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Stochastic Volatility: Univariate and Multivariate Extensions

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  • Éric Jacquier

    ()

  • Nicholas G. Polson
  • Peter E. Rossi

Abstract

Stochastic volatility models, aka SVOL, are more difficult to estimate than standard time-varying volatility models (ARCH). Advances in the literature now offer well tested estimators for a basic univariate SVOL model. However, the basic model is too restrictive for many economic and finance applications. The use of the basic model can lead to biased volatility forecasts especially around crucial periods of high volatility. We extend the basic SVOL needs to allow for the leverage effect, through a correlation between observable and variance errors, and fat-tails in the conditional distribution. We develop a Bayesian Markov Chain Monte Carlo algorithm for this extended model. We also provide an algorithm to analyze a multivariate factor SVOL model. The method simultaneously performs finite sample inference and smoothing. We document the performance of the estimator and show why the extensions are warranted. We provide the researcher with a range of model diagnostics, such as the identification of outliers for stochastic volatility models or the assessment of the normality of the conditional distribution. We implement this methodology on a number of univariate financial time series. There is strong evidence of (1) non-normal conditional distributions for most series, and (2) a leverage effect for stock returns. We illustrate the robustness of the results to the choice of the prior distributions. These results have policy implications on decisions based upon prediction of volatility, especially when dealing with tail prediction as in risk management. Les modèles de volatilité stochastique, alias SVOL, sont plus durs à estimer que les modèles traditionnels de type ARCH. La littérature récente offre des estimateurs éprouvés pour un modèle SVOL univarié de base. Ce modèle est trop contraignant pour une utilisation en économie financière. Les prévisions de volatilité qu'il produit peuvent etre biaisées, particulièrement quand la volatilité est élevée. Nous généralisons le modèle de base en y ajoutant des effets de levier par le biais d'une corrélation entre les chocs observables et de variance, et la possibilité de distributions conditionnelles à queues épaisses. Nous développons un algorithme bayésien à chaînes markoviennes de Monte Carlo. Nous développons aussi un algorithme pour l'analyse d'un modèle SVOL multivarié à facteurs. Ces estimateurs permettent une inférence en échantillon fini pour les paramètres et les volatilités. Nous documentons les performances de l'estimateur et montrons que les extensions sont nécessaires. Nous testons la normalité des distributions conditionnelles. Cette méthode est mise en oeuvre sur plusieurs séries financières. Il y a une forte évidence (1) de distributions conditionnelles à queues épaisses, et (2) d'effets de levier pour les actifs financiers. Les résultats sont robustes et ont d'importantes implications sur les décisions fondées sur les prédictions de volatilité, particulièrement pour la gestion de risques.

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Bibliographic Info

Paper provided by CIRANO in its series CIRANO Working Papers with number 99s-26.

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Date of creation: 01 Jul 1999
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Handle: RePEc:cir:cirwor:99s-26

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Keywords: Stochastic volatility; ARCH; MCMC algorithm; leverage effect; risk management; fat-tailed distributions; Volatilité stochastique; ARCH; algorithme MCMC; effets de levier; gestion de risque; distributions à queues épaisses;

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  1. Harvey, Andrew & Ruiz, Esther & Shephard, Neil, 1994. "Multivariate Stochastic Variance Models," Review of Economic Studies, Wiley Blackwell, vol. 61(2), pages 247-64, April.
  2. Eric Ghysels & Andrew Harvey & Éric Renault, 1995. "Stochastic Volatility," CIRANO Working Papers 95s-49, CIRANO.
  3. Jacquier, Eric & Polson, Nicholas G & Rossi, Peter E, 1994. "Bayesian Analysis of Stochastic Volatility Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 12(4), pages 371-89, October.
  4. John Geweke, 1992. "Priors for macroeconomic time series and their application," Discussion Paper / Institute for Empirical Macroeconomics 64, Federal Reserve Bank of Minneapolis.
  5. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
  6. Lawrence R. Glosten & Ravi Jagannathan & David E. Runkle, 1993. "On the relation between the expected value and the volatility of the nominal excess return on stocks," Staff Report 157, Federal Reserve Bank of Minneapolis.
  7. Andersen, Torben G. & Bollerslev, Tim & Christoffersen, Peter F. & Diebold, Francis X., 2005. "Volatility forecasting," CFS Working Paper Series 2005/08, Center for Financial Studies (CFS).
  8. Friedman, Moshe & Harris, Lawrence, 1998. "A Maximum Likelihood Approach for Non-Gaussian Stochastic Volatility Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 16(3), pages 284-91, July.
  9. Ronald Mahieu & Peter Schotman, 1994. "Stochastic volatility and the distribution of exchange rate news," Discussion Paper / Institute for Empirical Macroeconomics 96, Federal Reserve Bank of Minneapolis.
  10. repec:cup:etheor:v:10:y:1994:i:3-4:p:609-32 is not listed on IDEAS
  11. Sangjoon Kim & Neil Shephard, 1994. "Stochastic volatility: likelihood inference and comparison with ARCH models," Economics Papers 3., Economics Group, Nuffield College, University of Oxford.
  12. Melino, Angelo & Turnbull, Stuart M., 1990. "Pricing foreign currency options with stochastic volatility," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 239-265.
  13. John Geweke & Guofu Zhou, 1996. "Measuring the Pricing Error of the Arbitrage Pricing Theory," CEMA Working Papers 276, China Economics and Management Academy, Central University of Finance and Economics.
  14. Bollerslev, Tim & Chou, Ray Y. & Kroner, Kenneth F., 1992. "ARCH modeling in finance : A review of the theory and empirical evidence," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 5-59.
  15. Geweke, J, 1993. "Bayesian Treatment of the Independent Student- t Linear Model," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 8(S), pages S19-40, Suppl. De.
  16. John F. Geweke, 1994. "Bayesian comparison of econometric models," Working Papers 532, Federal Reserve Bank of Minneapolis.
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