Statistical Papers

, Volume 40, Issue 3, pp 297–321 | Cite as

Volatility and GMM — Monte Carlo studies and empirical estimations

  • Hartmut Nagel
  • Rainer Schöbel


In this article we examine small sample properties of a generalized method of moments (GMM) estimation using Monte Carlo simulations. We assume that the generated time series describe the stochastic variance rate of a stock index. we use a mean reverting square-root process to simulate the dynamics of this instantaneous variance rate. The time series obtained are used to estimate the parameters of the assumed variance rate process by applying GMM. Our results are described and compared to estimates from empirical data which consist of volatility as well as daily volume data of the German stock market. One of our main findings is that estimates of the mean reverting parameter that are not significantly different from zero do not necessarily imply a rejection of the hypothesis of a mean reverting behavior of the underlying stochastic process.


Time Series Volatility Stochastic Volatility Stock Index Longe Time Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Andrews, D. (1991). Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation.Econometrica, 59, 3, 817–858.CrossRefMathSciNetMATHGoogle Scholar
  2. Bühler, A., and A. Grünbichler (1996): Derivative Produkte auf Volatilität: Zum empirischen Verhalten eines Volatilitätsindex.Zeitschrift für Betriebswirtschaft, 66, 5, 607–625.Google Scholar
  3. Chan, K.C., G. A. Karolyi, F. A. Longstaff, and A. B. Sanders (1992). An Empirical Comparison of Alternative Models of the Short-Term Interest Rate.Journal of Finance, 47, 3, 1209–1227.CrossRefGoogle Scholar
  4. Cox, J. C., J. E. Ingersoll, and S. A. Ross (1985). A theory of the term structure of interest rates.Econometrica, 53, 2, 385–407.CrossRefMathSciNetGoogle Scholar
  5. Duffie, D. (1992). Dynamic Asset Pricing Theory. Princeton University Press, Oxford.Google Scholar
  6. Hansen, L. P. (1982). Large Sample Proporties of Generalized Method of Moments Estimators.Econometrica, 50, 4, 1029–1054.CrossRefMathSciNetMATHGoogle Scholar
  7. Heston, S. L. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options.The Review of Financial Studies 6, 327–343.CrossRefGoogle Scholar
  8. Jones, C. M., G. Kaul, and M. L. Lipson (1994). Transactions, Volume, and Volatility.The Review of Financial Studies, 7, 4, 631–651.CrossRefGoogle Scholar
  9. Newey, W. K. and K. D. West (1987). A Simple Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix.Econometrica, 55, 3, 703–708.CrossRefMathSciNetMATHGoogle Scholar
  10. Redelberger, T. (1994). Grundlagen und Konstruktion des VDAX-Volatilitätsindex der Deutsche Börse AG. Deutsche Börse AG, Frankfurt/Main.Google Scholar
  11. Stein, E. M., and J. C. Stein (1991): Stock Price Distributions with Stochastic Volatility: An Analytic Approach.The Review of Financial Studies, 4, 727–752.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1999

Authors and Affiliations

  • Hartmut Nagel
    • 1
  • Rainer Schöbel
    • 1
  1. 1.Wirtschaftswissenschaftliches Seminar Lehrstuhl für Betriebliche FinanzwirtschaftUniversität TübingenTübingenGermany

Personalised recommendations