Skip to main content
SpringerLink
Log in

Stochastic Volatility Models

  • Reference work entry
  • First Online:
The New Palgrave Dictionary of Economics

  • 41 Accesses

Abstract

Stochastic volatility (SV) is the main concept used in the fields of financial economics and mathematical finance to deal with the endemic time-varying volatility and codependence found in financial markets. Here I trace the origins of SV and provide links with the basic models used today in the literature. I briefly discuss some of the innovations in the second generation of SV models and discuss the literature on conducting inference for SV models. I talk about the use of SV to price options, and consider the connection of SV with realized volatility.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 6,499.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 8,499.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Bibliography

  • Andersen, T., and T. Bollerslev. 1998. Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review 39: 885–905.

    Article  Google Scholar 

  • Andersen, T.G., and B. Sørensen. 1996. GMM estimation of a stochastic volatility model: A Monte Carlo study. Journal of Business and Economic Statistics 14: 328–352.

    Google Scholar 

  • Andersen, T., T. Bollerslev, F. Diebold, and P. Labys. 2001. The distribution of exchange rate volatility. Journal of the American Statistical Association 96: 42–55. Correction published in vol. 98 (2003), p. 501.

    Article  Google Scholar 

  • Bandi, F., and J. Russell. 2003. Microstructure noise, realized volatility, and optimal sampling. Mimeo: Graduate School of Business, University of Chicago.

    Google Scholar 

  • Barndorff-Nielsen, O. 2001. Superposition of Ornstein–Uhlenbeck type processes. Theory of Probability and Its Applications 45: 175–194.

    Article  Google Scholar 

  • Barndorff-Nielsen, O., and N. Shephard. 2001. Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics (with discussion). Journal of the Royal Statistical Society, Series B 63: 167–241.

    Article  Google Scholar 

  • Barndorff-Nielsen, O., and N. Shephard. 2002. Econometric analysis of realised volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society, Series B 64: 253–280.

    Article  Google Scholar 

  • Barndorff-Nielsen, O., and N. Shephard. 2004a. Econometric analysis of realised covariation: High frequency covariance, regression and correlation in financial economics. Econometrica 72: 885–925.

    Article  Google Scholar 

  • Barndorff-Nielsen, O., and N. Shephard. 2004b. Power and bipower variation with stochastic volatility and jumps (with discussion). Journal of Financial Econometrics 2: 1–48.

    Article  Google Scholar 

  • Barndorff-Nielsen, O., and N. Shephard. 2006a. Variation, jumps, and high frequency data in financial econometrics. In Advances in economics and econometrics: Theory and applications, vol. 1, ed. R. Blundell, P. Torsten, and W. Newey. Cambridge: Cambridge University Press.

    Google Scholar 

  • Barndorff-Nielsen, O., and N. Shephard. 2006b. Econometrics of testing for jumps in financial economics using bipower variation. Journal of Financial Econometrics 4: 1–30.

    Article  Google Scholar 

  • Bates, D. 1996. Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options. Review of Financial Studies 9: 69–107.

    Article  Google Scholar 

  • Bates, D. 2000. Post-’97 crash fears in the S-&P 500 futures option market. Journal of Econometrics 94: 181–238.

    Article  Google Scholar 

  • Black, F. 1976. Studies of stock price volatility changes. Proceedings of the business and economic statistics section, American Statistical Association, 177–181.

    Google Scholar 

  • Black, F., and M. Scholes. 1972. The valuation of option contracts and a test of market efficiency. Journal of Finance 27: 399–418.

    Article  Google Scholar 

  • Black, F., and M. Scholes. 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81: 637–654.

    Article  Google Scholar 

  • Bochner, S. 1949. Diffusion equation and stochastic processes. Proceedings of the National Academy of Science of the United States of America 85: 369–370.

    Google Scholar 

  • Bollerslev, T., and H. Zhou. 2002. Estimating stochastic volatility diffusion using conditional moments of integrated volatility. Journal of Econometrics 109: 33–65.

    Article  Google Scholar 

  • Breidt, F., N. Crato, and P. de Lima. 1998. On the detection and estimation of long memory in stochastic volatility. Journal of Econometrics 83: 325–348.

    Article  Google Scholar 

  • Chernov, M., and E. Ghysels. 2000. A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation. Journal of Financial Economics 56: 407–458.

    Article  Google Scholar 

  • Chernov, M., A. Gallant, E. Ghysels, and G. Tauchen. 2003. Alternative models of stock price dynamics. Journal of Econometrics 116: 225–257.

    Article  Google Scholar 

  • Clark, P. 1973. A subordinated stochastic process model with fixed variance for speculative prices. Econometrica 41: 135–156.

    Article  Google Scholar 

  • Cochrane, J. 2001. Asset pricing. Princeton: Princeton University Press.

    Google Scholar 

  • Comte, F., and E. Renault. 1998. Long memory in continuous-time stochastic volatility models. Mathematical Finance 8: 291–323.

    Article  Google Scholar 

  • Comte, F., L. Coutin, and E. Renault. 2003. Affine fractional stochastic volatility models. Mimeo: University of Montreal.

    Google Scholar 

  • Cont, R., and P. Tankov. 2004. Financial modelling with jump processes. London: Chapman and Hall.

    Google Scholar 

  • Das, S., and R. Sundaram. 1999. Of smiles and smirks: A term structure perspective. Journal of Financial and Quantitative Analysis 34: 211–240.

    Article  Google Scholar 

  • Diebold, F., and M. Nerlove. 1989. The dynamics of exchange rate volatility: A multivariate latent factor ARCH model. Journal of Applied Econometrics 4: 1–21.

    Article  Google Scholar 

  • Doob, J. 1953. Stochastic processes. New York: Wiley.

    Google Scholar 

  • Duffie, D., J. Pan, and K. Singleton. 2000. Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68: 1343–1376.

    Article  Google Scholar 

  • Duffie, D., D. Filipovic, and W. Schachermayer. 2003. Affine processes and applications in finance. Annals of Applied Probability 13: 984–1053.

    Article  Google Scholar 

  • Elerian, O., S. Chib, and N. Shephard. 2001. Likelihood inference for discretely observed non-linear diffusions. Econometrica 69: 959–993.

    Article  Google Scholar 

  • Eraker, B. 2001. Markov chain Monte Carlo analysis of diffusion models with application to finance. Journal of Business and Economic Statistics 19: 177–191.

    Article  Google Scholar 

  • Eraker, B., M. Johannes, and N. Polson. 2003. The impact of jumps in returns and volatility. Journal of Finance 53: 1269–1300.

    Article  Google Scholar 

  • Fang, Y. 1996. Volatility modeling and estimation of high-frequency data with Gaussian noise. Ph.D. thesis. Sloan School of Management, MIT.

    Google Scholar 

  • Fiorentini, G., E. Sentana, and N. Shephard. 2004. Likelihood-based estimation of latent generalised ARCH structures. Econometrica 12: 1481–1517.

    Article  Google Scholar 

  • Gallant, A., and G. Tauchen. 1996. Which moments to match. Econometric Theory 12: 657–681.

    Article  Google Scholar 

  • Garcia, R., E. Ghysels, and E. Renault. 2006. The econometrics of option pricing. In Handbook of financial econometrics, ed. Y. Aït-Sahalia and L. Hansen. Amsterdam: North-Holland.

    Google Scholar 

  • Genon-Catalot, V., T. Jeantheau, and C. Larédo. 2000. Stochastic volatility as hidden Markov models and statistical applications. Bernoulli 6: 1051–1079.

    Article  Google Scholar 

  • Gourieroux, C., A. Monfort, and E. Renault. 1993. Indirect inference. Journal of Applied Econometrics 6: S85–S118.

    Article  Google Scholar 

  • Hansen, P., and A. Lunde. 2006. Realized variance and market microstructure noise (with discussion). Journal of Business and Economic Statistics 24: 127–161.

    Article  Google Scholar 

  • Harvey, A. 1998. Long memory in stochastic volatility. In Forecasting volatility in financial markets, ed. J. Knight and S. Satchell. Oxford: Butterworth-Heinemann.

    Google Scholar 

  • Harvey, A., E. Ruiz, and N. Shephard. 1994. Multivariate stochastic variance models. Review of Economic Studies 61: 247–264.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Hoffmann, M. 2002. Rate of convergence for parametric estimation in stochastic volatility models. Stochastic Processes and Their Application 97: 147–170.

    Article  Google Scholar 

  • Hull, J., and A. White. 1987. The pricing of options on assets with stochastic volatilities. Journal of Finance 42: 281–300.

    Article  Google Scholar 

  • Jacod, J. 1994. Limit of random measures associated with the increments of a Brownian semimartingale. Preprint No. 120. Paris: Laboratoire de Probabilitiés, Université Pierre et Marie Curie.

    Google Scholar 

  • Jacquier, E., N. Polson, and P. Rossi. 1994. Bayesian analysis of stochastic volatility models (with discussion). Journal of Business and Economic Statistics 12: 371–417.

    Google Scholar 

  • Jacquier, E., N. Polson, and P. Rossi. 2004. Stochastic volatility models: Univariate and multivariate extensions. Journal of Econometrics 122: 185–212.

    Article  Google Scholar 

  • Johnson, H. 1979. Option pricing when the variance rate is changing. Working paper. University of California, Los Angeles.

    Google Scholar 

  • Johnson, H., and D. Shanno. 1987. Option pricing when the variance is changing. Journal of Financial and Quantitative Analysis 22: 143–151.

    Article  Google Scholar 

  • Kim, S., N. Shephard, and S. Chib. 1998. Stochastic volatility: Likelihood inference and comparison with ARCH models. Review of Economic Studies 65: 361–393.

    Article  Google Scholar 

  • King, M., E. Sentana, and S. Wadhwani. 1994. Volatility and links between national stock markets. Econometrica 62: 901–933.

    Article  Google Scholar 

  • Mandelbrot, B. 1963. The variation of certain speculative prices. Journal of Business 36: 394–419.

    Article  Google Scholar 

  • Meddahi, N. 2001. An eigenfunction approach for volatility modeling, Cahiers de recherche No. 2001–29. Montreal: Department of Economics, University of Montreal.

    Google Scholar 

  • Melino, A., and S. Turnbull. 1990. Pricing foreign currency options with stochastic volatility. Journal of Econometrics 45: 239–265.

    Article  Google Scholar 

  • Merton, R. 1980. On estimating the expected return on the market: An exploratory investigation. Journal of Financial Economics 8: 323–361.

    Article  Google Scholar 

  • Nelson, D. 1991. Conditional heteroskedasticity in asset pricing: A new approach. Econometrica 59: 347–370.

    Article  Google Scholar 

  • Nicolato, E., and E. Venardos. 2003. Option pricing in stochastic volatility models of the Ornstein–Uhlenbeck type. Mathematical Finance 13: 445–466.

    Article  Google Scholar 

  • Officer, R. 1973. The variability of the market factor of the New York stock exchange. Journal of Business 46: 434–453.

    Article  Google Scholar 

  • Pastorello, S., V. Patilea, and E. Renault. 2003. Iterative and recursive estimation in structural non-adaptive models. Journal of Business and Economic Statistics 21: 449–509.

    Article  Google Scholar 

  • Phillips, P., and J. Yu. 2005. A two-stage realized volatility approach to the estimation for diffusion processes from discrete observations. Discussion paper no. 1523. Cowles Foundation, Yale University.

    Google Scholar 

  • Renault, E., and N. Touzi. 1996. Option hedging and implied volatilities in a stochastic volatility model. Mathematical Finance 6: 279–302.

    Article  Google Scholar 

  • Roberts, G., and O. Stramer. 2001. On inference for nonlinear diffusion models using the Hastings–Metropolis algorithms. Biometrika 88: 603–621.

    Article  Google Scholar 

  • Rosenberg, B. 1972. The behaviour of random variables with nonstationary variance and the distribution of security prices. Working paper 11, Graduate School of Business Administration, University of California, Berkeley. Reprinted in N. Shephard (2005).

    Google Scholar 

  • Shephard, N. 2005. Stochastic volatility: Selected readings. Oxford: Oxford University Press.

    Google Scholar 

  • Smith, A. 1993. Estimating nonlinear time series models using simulated vector autoregressions. Journal of Applied Econometrics 8: S63–S84.

    Article  Google Scholar 

  • Sørensen, M. 2000. Prediction based estimating equations. Econometrics Journal 3: 123–147.

    Article  Google Scholar 

  • Stein, E., and J. Stein. 1991. Stock price distributions with stochastic volatility: An analytic approach. Review of Financial Studies 4: 727–752.

    Article  Google Scholar 

  • Taylor, S. 1982. Financial returns modelled by the product of two stochastic processes – A study of daily sugar prices 1961–79. In Time series analysis: Theory and practice, vol. 1, ed. O. Anderson. Amsterdam: North-Holland.

    Google Scholar 

  • Wiggins, J. 1987. Option values under stochastic volatilities. Journal of Financial Economics 19: 351–372.

    Article  Google Scholar 

  • Yu, J. 2005. On leverage in a stochastic volatility model. Journal of Econometrics 127: 165–178.

    Article  Google Scholar 

  • Zhang, L., P. Mykland, and Y. Aït-Sahalia. 2005. A tale of two time scales: Determining integrated volatility with noisy high-frequency data. Journal of the American Statistical Association 100: 1394–1411.

    Article  Google Scholar 

  • Zhou, B. 1996. High-frequency data and volatility in foreign-exchange rates. Journal of Business and Economic Statistics 14: 45–52.

    Google Scholar 

Download references

Acknowledgment

My research is supported by the Economic and Social Science Research Council (UK) through the grant ‘High frequency financial econometrics based upon power variation’.

Author information

Authors and Affiliations

  1. http://link.springer.com/referencework/10.1057/978-1-349-95121-5

    Neil Shephard

Authors
  1. Neil Shephard
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Copyright information

© 2018 Macmillan Publishers Ltd.

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Shephard, N. (2018). Stochastic Volatility Models. In: The New Palgrave Dictionary of Economics. Palgrave Macmillan, London. https://doi.org/10.1057/978-1-349-95189-5_2756

Download citation

Publish with us

Policies and ethics

Search

Navigation