Skip to main content

Estimation of Continuous-Time Stochastic Volatility Models

  • Chapter
Palgrave Handbook of Econometrics

Abstract

This chapter reviews some of the key issues involved in estimating continuous-time stochastic volatility models. Such models have become popular recently because they provide a rich variety of alternative specifications which often lead to closed or semi-closed solutions in a variety of asset-pricing applications. An empirical comparison of various stochastic volatility models is also undertaken, along with a discussion of some directions for future research.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Aït-Sahalia, Y. (1999) Transition densities for interest rate and other nonlinear diffusions. Journal of Finance 54, 1361–95.

    Article  Google Scholar 

  • Aït-Sahalia, Y. (2002) Transition densities for interest rate and other nonlinear diffusions: a closed-form approximation approach. Econometrica 70, 223–62.

    Article  Google Scholar 

  • Aït-Sahalia, Y. (2007) Estimating continuous-time models using discretely sampled data. In R. Blundell, P. Torsten and W.K. Newey (eds.), Advances in Economics and Econometrics, Theory and Applications, Ninth World Congress. Cambridge: Cambridge University Press.

    Google Scholar 

  • Aït-Sahalia, Y. (2008) Closed-form likelihood expansions for multivariate diffusions. Annals of Statistics 36, 906–37.

    Article  Google Scholar 

  • Aït-Sahalia, Y. and R. Kimmel (2007) Maximum likelihood estimation of stochastic volatility models. Journal of Financial Economics 83, 413–52.

    Article  Google Scholar 

  • Aït-Sahalia, Y., P.A. Mykland and L. Zhang (2005) How often to sample a continuous-time process in the presence of market microstructure noise. Review of Financial Studies 18, 351–416.

    Article  Google Scholar 

  • Alizadeh, S., M.W. Brandt and F.X. Diebold (2002) Range-based estimation of stochastic volatility models. Journal of Finance 57, 1047–91.

    Article  Google Scholar 

  • Andersen, T.G., L. Benzoni and J. Lund (2002) An empirical investigation of continuous-time equity return models. Journal of Finance 57, 1239–84.

    Article  Google Scholar 

  • Andersen, T.G., T. Bollerslev, F.X. Diebold and P. Labys (2001) The distribution of exchange rate volatility. Journal of the American Statistical Association 96, 42–55.

    Article  Google Scholar 

  • Andersen, T.G., H. Chung and B.E. Sørensen (1999) Efficient method of moments estimation of a stochastic volatility model: a Monte Carlo study. Journal of Econometrics 91, 61–87.

    Article  Google Scholar 

  • Andersen, T.G. and J. Lund (1997) Estimating continuous-time stochastic volatility models of the short-term interest rate. Journal of Econometrics 77, 343–77.

    Article  Google Scholar 

  • Baillie, R.T. (2006) Modelling volatility. In T.C. Mills and K.D. Patterson (eds.), Palgrave Handbook of Econometrics. Volume 1: Econometric Theory, pp. 737–64. Basingstoke: Palgrave Macmillan.

    Google Scholar 

  • Bakshi, G., C. Cao and Z. Chen (1997) Empirical performance of alternative option pricing models. Journal of Finance 52, 2003–49.

    Article  Google Scholar 

  • Bakshi, G., N. Ju and H. Ou-Yang (2006) Estimation of continuous-time models with an application to equity volatility dynamics. Journal of Financial Economics 82, 227–49.

    Article  Google Scholar 

  • Bansal, R., A.R. Gallant, R. Hussey and G.E. Tauchen (1993) Computational aspects of nonparametric simulation estimation. In D.A. Belsley (ed.), Computational Techniques for Econometrics and Economic Analysis, pp. 3–22. Boston: Kluwer Academic Publishers.

    Google Scholar 

  • Bansal, R., A.R. Gallant, R. Hussey and G.E. Tauchen (1995) Nonparametric estimation of structural models for high-frequency currency market data. Journal of Econometrics 66, 251–87.

    Article  Google Scholar 

  • Barndorff-Nielsen, O.E. 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, 253–80.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Bates, D. (1996a) Testing option pricing models. In G.S. Maddala and C.R. Rao (eds.), Statistical Methods in Finance. Handbook of Statistics, Volume14, pp. 567–611. Amsterdam: Elsevier.

    Chapter  Google Scholar 

  • Bates, D. (1996b) 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-87 crash fears in S&P 500 futures options. Journal of Econometrics 94, 181–238.

    Article  Google Scholar 

  • Bates, D. (2006) Maximum likelihood estimation of latent affine processes. Review of Financial Studies 19, 909–65.

    Article  Google Scholar 

  • Black, F. (1976) Studies in stock price volatility changes. Proceedings of the 1976 Business Meeting of the Business and Economic Statistics Section, American Statistical Association, 177–81.

    Google Scholar 

  • Black, F., and M. Scholes (1973) The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–59.

    Article  Google Scholar 

  • Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–27.

    Article  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 

  • Bollerslev, T. and H. Zhou (2007) Expected stock returns and variance risk premia. Working Paper, Duke University.

    Google Scholar 

  • Brenner, M., E.Y. Ou and J.E. Zhang (2006) Hedging volatility risk. Journal of Banking and Finance 30, 811–21.

    Article  Google Scholar 

  • Broadie, M., M. Chernov and M. Johannes (2007) Model specification and risk premiums: the evidence from the futures options. Journal of Finance. Forthcoming.

    Google Scholar 

  • Carr, P. and L. Wu (2004) Time-changed Lévy processes and option pricing. Journal of Financial Economics 71, 113–41.

    Article  Google Scholar 

  • Carr, P. and L. Wu (2006) A tale of two indices. Journal of Derivatives 13, 13–29.

    Article  Google Scholar 

  • Carr, P. and L. Wu (2008) Variance risk premia. Review of Financial Studies. Forthcoming.

    Google Scholar 

  • Chacko, G. and L.M. Viceira (2003) Spectral GMM estimation of continuous-time processes. Journal of Econometrics 116, 259–92.

    Article  Google Scholar 

  • Chernov, M., A.R. Gallant, E. Ghysels and G. Tauchen (2003) Alternative models for stock price dynamics. Journal of Econometrics 116, 225–57.

    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 option valuation. Journal of Financial Economics 56, 407–58.

    Article  Google Scholar 

  • Chourdakis, K. (2002) Continuous-time regime switching models and applications in estimating processes with stochastic volatility and jumps. Working Paper 464, Queen Mary, University of London.

    Google Scholar 

  • Chourdakis, K. and G. Dotsis (2008) Maximum likelihood estimation and dynamic asset allocation with non-affine volatility processes. Working Paper, University of Essex.

    Google Scholar 

  • Christie, A. (1982) The stochastic behavior of common stock variances: value, leverage, and interest rate effects. Journal of Financial Economics 10, 407–32.

    Article  Google Scholar 

  • Christoffersen, P.F., K. Jacobs and K. Mimouni (2006) Models for S&P 500 dynamics: evidence from realized volatility, daily returns, and option prices. Working Paper, McGill University.

    Google Scholar 

  • Clark, P.K. (1973) A subordinated stochastic process model with finite variance for speculative prices. Econometrica 41, 135–56.

    Article  Google Scholar 

  • Detemple, J. and C. Osakwe (2000) The valuation of volatility options. European Finance Review 4, 21–50.

    Article  Google Scholar 

  • Dotsis, G., D. Psychoyios and G. Skiadopoulos (2007) An empirical comparison of continuous time models of implied volatility indices. Journal of Banking and Finance 31, 3584–603.

    Article  Google Scholar 

  • Duffie, D., J. Pan and K. Singleton (2000) Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–76.

    Article  Google Scholar 

  • Duffie, D. and K. Singleton (1993) Simulated moments estimation of Markov models of asset prices. Econometrica 61, 929–52.

    Article  Google Scholar 

  • Easley, D. and M. O’Hara (1992) Time and the process of security price adjustment. Journal of Finance 47, 577–605.

    Article  Google Scholar 

  • Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1006.

    Article  Google Scholar 

  • Eraker, B. (2004) Do stock prices and volatility jump? Reconciling evidence from spot and option prices. Journal of Finance 59, 1367–403.

    Article  Google Scholar 

  • Eraker, B., M. Johannes and N. Polson (2003) The impact of jumps in volatility and returns. Journal of Finance 53, 1269–300.

    Article  Google Scholar 

  • Fama, E.F. (1963) Mandelbrot and the stable Paretian distribution. Journal of Business 36, 420–9.

    Article  Google Scholar 

  • Fama, E.F. (1965) The behavior of stock market prices. Journal of Business 38, 34–105.

    Article  Google Scholar 

  • Feuerverger, A. (1990) An efficiency result for the empirical characteristic function in stationary time-series models. Canadian Journal of Statistics 18, 155–61.

    Article  Google Scholar 

  • Feuerverger, A. and P. McDunnough (1981a) On some Fourier methods for inference. Journal of the American Statistical Association 76, 379–87.

    Article  Google Scholar 

  • Feuerverger, A. and P. McDunnough (1981b) On the efficiency of empirical characteristic function procedures. Journal of the Royal Statistics Society, Series B 43, 20–7.

    Google Scholar 

  • Gallant, A.R., D.A. Hsieh and G.E. Tauchen (1997) Estimation of stochastic volatility models with diagnostics. Journal of Econometrics 81, 159–92.

    Article  Google Scholar 

  • Gallant, A.R. and G.E. Tauchen (1996) Which moments to match? Econometric Theory 12, 657–81.

    Article  Google Scholar 

  • Gallant, A.R. and G. Tauchen (1998) Reprojecting partially observed systems with application to interest rate diffusions. Journal of the American Statistical Association 93, 10–24.

    Article  Google Scholar 

  • Ghysels, E., A. Harvey and E. Renault (1996) Stochastic volatility. In G.S. Maddala and C.R. Rao (eds.), Statistical Methods in Finance. Handbook of Statistics, Volume 14, pp. 119–91. Amsterdam: Elsevier.

    Chapter  Google Scholar 

  • Glosten, L.R., R. Jagannathan and D. Runkle (1993) Relationship between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance 48, 1779–801.

    Article  Google Scholar 

  • Hansen, L.P. (1982) Large sample properties of generalized method of moments estimators. Econometrica 50, 1029–54.

    Article  Google Scholar 

  • Harvey, A., E. Ruiz and N. Shephard (1994) Multivariate stochastic variance models. Review of Economic Studies 61, 247–64.

    Article  Google Scholar 

  • Heston, S.L. (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6, 327–43.

    Article  Google Scholar 

  • Huang, S., Q. Liu and J. Yu (2007) Realized daily variance of S&P 500 cash index: a revaluation of stylized facts. Annals of Economics and Finance 8, 33–56.

    Google Scholar 

  • Hull, J.C. and A. White (1987) The pricing of options with stochastic volatility. Journal of Finance 42, 281–300.

    Article  Google Scholar 

  • Jacquier, E., N.G. Polson and P.E. Rossi (1994) Bayesian analysis of stochastic volatility models. Journal of Business and Economic Statistics 12, 371–89.

    Google Scholar 

  • Jiang, G.J. and J.L. Knight (2002) Estimation of continuous time processes via the empirical characteristic function. Journal of Business and Economic Statistics 20, 198–212.

    Article  Google Scholar 

  • Johannes, M. and N. Polson (2006) MCMC methods for financial econometrics. In Y. Aït-Sahalia and L. Hansen (eds.), Handbook of Financial Econometrics. Forthcoming.

    Google Scholar 

  • Johnson, H. and D. Shanno (1987) Option pricing when the variance is changing. Journal of Financial and Quantitative Analysis 22, 143–51.

    Article  Google Scholar 

  • Jones, C. (2003) The dynamics of stochastic volatility: evidence from underlying and options markets. Journal of Econometrics 116, 181–224.

    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, 361–93.

    Article  Google Scholar 

  • Lo, A.W. (1988) Maximum likelihood estimation of generalized Itô processes with discretely sampled data. Econometric Theory 4, 231–47.

    Article  Google Scholar 

  • Mandelbrot, B.B. (1963) The variation of certain speculative prices. Journal of Business 36, 394–416.

    Article  Google Scholar 

  • Melino, A. and S. Turnbull (1990) Pricing foreign currency options with stochastic volatility. Journal of Econometrics 45, 239–65.

    Article  Google Scholar 

  • Merton, R.C. (1973) Theory of rational option pricing. Bell Journal of Economics 4, 141–83.

    Article  Google Scholar 

  • Merton, R.C. (1976) Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3, 125–44.

    Article  Google Scholar 

  • Merton, R.C. (1980) On estimating the expected return on the market: an exploratory investigation. Journal of Financial Economics 8, 323–63.

    Article  Google Scholar 

  • Merton, R.C. (1990) Continuous-Time Finance. New York: Oxford University Press.

    Google Scholar 

  • Nelson, D.B. (1991) Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59, 347–70.

    Article  Google Scholar 

  • Pan, J. (2002) The jump-risk premia implicit in options: evidence from an integrated time-series study. Journal of Financial Economics 63, 3–50.

    Article  Google Scholar 

  • Pedersen, A. (1995a) A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scandinavian Journal of Statistics 22, 55–71.

    Google Scholar 

  • Pedersen, A. (1995b) Consistency and asymptotic normality of an approximate maximum likelihood estimator for discretely observed diffusion processes. Bernoulli 1, 257–79.

    Article  Google Scholar 

  • Psychoyios, D., G. Dotsis and R.N. Markellos (2007) Ajump diffusion model for VIX options and futures. Working Paper, Athens University of Economics and Business.

    Google Scholar 

  • Psychoyios, D., G. Skiadopoulos and P. Alexakis (2003) A review of stochastic volatility processes: properties and implications, Journal of Risk Finance 4(3), 43–60.

    Article  Google Scholar 

  • Santa-Clara, P. (1995) Simulated likelihood estimation of diffusions with an application to the short term interest rate. Ph.D. dissertation, INSEAD.

    Google Scholar 

  • Schwert, G.W. (1989) Why does market volatility change over time? Journal of Finance 44, 1115–53.

    Article  Google Scholar 

  • Scott, L. (1987) Option pricing when the variance changes randomly: theory, estimation, and an application. Journal of Financial and Quantitative Analysis 22, 419–38.

    Article  Google Scholar 

  • Singleton, K. (2001) Estimation of affine asset pricing models using the empirical characteristic function. Journal of Econometrics 102, 111–41.

    Article  Google Scholar 

  • Stein, E. and J. Stein (1991) Stock price distributions with stochastic volatility: an analytic approach. Review of Financial Studies 4, 727–52.

    Article  Google Scholar 

  • Sundaresan, S.M. (2000) Continuous time methods in finance: a review and an assessment. Journal of Finance 55, 1569–622.

    Article  Google Scholar 

  • Vasicek, O. (1977) An equilibrium characterization of the term structure. Journal of Financial Economics 5, 177–88.

    Article  Google Scholar 

  • Wiggins, J. (1987) Option values under stochastic volatility: theory and empirical estimates. Journal of Financial Economics 19, 351–72.

    Article  Google Scholar 

  • Zhang, L., P.A. Mykland and Y. Aït-Sahalia (2005) Edgeworth expansions for realized volatility and related estimators. Technical Report No. 556, University of Chicago, Department of Statistics.

    Book  Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Copyright information

© 2009 George Dotsis, Raphael N. Markellos and Terence C. Mills

About this chapter

Cite this chapter

Dotsis, G., Markellos, R.N., Mills, T.C. (2009). Estimation of Continuous-Time Stochastic Volatility Models. In: Mills, T.C., Patterson, K. (eds) Palgrave Handbook of Econometrics. Palgrave Macmillan, London. https://doi.org/10.1057/9780230244405_19

Download citation

Publish with us

Policies and ethics