Biblioteca de la Universidad Complutense de Madrid

Estimating the Leverage Parameter of Continuous-time Stochastic Volatility Models Using High Frequency S&P 500 and VIX


Ishida , Isao y McAleer, Michael y Oya, Kosuke (2011) Estimating the Leverage Parameter of Continuous-time Stochastic Volatility Models Using High Frequency S&P 500 and VIX. [ Documentos de trabajo del Instituto Complutense de Análisis Económico (ICAE); nº 17, 2011, ] (No publicado)

Vista previa
Creative Commons License
Esta obra está bajo una licencia de Creative Commons: Reconocimiento - No comercial.


URL Oficial:


This paper proposes a new method for estimating continuous-time stochastic volatility (SV) models for the S&P 500 stock index process using intraday high-frequency observations of both the S&P 500 index and the Chicago Board of Exchange (CBOE) implied (or expected) volatility index (VIX). Intraday high-frequency observations data have become readily available for an increasing number of financial assets and their derivatives in recent years, but it is well known that attempts to directly apply popular continuous-time models to short intraday time intervals, and estimate the parameters using such data, can lead to nonsensical estimates due to severe intraday seasonality. A primary purpose of the paper is to provide a fraework for using intraday high frequency data of both the index estimate, in particular, for improving the estimation accuracy of the leverage parameter, p, that is, the correlation between the two Brownian motions driving the diffusive components of the price process and its spot variance process, respectively. As a special case, we focus on Heston’s (1993) square-root SV model, and propose the realized leverage estimator for p, noting that, under this model without measurement errors, the “realized leverage,” or the realized covariation of the price and VIX processes divided by the product of the realized volatilities of the two processes, is in-fill consistent for p. Finite sample simulation results show that the proposed estimator delivers more accurate estimates of the leverage parameter than do existing methods.

Tipo de documento:Documento de trabajo o Informe técnico
Información Adicional:

JEL Classifications: G13, G17, G32.

The authors are most grateful to two referees for helpful comments and suggestions. The first author wishes to thank Yusho Kaguraoka, Toshiaki Watanabe, and participants at the 2010 Annual Meeting of the Nippon Finance Association, the CSFI Nakanoshima Workshop 2009, and the Hiroshima University of Economics Financial Econometrics Workshop 2010 for valuable comments, and the Japan Society for the Promotion of Science (Grants-in-Aid for Scientific Research No. 20530265) for financial support. The second author is most grateful for the financial support of the Australian Research Council, National Science Council, Taiwan, and the Japan Society for the Promotion of Science. The third author is thankful for Grants-in-Aid for Scientific Research No. 22243021 from the Japan Society for the Promotion of Science.

Palabras clave:Continuous time, High frequency data, Stochastic volatility, S&P 500, Implied volatility, VIX.
Materias:Ciencias Sociales > Economía > Econometría
Ciencias Sociales > Economía > Indicadores económicos
Título de serie o colección:Documentos de trabajo del Instituto Complutense de Análisis Económico (ICAE)
Código ID:12807

Aït-Sahalia, Y., and R. Kimmel, 2007, Maximum likelihood estimation of stochastic volatility models, Journal of Financial Economics 83, 413-452.

Andersen, T.G., L. Benzoni, and J. Lund, 2002, Estimating jump-diffusions for equity returns, Journal of Finance 57, 1239-84.

Andersen, T.G., and T. Bollerslev, 1997, Intraday periodicity and volatility persistence in financial markets, Journal of Empirical Finance 4, 115-158.

Bakshi, G., N. Ju, and H. Ou-Yang, 2006, Estimation of continuous-time models with an application to equity volatility dynamics, Journal of Financial Economics 82, 227-249.

Barndorff-Nielsen, O.E., S.E. Graversen, J. Jacod, and N. Shephard, 2006, Limit theorems for bipower variation in financial econometrics, Econometric Theory 22, 677-719.

Barndorff-Nielsen, O.E., and N. Shephard, 2004, Econometric analysis of realized covariation: High frequency based covariance, regression, and correlation in financial economics, Econometrica 72, 885-925.

Black, F., 1976, Studies in stock price volatility changes, Proceedings of the 1976 Meetings of the American Statistical Association, Business and Economics Statistics, 177-181.

Bollerslev, T., M. Gibson, and H. Zhou, 2011, Dynamic estimation of volatility risk premia and investor risk aversion from option-implied and realized volatilities, Journal of Econometrics 160, 235-245.

Bollerslev, T., J. Litvinova, and G. Tauchen, 2006, Leverage and volatility feedback effects in high-frequency data, Journal of Financial Econometrics 4, 353-384.

Bollerslev, T., and H. Zhou, 2002, Estimating stochastic volatility diffusion using conditional moments of integrated volatility, Journal of Econometrics 109, 33-65 (2004, Corrigendum, Journal of Econometrics 119, 221-222).

Britten-Jones, M., and A. Neuberger, 2000, Option prices, implied price processes, and stochastic volatility, Journal of Finance 55, 839-866.

CBOE, 2009, The CBOE volatility index - VIX, CBOE website.

Chernov, M., A.R. Gallant, E. Ghysels, and G. Tauchen, 2003, Alternative models for stock price dynamics, Journal of Econometrics 116, 225-257.

Corradi, V., and W. Distaso, 2006, Semiparametric comparison of stochastic volatility models using realised measures, Review of Economic Studies 73, 635-667.

Das, S.R., and R.K. Sundaram, 1999, Of smiles and smirks: A term structure perspective, Journal of Financial and Quantitative Analysis 34, 211-234.

Demeterfi, K., E. Derman, M. Kamal and J. Zhou, 1999, More than you ever wanted to know about volatility swaps, Goldman Sachs Quantitative Strategies Research Notes.

Dotsis, G., D. Psychoyios, and G. Skiadopoulos, 2007, An empirical comparison of continuous-time models of implied volatility indices, Journal of Banking and Finance 31, 3584-3603.

Duan, J.-C., and C.-Y. Yeh, 2010, Jump and volatility risk premiums implied by VIX, Journal of Economic Dynamics & Control 34, 2232-2244.

Engle, R.F., and I. Ishida, 2002, Modeling variance of variance: The square-root, the affine, and the CEV GARCH models, Working Paper, New York University.

Eraker, B., 2004, Do stock prices and volatility jump? Reconciling evidence from spot and option prices, Journal of Finance 59, 1367-1403.

Eraker, B., M. Johannes, and N. Polson, 2003, The role of jumps in returns and volatility, Journal of Finance 58, 1269-1300.

Fukasawa, M., I. Ishida, N. Maghrebi, K. Oya, M. Ubukata, and K. Yamazaki, 2010, Model-free implied volatility: From surface to index, forthcoming in International Journal of Theoretical and Applied Finance.

Garcia, R., M.-A. Lewis, S. Pastorello, and É. Renault, 2011, Estimation of objective and risk-neutral distributions based on moments of integrated volatility, Journal of Econometrics 160, 22-32.

Heston, S., 1993, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies 6, 327-343.

Jacod, J., and V. Todorov, 2010, Do price and volatility jump together, Annals of Applied Probability 20, 1425-1469.

Jiang, G., and Y. Tian, 2005, The model-free implied volatility and its information content, Review of Financial Studies 18, 1305-1342.

Jones, C., 2003, The dynamics of stochastic volatility: Evidence from underlying and options markets, Journal of Econometrics 116, 181-224.

Lee, S.S., and P.A. Mykland, 2008, Jumps in financial markets: A new nonparametric test and jump dynamics, Review of Financial Studies 21, 2535-2563.

Nelson, D.B., 1990, ARCH models as diffusion approximation, Journal of Econometrics 45, 7-38.

Pan, J., 2002, The jump-risk premia implicit in options: Evidence from an integrated time series, Journal of Financial Economics 63, 3-50.

Rydberg, T.H., and N. Shephard, 2003, Dynamics of trade-by-trade price movements: Decomposition and models, Journal of Financial Econometrics 1, 2-25.

Vortelinos, D.I., 2010, The properties of realized correlation: Evidence from the French, German and Greek equity markets, Quarterly Review of Economics and Finance 50, 273-290.

Depositado:01 Jun 2011 07:25
Última Modificación:05 Feb 2016 10:44

Sólo personal del repositorio: página de control del artículo