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Stochastic Volatility and Time Deformation: An Application to Trading Volume and Leverage Effects

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  • Eric Ghysels
  • Joanna Jasiak

Abstract

In this paper, we study stochastic volatility models with time deformation. Such processes relate to early work by Mandelbrot and Taylor (1967), Clark (1973), Tauchen and Pitts (1983), among others. In our setup, the latent process of stochastic volatility evolves in a operational time which differs from calendar time. The time deformation can be determined by past volume of trade, past price changes, possibly with an asymmetric leverage effect, and other variables setting the pace of information arrival. The econometric specification exploits the state-space approach for stochastic volatility models proposed by Harvey, Ruiz and Shephard (1994) as well as matching moment estimation procedures using SNP densities of stock returns and trading volume estimated by Gallant, Rossi and Tauchen (1992). Daily data on the price changes and volume of trade of the S&P 500 over a 1950-1987 sample are investigated. Supporting evidence for a time deformation representation is found and its impact on the behaviour of price series and volume is analyzed. We find that increases in volume accelerate operational time, resulting in volatility being less persistent and subject to shocks with a higher innovation variance. Downward price movements have similar effects while upward price movements increase persistence in volatility and decrease the dispersion of shocks by slowing down the operational time clock. We present the basic model as well as several extensions, in particular, we formulate and estimate a bivariate return-volume stochastic volatility model with time deformation. The latter is examined through bivariate impulse response profiles following the example of Gallant, Rossi and Tauchen (1993). Nous proposons un modèle de volatilité stochastique avec déformation du temps suite aux travaux par Mandelbrot et Taylor (1967), Clark (1973), Tauchen et Pitts (1983) et autres. La volatilité est supposée être un processus qui évolue dans un temps déformé déterminé par l'arrivée de l'information sur le marché d'actifs financiers. Des séries telles que le volume de transaction et le rendement passé sont utilisées pour identifier la correspondance entre le temps calendrier et opérationnel. Le modèle est estimé soit par la procédure de pseudo maximum de vraisemblance comme proposé par Harvey et al. (1994), soit par des méthodes d'inférence indirecte utilisant la densité SNP de Gallant, Rossi et Tauchen (1992). Dans la partie empirique, nous utilisons des données journalières de la bourse de New York. Un modèle univarié de volatilité stochastique ainsi qu'un modèle bivarié de volume et rendements avec déformation du temps sont analysés.

Suggested Citation

  • Eric Ghysels & Joanna Jasiak, 1995. "Stochastic Volatility and Time Deformation: An Application to Trading Volume and Leverage Effects," CIRANO Working Papers 95s-31, CIRANO.
  • Handle: RePEc:cir:cirwor:95s-31
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    Citations

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    Cited by:

    1. Robert F. Engle & Jeffrey R. Russell, 1994. "Forecasting Transaction Rates: The Autoregressive Conditional Duration Model," NBER Working Papers 4966, National Bureau of Economic Research, Inc.
    2. Ghysels, E. & Gourieroux, C. & Jasiak, J., 1995. "Market Time and Asset Price Movements: Theory and Estimation," Cahiers de recherche 9536, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    3. Ming Liu & Harold H. Zhang, "undated". "Specification Tests in the Efficient Method of Moments Framework with Application to the Stochastic Volatility Models," Computing in Economics and Finance 1997 93, Society for Computational Economics.
    4. Alfonso Dufour & Robert F. Engle, 2000. "Time and the Price Impact of a Trade," Journal of Finance, American Finance Association, vol. 55(6), pages 2467-2498, December.
    5. Bossaerts, P. & Ghysels, E. & Gourieroux, C., 1996. "Arbitrage-Based Pricing when Volatility is Stochastic," Cahiers de recherche 9615, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    6. Carmen Broto & Esther Ruiz, 2004. "Estimation methods for stochastic volatility models: a survey," Journal of Economic Surveys, Wiley Blackwell, vol. 18(5), pages 613-649, December.
    7. Carrasco, Marine & Chernov, Mikhaël & Florens, Jean-Pierre & Ghysels, Eric, 2000. "Efficient Estimation of Jump Diffusions and General Dynamic Models with a Continuum of Moment Conditions," IDEI Working Papers 116, Institut d'Économie Industrielle (IDEI), Toulouse, revised 2002.
    8. Eric Ghysels & Joanna Jasiak, 1997. "GARCH for Irregularly Spaced Data: The ACD-GARCH Model," CIRANO Working Papers 97s-06, CIRANO.
    9. Eric Ghysels & Christian Gouriéroux & Joanna Jasiak, 1996. "Kernel Autocorrelogram for Time Deformed Processes," CIRANO Working Papers 96s-19, CIRANO.
    10. Tim Bollerslev & Eric Ghysels, 1994. "On Periodic Autogressive Conditional Heteroskedasticity," CIRANO Working Papers 94s-03, CIRANO.
    11. Yves Sprumont, 1998. "On the Game-Theoretic Structure of Public-Good Economies," International Journal of Game Theory, Springer;Game Theory Society, vol. 26(4), pages 455-472.
    12. Andersen, Torben G, 1996. " Return Volatility and Trading Volume: An Information Flow Interpretation of Stochastic Volatility," Journal of Finance, American Finance Association, vol. 51(1), pages 169-204, March.
    13. P. Gagliardini & C. Gourieroux, 2008. "Duration time-series models with proportional hazard," Journal of Time Series Analysis, Wiley Blackwell, vol. 29(1), pages 74-124, January.
    14. Barndorff-Nielsen, Ole E. & Shephard, Neil, 2006. "Impact of jumps on returns and realised variances: econometric analysis of time-deformed Levy processes," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 217-252.
    15. Griffin, J.E. & Steel, M.F.J., 2006. "Inference with non-Gaussian Ornstein-Uhlenbeck processes for stochastic volatility," Journal of Econometrics, Elsevier, vol. 134(2), pages 605-644, October.
    16. Eric Ghysels & Christian Gouriéroux & Joanna Jasiak, 1995. "Trading Patterns, Time Deformation and Stochastic Volatility in Foreign Exchange Markets," CIRANO Working Papers 95s-42, CIRANO.
    17. Eisler, Z. & Kertész, J., 2004. "Multifractal model of asset returns with leverage effect," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 343(C), pages 603-622.
    18. Zoltan Eisler & Janos Kertesz, 2004. "Multifractal model of asset returns with leverage effect," Papers cond-mat/0403767, arXiv.org, revised May 2004.

    More about this item

    Keywords

    Stochastic volatility; Trading volume; Volatilité stochastique ; Volume de transaction;

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General

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