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Stochastic Volatility: Univariate and Multivariate Extensions

  • Eric Jacquier

    ()

    (Boston College)

  • Nicholas G. Polson

    ()

    (University of Chicago)

  • Peter Rossi

    ()

    (University of Chicago)

Discrete time stochastic volatility models (hereafter SVOL) are noticeably more difficult to estimate than the successful ARCH family of models. In this paper we demonstrate efficient estimation and prediction for a number of univariate and multivariate SVOL models. Namely, we model fat-tailed and skewed conditional distributions, correlated errors distributions (leverage effect), and two multivariate models, a stochastic factor-structure model and a stochastic discount dynamic model. These extensions to the basic model are needed if one wants, for example, to compare SVOL models with ARCH-style models or to implement option pricing and portfolio selection under stochastic volatility. We specify the models as a hierarchy of conditional probability distributions: Pr(data | volatilities), Pr(volatilities | parameters) and Pr(parameters). This conceptually simple methodology provides a natural environment for the construction of stochastic volatility models that depart from standard distributional assumptions. Given a model and the data, inference and prediction are based on the joint posterior distribution of the volatilities and the parameters that we simulate via Markov chain Monte Carlo (MCMC) methods. Our approach also provides a sensitivity analysis for parameter inference and an outlier diagnostic. We estimate the model for several financial time series and find that the extensions considered are indeed needed. For the SVOL model we find strong evidence of non-normal conditional distributions for stock returns and exchange rates. We also find evidence of correlated errors for stock returns.

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Paper provided by Society for Computational Economics in its series Computing in Economics and Finance 1999 with number 112.

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Date of creation: 01 Mar 1999
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Handle: RePEc:sce:scecf9:112
Contact details of provider: Postal: CEF99, Boston College, Department of Economics, Chestnut Hill MA 02467 USA
Fax: +1-617-552-2308
Web page: http://fmwww.bc.edu/CEF99/

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  1. 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.
  2. 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.
  3. 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).
  4. Jacquier, Eric & Polson, Nicholas G & Rossi, Peter E, 2002. "Bayesian Analysis of Stochastic Volatility Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(1), pages 69-87, January.
  5. repec:cup:etheor:v:10:y:1994:i:3-4:p:609-32 is not listed on IDEAS
  6. Ghysels, E. & Harvey, A. & Renault, E., 1996. "Stochastic Volatility," Cahiers de recherche 9613, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
  7. John Geweke & Guofu Zhou, 1995. "Measuring the pricing error of the arbitrage pricing theory," Staff Report 189, Federal Reserve Bank of Minneapolis.
  8. Glosten, Lawrence R & Jagannathan, Ravi & Runkle, David E, 1993. " On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks," Journal of Finance, American Finance Association, vol. 48(5), pages 1779-1801, December.
  9. Geweke, John, 1994. "Priors for Macroeconomic Time Series and Their Application," Econometric Theory, Cambridge University Press, vol. 10(3-4), pages 609-632, August.
  10. Harvey, Andrew & Ruiz, Esther & Shephard, Neil, 1994. "Multivariate Stochastic Variance Models," Review of Economic Studies, Wiley Blackwell, vol. 61(2), pages 247-64, April.
  11. John F. Geweke, 1994. "Bayesian comparison of econometric models," Working Papers 532, Federal Reserve Bank of Minneapolis.
  12. Sangjoon Kim & Neil Shephard, 1994. "Stochastic volatility: likelihood inference and comparison with ARCH models," Economics Papers 3., Economics Group, Nuffield College, University of Oxford.
  13. 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.
  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. Melino, Angelo & Turnbull, Stuart M., 1990. "Pricing foreign currency options with stochastic volatility," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 239-265.
  16. 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.
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