IDEAS home Printed from
MyIDEAS: Login to save this paper or follow this series

Maximum Likelihood Estimation of Stochastic Volatility Models

  • Yacine Ait-Sahalia
  • Robert Kimmel

We develop and implement a new method for maximum likelihood estimation in closed-form of stochastic volatility models. Using Monte Carlo simulations, we compare a full likelihood procedure, where an option price is inverted into the unobservable volatility state, to an approximate likelihood procedure where the volatility state is replaced by the implied volatility of a short dated at-the-money option. We find that the approximation results in a negligible loss of accuracy. We apply this method to market prices of index options for several stochastic volatility models, and compare the characteristics of the estimated models. The evidence for a general CEV model, which nests both the affine model of Heston (1993) and a GARCH model, suggests that the elasticity of variance of volatility lies between that assumed by the two nested models.

If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.

File URL:
Download Restriction: no

Paper provided by National Bureau of Economic Research, Inc in its series NBER Working Papers with number 10579.

in new window

Date of creation: Jun 2004
Date of revision:
Publication status: published as Ait-Sahalia, Yacine and Robert Kimmel. "Maximum Likelihood Estimation of Stochastic Volatility Models." Journal of Financial Economics 83, 2 (February 2007): 413-52.
Handle: RePEc:nbr:nberwo:10579
Note: AP
Contact details of provider: Postal: National Bureau of Economic Research, 1050 Massachusetts Avenue Cambridge, MA 02138, U.S.A.
Phone: 617-868-3900
Web page:

More information through EDIRC

References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:

as in new window
  1. Ledoit, Olivier & Santa-Clara, Pedro & Yan, Shu, 2002. "Relative Pricing of Options with Stochastic Volatility," University of California at Los Angeles, Anderson Graduate School of Management qt7jp8f42t, Anderson Graduate School of Management, UCLA.
  2. 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.
  3. Yacine Ait-Sahalia, 2002. "Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-form Approximation Approach," Econometrica, Econometric Society, vol. 70(1), pages 223-262, January.
  4. Harvey, Andrew C & Shephard, Neil, 1996. "Estimation of an Asymmetric Stochastic Volatility Model for Asset Returns," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(4), pages 429-34, October.
  5. Gallant, A. Ronald & Tauchen, George, 1996. "Which Moments to Match?," Econometric Theory, Cambridge University Press, vol. 12(04), pages 657-681, October.
  6. Jones, Christopher S., 2003. "The dynamics of stochastic volatility: evidence from underlying and options markets," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 181-224.
  7. Singleton, Kenneth J., 2001. "Estimation of affine asset pricing models using the empirical characteristic function," Journal of Econometrics, Elsevier, vol. 102(1), pages 111-141, May.
  8. Ruiz, Esther, 1994. "Quasi-maximum likelihood estimation of stochastic volatility models," Journal of Econometrics, Elsevier, vol. 63(1), pages 289-306, July.
  9. Jiang, George J & Knight, John L, 2002. "Estimation of Continuous-Time Processes via the Empirical Characteristic Function," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(2), pages 198-212, April.
  10. Bates, David S., 2000. "Post-'87 crash fears in the S&P 500 futures option market," Journal of Econometrics, Elsevier, vol. 94(1-2), pages 181-238.
  11. Torben G. Andersen & Tim Bollerslev & Nour Meddahi, 2002. "Analytic Evaluation of Volatility Forecasts," CIRANO Working Papers 2002s-90, CIRANO.
  12. Neil Shephard, 2005. "Stochastic Volatility," Economics Papers 2005-W17, Economics Group, Nuffield College, University of Oxford.
  13. Nour Meddahi, 2001. "An Eigenfunction Approach for Volatility Modeling," CIRANO Working Papers 2001s-70, CIRANO.
  14. Wiggins, James B., 1987. "Option values under stochastic volatility: Theory and empirical estimates," Journal of Financial Economics, Elsevier, vol. 19(2), pages 351-372, December.
  15. Stephen J. Taylor, 1994. "Modeling Stochastic Volatility: A Review And Comparative Study," Mathematical Finance, Wiley Blackwell, vol. 4(2), pages 183-204.
  16. Stein, Jeremy, 1989. " Overreactions in the Options Market," Journal of Finance, American Finance Association, vol. 44(4), pages 1011-23, September.
  17. 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.
  18. Tim Bollerslev & Hao Zhou, 2001. "Estimating stochastic volatility diffusion using conditional moments of integrated volatility," Finance and Economics Discussion Series 2001-49, Board of Governors of the Federal Reserve System (U.S.).
  19. Chesney, Marc & Scott, Louis, 1989. "Pricing European Currency Options: A Comparison of the Modified Black-Scholes Model and a Random Variance Model," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 24(03), pages 267-284, September.
  20. Yacine Aït-Sahalia & Andrew W. Lo, 1998. "Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices," Journal of Finance, American Finance Association, vol. 53(2), pages 499-547, 04.
  21. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
  22. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-52.
  23. Yacine Aït-Sahalia, 1999. "Transition Densities for Interest Rate and Other Nonlinear Diffusions," Journal of Finance, American Finance Association, vol. 54(4), pages 1361-1395, 08.
  24. Torben G. Andersen & Luca Benzoni & Jesper Lund, 2001. "An Empirical Investigation of Continuous-Time Equity Return Models," NBER Working Papers 8510, National Bureau of Economic Research, Inc.
  25. Charles Quanwei Cao & Gurdip S. Bakshi & Zhiwu Chen, 1999. "Do Call Prices and the Underlying Stock Always Move in the Same Direction?," Yale School of Management Working Papers ysm125, Yale School of Management.
  26. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-43.
  27. Chacko, George & Viceira, Luis M., 2003. "Spectral GMM estimation of continuous-time processes," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 259-292.
  28. Yacine Ait--Sahalia & Per A. Mykland, 2003. "The Effects of Random and Discrete Sampling when Estimating Continuous--Time Diffusions," Econometrica, Econometric Society, vol. 71(2), pages 483-549, March.
  29. Yacine Ait-Sahalia, 2002. "Closed-Form Likelihood Expansions for Multivariate Diffusions," NBER Working Papers 8956, National Bureau of Economic Research, Inc.
  30. Eraker, Bjorn, 2001. "MCMC Analysis of Diffusion Models with Application to Finance," Journal of Business & Economic Statistics, American Statistical Association, vol. 19(2), pages 177-91, April.
  31. repec:cup:etheor:v:12:y:1996:i:4:p:657-81 is not listed on IDEAS
  32. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
  33. Pan, Jun, 2002. "The jump-risk premia implicit in options: evidence from an integrated time-series study," Journal of Financial Economics, Elsevier, vol. 63(1), pages 3-50, January.
  34. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
Full references (including those not matched with items on IDEAS)

This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

When requesting a correction, please mention this item's handle: RePEc:nbr:nberwo:10579. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: ()

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

If references are entirely missing, you can add them using this form.

If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

This information is provided to you by IDEAS at the Research Division of the Federal Reserve Bank of St. Louis using RePEc data.