Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-01-13T22:57:18.421Z Has data issue: false hasContentIssue false
  • English
  • Français

ESTIMATION OF VOLATILITY FUNCTIONS IN JUMP DIFFUSIONS USING TRUNCATED BIPOWER INCREMENTS

Published online by Cambridge University Press:  08 October 2020

Jihyun Kim ,
Joon Y. Park  and
Bin Wang
Jihyun Kim*
Affiliation:
Toulouse School of Economics University of Toulouse Capitole
Joon Y. Park
Affiliation:
Indiana University Sungkyunkwan University
Bin Wang
Affiliation:
Harbin Institute of Technology, Shenzhen
*
Address correspondence to Jihyun Kim, Toulouse School of Economics, Toulouse 31000, France; e-mail: jihyun.kim@tse-fr.eu.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article, we introduce and analyze a new methodology to estimate the volatility functions of jump diffusion models. Our methodology relies on the standard kernel estimation technique using truncated bipower increments. The relevant asymptotics are fully developed, allowing for the time span to increase as well as the sampling interval to decrease, and accommodate both stationary and nonstationary recurrent processes. We evaluate the performance of our estimators by simulation and provide some illustrative empirical analyses.

Type
ARTICLES
Information
Econometric Theory , Volume 37 , Issue 5 , October 2021 , pp. 926 - 958
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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 thank the Editor Peter C.B. Phillips, the Co-Editor Dennis Kristensen, and three anonymous referees for many helpful comments. For useful discussions, we are also grateful to Yacine Aït-Sahalia, Yoosoon Chang, Nour Meddahi, Mathieu Rosenbaum, Roberto Renò, Jun Yu, and the participants at 2015 Toulouse Financial Econometrics Conference, 2016 Princeton-QUT-SJTU-SMU Econometrics Conference and 2017 Asian Meeting of Econometric Society. Jihyun Kim is grateful to the French Government and the ANR for support under the Investissements d’Avenir program, grant ANR-17-EURE-0010. Bin Wang gratefully acknowledges financial support provided to him by Shenzhen Key Research Base of Humanities and Social Sciences.

References

REFERENCES

Aït-Sahalia, Y. (2002) Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach. Econometrica 70, 223–262.CrossRefGoogle Scholar
Aït-Sahalia, Y. & Jacod, J. (2009) Testing for jumps in a discretely observed process. Annals of Statistics 37, 184–222.CrossRefGoogle Scholar
Aït-Sahalia, Y. & Mykland, P.A. (2003) The effects of random and discrete sampling when estimating continuous-time diffusions. Econometrica 71, 483–549.CrossRefGoogle Scholar
Aït-Sahalia, Y. & Mykland, P.A. (2004) Estimators of diffusions with randomly spaced discrete observations, A general theory. Annals of Statistics 32, 2186–2222.CrossRefGoogle Scholar
Aït-Sahalia, Y. & Park, J.Y. (2016) Bandwidth selection & asymptotic properties of local nonparametric estimators in possibly nonstationary continuous-time models. Journal of Econometrics 192, 119–138.CrossRefGoogle Scholar
Andersen, T.G., Benzoni, L., & Lund, J. (2004) Stochastic volatility, mean drift, and jumps in the short-term interest rate. Working Paper, Northwestern University.Google Scholar
Andersen, T.G., Dobrev, D., & Schaumburg, E. (2012) Jump-robust volatility estimation using nearest neighbor truncation. Journal of Econometrics 169, 75–93.CrossRefGoogle Scholar
Bandi, F.M. & Nguyen, T.H. (2003) On the functional estimation of jump-diffusion models. Journal of Econometrics 116, 293–328.CrossRefGoogle Scholar
Bandi, F.M. & Phillips, P.C.B. (2003) Fully nonparametric estimation of scalar diffusion models. Econometrica 71, 241–283.CrossRefGoogle Scholar
Bandi, F.M. & Renò, R. (2018) Nonparametric stochastic volatility. Econometric Theory 34, 1207–1255.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E., Graversen, S.E., Jacod, J., Podolskij, M., & Shephard, N. (2005) A central limit theorem for realised power and bipower variations of continuous semimartingales. In Kabanov, Y. & Lipster, R., (eds.), From Stochastic Calculus to Mathematical Finance, Festschrift for Albert Shiryaev, pp. 33–68. Springer.Google Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2003) Realized power variation and stochastic volatility. Bernoulli 9, 243–265.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2004) Power and bipower variation with stochastic volatility and jumps. Journal of Financial Econometrics 2, 1–37.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2006) Econometrics of testing for jumps in financial economics using bipower variation. Journal of Financial Econometrics 4, 1–30.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E., Shephard, N., & Winkel, M. (2006) Limit theorems for multipower variation in the presence of jumps. Stochastic Processes and their Applications 5, 796–806.CrossRefGoogle Scholar
Bollerslev, T. & Todorov, V. (2011) Estimation of jump tails. Econometrica 79, 1727–1783.Google Scholar
Bosq, D. (1998) Nonparametric Statistics for Stochastic Processes. Springer.CrossRefGoogle Scholar
Boudt, K., Croux, C., & Laurent, S. (2011) Outlyingness weighted covariation. Journal of Financial Econometrics 9, 657–684.CrossRefGoogle Scholar
Chang, J. & Chen, S.X. (2011) On the approximate maximum likelihood estimation for diffusion processes. Annals of Statistics 39, 2820–2851.CrossRefGoogle Scholar
Chen, S.X., Gao, J., & Tang, C.Y. (2008) A test for model specification of diffusion processes. Annals of Statistics 36, 167–198.CrossRefGoogle Scholar
Christensen, K., Oomen, R., & Podolskij, M. (2010) Realised quantile-based estimation of the integrated variance. Journal of Econometrics 159, 74–98.CrossRefGoogle Scholar
Corsi, F., Pirino, D., & Renò, R. (2010) Threshold bipower variation and the impact of jumps on volatility forecasting. Journal of Econometrics 159, 276–288.CrossRefGoogle Scholar
Fan, J. & Zhang, C. (2003) A reexamination of diffusion estimators with applications to financial model validation. Journal of the American Statistical Association 98, 118–134.CrossRefGoogle Scholar
Höpfner, R. & Löcherbach, E. (2003) Limit Theorems for Null Recurrent Markov Processes. American Mathematical Society.CrossRefGoogle Scholar
Jacod, J. (2008) Asymptotic properties of realized power variations and related functionals of semimartingales. Stochastic Processes and their Applications 118, 517–559.CrossRefGoogle Scholar
Jacod, J. (2012) Statistics and high frequency data. In Kessler, M., Lindner, A., & Sorensen, M. (eds.), Statistical Methods for Stochastic Differential Equations, pp. 191–310. Chapman & Hall.Google Scholar
Jacod, J. & Rosenbaum, M. (2013) Quarticity and other functionals of volatility: Efficient estimation. Annals of Statistics 41, 1462–1484.CrossRefGoogle Scholar
Jacod, J. & Shiryaev, A.N. (2003) Limit Theorems for Stochastic Processes. Springer.CrossRefGoogle Scholar
Jeong, M. & Park, J.Y. (2016) An asymptotic theory of jump diffusion model. Working Paper, Indiana University.Google Scholar
Kanaya, S. (2016) Uniform convergence rates of kernel-based nonparametric estimators for continuous time diffusion processes: A damping function approach. Econometric Theory 32, 1–41.Google Scholar
Kanaya, S. & Kristensen, D. (2016) Estimation of stochastic volatility models by nonparametric filtering. Econometric Theory 32, 861–916.CrossRefGoogle Scholar
Kim, J. & Park, J.Y. (2017a) Asymptotics for recurrent diffusions with application to high frequency regression. Journal of Econometrics 196, 27–54.CrossRefGoogle Scholar
Kim, J. & Park, J.Y. (2017b) Unit root, mean reversion and nonstationarity in Financial Time Series. Working Paper, Indiana University.Google Scholar
Kristensen, D. (2010) Nonparametric filtering of the realised spot volatility: A kernel-based approach. Econometric Theory 26, 60–93.CrossRefGoogle Scholar
Mancini, C. (2009) Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps. Scandinavian Journal of Statistics 36, 270–296.CrossRefGoogle Scholar
Mancini, C. & Renò, R. (2011) Threshold estimation of Markov models with jumps and interest rate modeling. Journal of Econometrics 160, 77–92.CrossRefGoogle Scholar
Park, J.Y. & Wang, B. (2018) Nonparametric estimation of jump diffusion models. Working Paper, Indiana University.Google Scholar
Protter, P.E. (2005) Stochastic Integration and Differential Equations. Springer.CrossRefGoogle Scholar
Renò, R. (2006) Nonparametric estimation of stochastic volatility models. Economics Letters 90, 390–395.CrossRefGoogle Scholar
Renò, R. (2008) Nonparametric estimation of the diffusion coefficient of stochastic volatility models. Econometric Theory 24, 1174–1206.CrossRefGoogle Scholar
Revuz, D. & Yor, M. (1998) Continuous Martingales and Brownian Motion. Springer.Google Scholar
Touati, A. (1987) Théorèmes limites pour des processus de Markov récurrents. Comptes Rendus de I’Académie des Sciences-Series I-Mathematics 305, 841–844.Google Scholar
Ueltzhöfer, F.A.J. (2013) On non-parametric estimation of the Lévy kernel of Markov processes. Stochastic Processes and their Applications 123, 3663–3709.CrossRefGoogle Scholar
Yu, C., Fang, Y., Li, Z., Zhang, B., & Zhao, X. (2014) Non-parametric estimation of high-frequency spot volatility for Brownian semimartingale with jumps. Journal of Time Series Analysis 35, 572–591.CrossRefGoogle Scholar
Supplementary material: PDF

Kim et al. Supplementary Materials

Kim et al. Supplementary Materials

Download Kim et al. Supplementary Materials(PDF)
PDF 250.6 KB