The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Stochastic Volatility Models

  • Neil Shephard
Reference work entry
DOI: https://doi.org/10.1057/978-1-349-95189-5_2756

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.

Keywords

Asset pricing Black–Scholes–Merton prices Brownian motion Dambis–Dubins–Schwartz theorem Financial econometrics Generalized method of moments (GMM) estimators Kalman filter Markov chain Monto Carlo (MCMC) methods Markov processes Martingales Multivariate models Option pricing theory Options Probability Quadratic variation (QV) process Realized volatility Stochastic discount factor (SDF) approach Stochastic volatility 
This is a preview of subscription content, log in to check access

Notes

Acknowledgment

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

Bibliography

  1. Andersen, T., and T. Bollerslev. 1998. Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review 39: 885–905.CrossRefGoogle Scholar
  2. 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
  3. 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.CrossRefGoogle Scholar
  4. Bandi, F., and J. Russell. 2003. Microstructure noise, realized volatility, and optimal sampling. Mimeo: Graduate School of Business, University of Chicago.Google Scholar
  5. Barndorff-Nielsen, O. 2001. Superposition of Ornstein–Uhlenbeck type processes. Theory of Probability and Its Applications 45: 175–194.CrossRefGoogle Scholar
  6. 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.CrossRefGoogle Scholar
  7. 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.CrossRefGoogle Scholar
  8. 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.CrossRefGoogle Scholar
  9. 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.CrossRefGoogle Scholar
  10. 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
  11. 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.CrossRefGoogle Scholar
  12. Bates, D. 1996. Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options. Review of Financial Studies 9: 69–107.CrossRefGoogle Scholar
  13. Bates, D. 2000. Post-’97 crash fears in the S-&P 500 futures option market. Journal of Econometrics 94: 181–238.CrossRefGoogle Scholar
  14. Black, F. 1976. Studies of stock price volatility changes. Proceedings of the business and economic statistics section, American Statistical Association, 177–181.Google Scholar
  15. Black, F., and M. Scholes. 1972. The valuation of option contracts and a test of market efficiency. Journal of Finance 27: 399–418.CrossRefGoogle Scholar
  16. Black, F., and M. Scholes. 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81: 637–654.CrossRefGoogle Scholar
  17. 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
  18. Bollerslev, T., and H. Zhou. 2002. Estimating stochastic volatility diffusion using conditional moments of integrated volatility. Journal of Econometrics 109: 33–65.CrossRefGoogle Scholar
  19. 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.CrossRefGoogle Scholar
  20. 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.CrossRefGoogle Scholar
  21. Chernov, M., A. Gallant, E. Ghysels, and G. Tauchen. 2003. Alternative models of stock price dynamics. Journal of Econometrics 116: 225–257.CrossRefGoogle Scholar
  22. Clark, P. 1973. A subordinated stochastic process model with fixed variance for speculative prices. Econometrica 41: 135–156.CrossRefGoogle Scholar
  23. Cochrane, J. 2001. Asset pricing. Princeton: Princeton University Press.Google Scholar
  24. Comte, F., and E. Renault. 1998. Long memory in continuous-time stochastic volatility models. Mathematical Finance 8: 291–323.CrossRefGoogle Scholar
  25. Comte, F., L. Coutin, and E. Renault. 2003. Affine fractional stochastic volatility models. Mimeo: University of Montreal.Google Scholar
  26. Cont, R., and P. Tankov. 2004. Financial modelling with jump processes. London: Chapman and Hall.Google Scholar
  27. Das, S., and R. Sundaram. 1999. Of smiles and smirks: A term structure perspective. Journal of Financial and Quantitative Analysis 34: 211–240.CrossRefGoogle Scholar
  28. 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.CrossRefGoogle Scholar
  29. Doob, J. 1953. Stochastic processes. New York: Wiley.Google Scholar
  30. Duffie, D., J. Pan, and K. Singleton. 2000. Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68: 1343–1376.CrossRefGoogle Scholar
  31. Duffie, D., D. Filipovic, and W. Schachermayer. 2003. Affine processes and applications in finance. Annals of Applied Probability 13: 984–1053.CrossRefGoogle Scholar
  32. Elerian, O., S. Chib, and N. Shephard. 2001. Likelihood inference for discretely observed non-linear diffusions. Econometrica 69: 959–993.CrossRefGoogle Scholar
  33. Eraker, B. 2001. Markov chain Monte Carlo analysis of diffusion models with application to finance. Journal of Business and Economic Statistics 19: 177–191.CrossRefGoogle Scholar
  34. Eraker, B., M. Johannes, and N. Polson. 2003. The impact of jumps in returns and volatility. Journal of Finance 53: 1269–1300.CrossRefGoogle Scholar
  35. Fang, Y. 1996. Volatility modeling and estimation of high-frequency data with Gaussian noise. Ph.D. thesis. Sloan School of Management, MIT.Google Scholar
  36. Fiorentini, G., E. Sentana, and N. Shephard. 2004. Likelihood-based estimation of latent generalised ARCH structures. Econometrica 12: 1481–1517.CrossRefGoogle Scholar
  37. Gallant, A., and G. Tauchen. 1996. Which moments to match. Econometric Theory 12: 657–681.CrossRefGoogle Scholar
  38. 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
  39. Genon-Catalot, V., T. Jeantheau, and C. Larédo. 2000. Stochastic volatility as hidden Markov models and statistical applications. Bernoulli 6: 1051–1079.CrossRefGoogle Scholar
  40. Gourieroux, C., A. Monfort, and E. Renault. 1993. Indirect inference. Journal of Applied Econometrics 6: S85–S118.CrossRefGoogle Scholar
  41. Hansen, P., and A. Lunde. 2006. Realized variance and market microstructure noise (with discussion). Journal of Business and Economic Statistics 24: 127–161.CrossRefGoogle Scholar
  42. 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
  43. Harvey, A., E. Ruiz, and N. Shephard. 1994. Multivariate stochastic variance models. Review of Economic Studies 61: 247–264.CrossRefGoogle Scholar
  44. 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.CrossRefGoogle Scholar
  45. Hoffmann, M. 2002. Rate of convergence for parametric estimation in stochastic volatility models. Stochastic Processes and Their Application 97: 147–170.CrossRefGoogle Scholar
  46. Hull, J., and A. White. 1987. The pricing of options on assets with stochastic volatilities. Journal of Finance 42: 281–300.CrossRefGoogle Scholar
  47. 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
  48. 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
  49. Jacquier, E., N. Polson, and P. Rossi. 2004. Stochastic volatility models: Univariate and multivariate extensions. Journal of Econometrics 122: 185–212.CrossRefGoogle Scholar
  50. Johnson, H. 1979. Option pricing when the variance rate is changing. Working paper. University of California, Los Angeles.Google Scholar
  51. Johnson, H., and D. Shanno. 1987. Option pricing when the variance is changing. Journal of Financial and Quantitative Analysis 22: 143–151.CrossRefGoogle Scholar
  52. Kim, S., N. Shephard, and S. Chib. 1998. Stochastic volatility: Likelihood inference and comparison with ARCH models. Review of Economic Studies 65: 361–393.CrossRefGoogle Scholar
  53. King, M., E. Sentana, and S. Wadhwani. 1994. Volatility and links between national stock markets. Econometrica 62: 901–933.CrossRefGoogle Scholar
  54. Mandelbrot, B. 1963. The variation of certain speculative prices. Journal of Business 36: 394–419.CrossRefGoogle Scholar
  55. Meddahi, N. 2001. An eigenfunction approach for volatility modeling, Cahiers de recherche No. 2001–29. Montreal: Department of Economics, University of Montreal.Google Scholar
  56. Melino, A., and S. Turnbull. 1990. Pricing foreign currency options with stochastic volatility. Journal of Econometrics 45: 239–265.CrossRefGoogle Scholar
  57. Merton, R. 1980. On estimating the expected return on the market: An exploratory investigation. Journal of Financial Economics 8: 323–361.CrossRefGoogle Scholar
  58. Nelson, D. 1991. Conditional heteroskedasticity in asset pricing: A new approach. Econometrica 59: 347–370.CrossRefGoogle Scholar
  59. Nicolato, E., and E. Venardos. 2003. Option pricing in stochastic volatility models of the Ornstein–Uhlenbeck type. Mathematical Finance 13: 445–466.CrossRefGoogle Scholar
  60. Officer, R. 1973. The variability of the market factor of the New York stock exchange. Journal of Business 46: 434–453.CrossRefGoogle Scholar
  61. 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.CrossRefGoogle Scholar
  62. 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
  63. Renault, E., and N. Touzi. 1996. Option hedging and implied volatilities in a stochastic volatility model. Mathematical Finance 6: 279–302.CrossRefGoogle Scholar
  64. Roberts, G., and O. Stramer. 2001. On inference for nonlinear diffusion models using the Hastings–Metropolis algorithms. Biometrika 88: 603–621.CrossRefGoogle Scholar
  65. 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
  66. Shephard, N. 2005. Stochastic volatility: Selected readings. Oxford: Oxford University Press.Google Scholar
  67. Smith, A. 1993. Estimating nonlinear time series models using simulated vector autoregressions. Journal of Applied Econometrics 8: S63–S84.CrossRefGoogle Scholar
  68. Sørensen, M. 2000. Prediction based estimating equations. Econometrics Journal 3: 123–147.CrossRefGoogle Scholar
  69. Stein, E., and J. Stein. 1991. Stock price distributions with stochastic volatility: An analytic approach. Review of Financial Studies 4: 727–752.CrossRefGoogle Scholar
  70. 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
  71. Wiggins, J. 1987. Option values under stochastic volatilities. Journal of Financial Economics 19: 351–372.CrossRefGoogle Scholar
  72. Yu, J. 2005. On leverage in a stochastic volatility model. Journal of Econometrics 127: 165–178.CrossRefGoogle Scholar
  73. 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.CrossRefGoogle Scholar
  74. Zhou, B. 1996. High-frequency data and volatility in foreign-exchange rates. Journal of Business and Economic Statistics 14: 45–52.Google Scholar

Copyright information

© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Neil Shephard
    • 1
  1. 1.