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ESTIMATING THE VOLATILITY OCCUPATION TIME VIA REGULARIZED LAPLACE INVERSION

Published online by Cambridge University Press:  25 May 2015

Jia Li,
Viktor Todorov and
George Tauchen
Jia Li*
Affiliation:
Duke University
Viktor Todorov
Affiliation:
Northwestern University
George Tauchen
Affiliation:
Duke University
*
*Address correspondence to Jia Li, Department of Economics, Duke University, Durham, NC 27708; e-mail: jl410@duke.edu.
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Abstract

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We propose a consistent functional estimator for the occupation time of the spot variance of an asset price observed at discrete times on a finite interval with the mesh of the observation grid shrinking to zero. The asset price is modeled nonparametrically as a continuous-time Itô semimartingale with nonvanishing diffusion coefficient. The estimation procedure contains two steps. In the first step we estimate the Laplace transform of the volatility occupation time and, in the second step, we conduct a regularized Laplace inversion. Monte Carlo evidence suggests that the proposed estimator has good small-sample performance and in particular it is far better at estimating lower volatility quantiles and the volatility median than a direct estimator formed from the empirical cumulative distribution function of local spot volatility estimates. An empirical application shows the use of the developed techniques for nonparametric analysis of variation of volatility.

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Type
ARTICLES
Information
Econometric Theory , Volume 32 , Issue 5 , October 2016, pp. 1253 - 1288
Copyright
Copyright © Cambridge University Press 2015 

References

REFERENCES

Aït-Sahalia, Y. & Jacod, J. (2009) Estimating the degree of activity of jumps in high frequency financial data. Annals of Statistics 37, 2202–2244.CrossRefGoogle Scholar
Andersen, T. G., Bollerslev, T., Christoffersen, P. F., & Diebold, F. X. (2013) Financial risk measurement for financial risk management. In Constanides, G., Harris, M., and Stulz, R. (Eds.), Handbook of the Economics of Finance, Vol.II. Elsevier Science B.V.Google Scholar
Barndorff-Nielsen, O. & Shephard, N. (2004) Power and Bipower Variation with Stochastic Volatility and Jumps. Journal of Financial Econometrics 2, 1–37.CrossRefGoogle Scholar
Barndorff-Nielsen, O. & Shephard, N. (2006) Econometrics of Testing for Jumps in Financial Economics using Bipower Variation. Journal of Financial Econometrics 4, 1–30.CrossRefGoogle Scholar
Black, F. (1976) Studies of stock price volatility changes. Proceedings of the Business and Economics Section of the American Statistical Association, 177–181.Google Scholar
Bollerslev, T. & Todorov, V. (2011) Estimation of Jump Tails. Economtrica 79, 1727–17783.Google Scholar
Carrasco, M. & Florens, J.-P. (2011) A Spectral Method for Deconvolving a Density. Econometric Theory 27(3), pp. 546–581.CrossRefGoogle Scholar
Carrasco, M., Florens, J.-P., & Renault, E. (2007) Linear inverse problems in structural econometrics estimation based on spectral decomposition and regularization. In Handbook of Econometrics, Volume 6B. Elsevier.Google Scholar
Chernozhukov, V., Fernandez-Val, I., & Galichon, A. (2010) Quantile and Probability Curves without Crossing. Econometrica 78, 1093–1125.Google Scholar
Comte, F. & Renault, E. (1996) Long memory continuous time models. Journal of Econometrics 73, 101–149.Google Scholar
Comte, F. & Renault, E. (1998) Long memory in continuous-time stochastic volatility models. Mathematical Finance 8, 291–323.Google Scholar
Cont, R. & da Fonseca, J. (2002) Dynamics of implied volatility surfaces. Quantitative Finance 2(1), 45–60.Google Scholar
Eisenbaum, N. & Kaspi, H. (2007) On the Continuity of Local Times of Borel Right Markov Processes. Annals of Probability 35, 915–934.Google Scholar
Foster, D. & Nelson, D. B. (1996) Continuous record asymptotics for rolling sample variance estimators. Econometrica 64, 139–174.Google Scholar
Geman, D. & Horowitz, J. (1980) Occupation Densities. Annals of Probability 8, 1–67.Google Scholar
Härdle, W. & Linton, O. (1994) Applied nonparametric methods. In Engle, R. F. and McFadden, D. L. (Eds.), Handbook of Econometrics, Volume 4, Chapter 38. Elsevier.Google Scholar
Hardy, G., Littlewood, J., & Polya, G. (1952) Inequalities. Cambridge University Press.Google Scholar
Ibragimov, I. & Has’minskii, R. (1981) Statistical Estimation: Asymptotic Theory. Berlin: Springer.CrossRefGoogle Scholar
Jacod, J. & Protter, P. (2012) Discretization of Processes. Springer.CrossRefGoogle Scholar
Jacod, J. & Reiß, M. (2014) A remark on the rates of convergence for integrated volatility estimation in the presence of jumps. Annals of Statistics, 42(3), 1131–1144.Google Scholar
Jacod, J. & Rosenbaum, M. (2013) Quarticity and Other Functionals of Volatility: Efficient Estimation. Annals of Statistics 41, 1462–1484.CrossRefGoogle Scholar
Kristensen, D. (2010) Nonparametric filtering of the realized spot volatility: A kernel-based approach. Econometric Theory 26(1), 60–93.Google Scholar
Kryzhniy, V. (2003a) Direct Regularization of the Inversion of Real-Valued Laplace Transforms. Inverse Problems 19, 573–583.CrossRefGoogle Scholar
Kryzhniy, V. (2003b) Regularized Inversion of Integral Transformations of Mellin Convolution Type. Inverse Problems 19, 1227–1240.CrossRefGoogle Scholar
Ledoux, M. & Talagrand, M. (1991) Probability in Banach Spaces. Springer.Google Scholar
Li, J., Todorov, V., & Tauchen, G. (2013) Volatility occupation times. Annals of Statistics 41, 1865–1891.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
Mancini, C. (2009) Non-parametric Threshold Estimation for Models with Stochastic Diffusion Coefficient and Jumps. Scandinavian Journal of Statistics 36, 270–296.CrossRefGoogle Scholar
Marcus, M. B. & Rosen, J. (2006) Markov Processes, Gaussian Processes, and Local Times. Cambridge University Press.CrossRefGoogle Scholar
Protter, P. (2004) Stochastic Integration and Differential Equations, 2nd ed. Springer-Verlag.Google Scholar
Renault, E., Sarisoy, C., & Werker, B. J. (2014) Efficient estimation of integrated volatility and related processes. Technical report, Brown University.CrossRefGoogle Scholar
Sato, K. (1999) Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Tikhonov, A. & Arsenin, V. (1977) Solutions of Ill-Posed Problems. Winston.Google Scholar
Todorov, V. & Tauchen, G. (2012a) Inverse Realized Laplace Transforms for Nonparametric Volatility Density Estimation in Jump-Diffusions. Journal of the American Statistical Association 107, 622–635.Google Scholar
Todorov, V. & Tauchen, G. (2012b) The Realized Laplace Transform of Volatility. Economtrica 80, 1105–1127.Google Scholar
van der Vaart, A. W. (1998) Asymptotic Statistics. Cambridge University Press.Google Scholar