(6), 1343–1376.CrossRefGoogle Scholar
Engle, R.F. (2004) Risk and volatility: Econometric models and financial practice. American Economic Review 94(3), 405–420.CrossRefGoogle Scholar
Eraker, B. (2004) Do stock prices and volatility jump? Reconciling evidence from spot and option prices. The Journal of Finance 59(3), 1367–1403.CrossRefGoogle Scholar
Eraker, B., Johannes, M., & Polson, N. (2003) The impact of jumps in volatility and returns. The Journal of Finance 58(3), 1269–1300.CrossRefGoogle Scholar
Foster, D.P. & Nelson, D.B. (1996) Continuous record asymptotics for rolling sample variance estimators. Econometrica 64(1), 139–174.CrossRefGoogle Scholar
Geweke, J. & Porter-Hudak, S. (1983) The estimation and application of long memory time series models. Journal of Time Series Analysis 4(4), 221–238.CrossRefGoogle Scholar
Granger, C. (1980) Long memory relationships and the aggregation of dynamic models. Journal of Econometrics 14(2), 227–238.CrossRefGoogle Scholar
Harrison, M. & Kreps, D. (1979) Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory 20(3), 381–408.CrossRefGoogle Scholar
Jacod, J. (2008) Asymptotic properties of realized power variations and related functionals of semimartingales. Stochastic Processes and their Applications 118(4), 517–559.CrossRefGoogle Scholar
Jacod, J., Li, Y., & Zheng, X. (2017) Statistical properties of microstructure noise. Econometrica 85(4), 1133–1174.CrossRefGoogle Scholar
Jacod, J., Li, Y., & Zheng, X. (2019) Estimating the integrated volatility with tick observations. Journal of Econometrics 208(1), 80–100.CrossRefGoogle Scholar
Jacod, J. & Protter, P. (2012) Discretization of Processes. Springer.CrossRefGoogle Scholar
Jacod, J. & Rosenbaum, M. (2013) Quarticity and other functionals of volatility: Efficient estimation. Annals of Statistics 41(3), 1462–1484.CrossRefGoogle Scholar
Jacod, J. & Rosenbaum, M. (2015) Estimation of volatility functionals: The case of a a square root n window. In Friz, P.K., Gatheral, J., Gulisashvili, A., Jacquier, A., & Teichmann, J. (eds.), Large Deviations and Asymptotic Methods in Finance, pp. 559–590. Springer International Publishing.CrossRefGoogle Scholar
Jacod, J. & Shiryaev, A.N. (2003) Limit Theorems for Stochastic Processes, 2nd edition. Springer-Verlag.CrossRefGoogle Scholar
Jeganathan, P. (1982) On the asymptotic theory of estimation when the limit of the log-likelihood is mixed normal. Sankhya: The Indian Journal of Statistics, Series A (1961–2002) 44(2), 173–212.Google Scholar
Jeganathan, P. (1983) Some asymptotic properties of risk functions when the limit of the experiment is mixed normal. Sankhya: The Indian Journal of Statistics, Series A (1961–2002) 45(1), 66–87.Google Scholar
Kalnina, I. & Xiu, D. (2017) Nonparametric estimation of the leverage effect: A trade-off between robustness and efficiency. Journal of the American Statistical Association 112(517), 384–396.CrossRefGoogle Scholar
Kristensen, D. (2010) Nonparametric filtering of the realized spot volatility: A kernel-based approach. Econometric Theory 26(1), 60–93.CrossRefGoogle Scholar
Li, J., Todorov, V., & Tauchen, G. (2016) Inference theory on volatility functional dependencies. Journal of Econometrics 193(1), 17–34.CrossRefGoogle Scholar
Li, J., Todorov, V., & Tauchen, G. (2017) Adaptive estimation of continuous-time regression models using high-frequency data. Journal of Econometrics 200(1), 36–47.CrossRefGoogle Scholar
Li, J. & Xiu, D. (2016) Generalized method of integrated moments for high-frequency data. Econometrica 84(4), 1613–1633.CrossRefGoogle Scholar
Mancini, C. (2001) Disentangling the jumps of the diffusion in a geometric jumping Brownian motion. Giornale dell’Istituto Italiano degli Attuari LXIV, 19–47.Google Scholar
McCloskey, A. & Perron, P. (2013) Memory parameter estimation in the presence of level shifts and deterministic trends. Econometric Theory 29(6), 1196–1237.CrossRefGoogle Scholar
Merton, R.C. (1980) On estimating the expected return on the market. Journal of Financial Economics 8(4), 323–361.CrossRefGoogle Scholar
Mykland, P. & Zhang, L. (2006) ANOVA for diffusions and Ito processes. Annals of Statistics 34(4), 1931–1963.CrossRefGoogle Scholar
Mykland, P. & Zhang, L. (2009) Inference for continuous semimartingales observed at high frequency. Econometrica 77(5), 1403–1445.Google Scholar
Nualart, D. (2005) The Malliavin Calculus and Related Topics, 2nd edition. Springer-Verlag.Google Scholar
Renault, E., Sarisoy, C., & Werker, B.J. (2017) Efficient estimation of integrated volatility and related processes. Econometric Theory 33(2), 439–478.CrossRefGoogle Scholar
Shepard, N. (2005) Stochastic Volatility: Selected Readings (Advanced Texts in Econometrics). Oxford University Press.Google Scholar
Singleton, K.J. (2006) Empirical Dynamic Asset Pricing: Model Specification and Econometric Assessment. Princeton University Press.Google Scholar
Stein, C. (1956) Efficient nonparametric testing and estimation. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, pp. 187–195. University of California Press.Google Scholar
Todorov, V. & Tauchen, G. (2012) The realized Laplace transform of volatility. Econometrica 80(3), 1105–1127.Google Scholar
Todorov, V., Tauchen, G., & Grynkiv, I. (2011) Realized Laplace transforms for estimation of jump diffusive volatility models. Journal of Econometrics 164(2), 367–381.CrossRefGoogle Scholar
van der Vaart, A. & Wellner, J. (1996) Weak Convergence and Empirical Processes. Springer-Verlag.CrossRefGoogle Scholar
Zhang, L. (2011) Estimating covariation: Epps effect and microstructure noise. Journal of Econometrics 160(1), 33–47.CrossRefGoogle Scholar
Zhang, L., Mykland, P.A., & Aït-Sahalia, Y. (2005) A tale of two time scales: Determining integrated volatility with noisy high-frequency data. Journal of the American Statistical Association 100(472), 1394–1411.CrossRefGoogle Scholar