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Moment–Based Estimation of Stochastic Volatility Models

  • Eric RenaultEmail author
Chapter

Abstract

This chapter reviews the possible uses of the Generalized Method of Moments (GMM) to estimate Stochastic Volatility (SV) models. A primary attraction of the method of moments technique is that it is well suited for identifying and estimating volatility models without a complete parametric specification of the probability distributions. Moreover, simulation-based methods of moments are able to exploit a variety of moments, while avoiding limitations due to a lack of closed form expressions. The chapter first highlights the suitability of GMM for popular regression models of volatility forecasting. Then, it reviews the implications of the SV model specification in terms of higher order moments: skewness, kurtosis, variance of the variance, leverage and feedback effects. The chapter examines the ability of a continuous time version of SV models to accommodate data from other sources like option prices or high frequency data on returns and transactions dates. Simulation-based methods are particularly useful for studying continuous time models due to the frequent lack of closed form expressions for their discrete time dynamics. These simulation-based methods of moments are presented within the unifying framework of indirect inference with a special emphasis on misspecification. Likely misspecification of the parametric model used for simulation requires a parsimonious and well-focused choice of the moments to match.

Keywords

Option Price Stochastic Volatility Implied Volatility Empirical Likelihood Stochastic Volatility Model 
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.

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References

  1. Aguilar, M. and Renault, E. (2008): A Moment-based Test for GARCH against Stochastic Volatility. Working Paper UNC. Google Scholar
  2. Altonji, J.G. and Segal, L.M. (1996): Small Sample Bias in GMM Estimation of Covariance Structures. Journal of Business and Economic Statistics 14, 353–366.CrossRefGoogle Scholar
  3. Andersen, T. G. (1994): Stochastic Autoregressive Volatility: A Framework for Volatility Modeling. Mathematical Finance 4, 75–102.zbMATHGoogle Scholar
  4. Andersen, T.G. and Bollerslev, T. (1997): Intraday Periodicity and Volatility Persistence in Financial Markets. Journal of Empirical Finance 4, 115–158.CrossRefGoogle Scholar
  5. Andersen, T.G., Bollerslev, T. and Diebold, F.X. (2006): Parametric and Nonparametric Volatility Measurement. In: Aït-Sahalia, Y. and Hansen, L.P. (Eds.): Handbook of Financial Econometrics. Elsevier Science B.V., Amsterdam forthcoming.Google Scholar
  6. Andersen, T.G., Bollerslev, T. and Diebold, F.X. (2007): Roughing It Up: Including Jump Components in Measuring, Modeling and Forecasting Asset Return Volatility. Review of Economics and Statistics 89, 701–720.CrossRefGoogle Scholar
  7. Andersen, T., Bollerslev, T., Diebold, F.X. and Ebens, H. (2001): The distribution of stock return volatility. Journal of Financial Economics 61, 43–76.CrossRefGoogle Scholar
  8. Andersen, T., Bollerslev, T., Diebold, F.X. and Labys, P. (2001): The distribution of exchange rate volatility. Journal of American Statistical Association 96, 42–55.zbMATHCrossRefMathSciNetGoogle Scholar
  9. Andersen, T.G., Bollerslev, T., Diebold, F.X. and Labys, P. (2003): Modeling and forecasting realized volatility. Econometrica 71, 579–626.zbMATHCrossRefMathSciNetGoogle Scholar
  10. Andersen T.G., Bollerslev T., Diebold F.X. and Vega C. (2003): Micro Effects of Macro Announcements: Real-Time Price Discovery in Foreign Exchange. The American Economic Review 93, 38–62.CrossRefGoogle Scholar
  11. Andersen T.G., Bollerslev T., Diebold F.X. and Vega C. (2007): Real-time price discovery in global stock, bond and foreign exchange markets. Journal of International Economics 73, 251–277.CrossRefGoogle Scholar
  12. Andersen T.G., Bollerslev T., and Meddahi N. (2005): Correcting the Errors: Volatility Forecast Evaluation Using High-Frequency Data and Realized Volatilities. Econometrica 73, 279–296zbMATHCrossRefMathSciNetGoogle Scholar
  13. Andersen, T.G. and Lund, J. (1997): Estimating continuous–time stochastic volatility models. Journal of Econometrics 77, 343–379.zbMATHCrossRefGoogle Scholar
  14. Andersen, T.G. and Sorensen, B.E. (1996): GMM estimation of a stochastic volatility model: a Monte Carlo study. Journal of Business and Economic Statistics 14, 328–352.CrossRefGoogle Scholar
  15. Andreou, E. and Ghysels, E. (2002): Rolling–Sample Volatility Estimators: Some New Theoretical Simulation and Empirical Results. Journal of Business and Economic Statistics 20, 363–376.CrossRefMathSciNetGoogle Scholar
  16. Andrews, D. (1993): Exactly median unbiased estimation of first order autoregressive/unit root models. Econometrica 61, 139–65.zbMATHCrossRefMathSciNetGoogle Scholar
  17. Antoine B., Bonnal H. and Renault E. (2007): On the Efficient Use of the Informational Content of Estimating Equations: Implied Probabilities and Euclidean Empirical Likelihood. Journal of Econometrics 138, 461–487.CrossRefMathSciNetGoogle Scholar
  18. Barndorff–Nielsen, O.E and Shephard, N. (2001): Non–Gaussian Ornstein–Uhlenbeck–based models and some of their uses in financial economics. Journal of the Royal Statistical Society Series B 63, 167–241.zbMATHCrossRefMathSciNetGoogle Scholar
  19. Barndorff–Nielsen, O.E. and Shephard, N. (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.zbMATHCrossRefMathSciNetGoogle Scholar
  20. Barndorff–Nielsen, O.E and Shephard, N. (2004): Power and bipower variation with stochastic jumps. Journal of Financial Econometrics 2, 1–48.CrossRefGoogle Scholar
  21. Barndorff–Nielsen, O.E. and Shephard, N. (2007): Variations, jumps, market frictions and high frequency data in financial econometrics. In: Blundell, R., Torsten, P. and Newey, W.K. (Eds.): Advances in Economics and Econometrics, Theory and Applications, Ninth World Congress. Econometric Society Monographs, Cambridge University Press.Google Scholar
  22. Black, F. (1976): The pricing of commodity contracts. Journal of Financial Economics 3, 167–79.CrossRefGoogle Scholar
  23. Black, F. and Scholes, M. (1973): The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–659.CrossRefGoogle Scholar
  24. Bollerslev T,, (1986): Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics 31, 307–327.zbMATHCrossRefMathSciNetGoogle Scholar
  25. Bollerslev, T., Gibson, M. and Zhou, H. (2004): Dynamic estimation of volatility risk premia and investor risk aversion from option–implied and realized volatilities. Finance and Economics Discussion Series, Divisions of Research and Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C..Google Scholar
  26. Bollerslev, T., Litvinova, J. and Tauchen, G. (2006): Leverage and Volatility Feedback Effects in High–Frequency Data. Journal of Financial Econometrics 4, 353–384.CrossRefGoogle Scholar
  27. Bollerslev, T. and Zhou, H. (2002): Estimating stochastic volatility diffusion using conditional moments of the integrated volatility. Journal of Econometrics 109, 33–65.zbMATHCrossRefMathSciNetGoogle Scholar
  28. Carnero, M.A, Pena, D. and Ruiz, E. (2004): Persistence and Kurtosis in GARCH and Stochastic Volatility Models. Journal of Financial Econometrics 2, 319–342.CrossRefGoogle Scholar
  29. Carrasco, M. and Florens, J.P. (2002): Simulation based method of moments and efficiency. Journal of Business and Economic Statistics 20, 482–492.CrossRefMathSciNetGoogle Scholar
  30. Carrasco M, Chernov M., Florens J.P. and Ghysels E. (2007): Efficient Estimation of Jump Diffusions and General Dynamic Models with a Continuum of Moment Conditions. Journal of Econometrics 140, 529–573.CrossRefMathSciNetGoogle Scholar
  31. Chen, X. and Ghysels, E. (2008): News – Good or Bad – and its impact on volatility predictions over multiple horizons. Discussion Paper, UNC. Google Scholar
  32. Chernov, M. and Ghysels, E. (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
  33. Clark, P.K. (1973): A subordinated stochastic process with fixed variance for speculative prices. Econometrica 41, 135–156.zbMATHCrossRefMathSciNetGoogle Scholar
  34. Corradi, V. and Distaso, W. (2006): Semiparametric comparison of stochastic volatility models using realized measures. Review of Economic Studies 73, 635–667.zbMATHCrossRefMathSciNetGoogle Scholar
  35. Cox, D., Ingersoll, J.E. and Ross, S. (1985): A theory of the Term Structure of Interest Rates. Econometrica 53, 385–407.CrossRefMathSciNetGoogle Scholar
  36. Diebold, F.X. and Nerlove, M. (1989): The Dynamics of Exchange Rate Volatility: A Multivariate Latent Factor ARCH Model. Journal of Applied Econometrics 4, 1–22.CrossRefGoogle Scholar
  37. Doz, C. and Renault, E. (2006): Factor Stochastic Volatility in Mean Models: a GMM approach. Econometric Reviews 25, 275–309.zbMATHCrossRefMathSciNetGoogle Scholar
  38. Dridi, R., Guay, A. and Renault, E. (2007): Indirect Inference and Calibration of Dynamic Stochastic General Equilibrium Models. Journal of Econometrics 136, 397–430.CrossRefMathSciNetGoogle Scholar
  39. Drost, F.C. and Nijman, T.E. (1993): Temporal aggregation of GARCH processes. Econometrica 61, 909–927.zbMATHCrossRefMathSciNetGoogle Scholar
  40. Drost, F. and Werker, B. (1996): Closing the GARCH gap: continuous time GARCH modeling. Journal of Econometrics 74, 31–58.zbMATHCrossRefMathSciNetGoogle Scholar
  41. Duffie, D., Pan, J. and Singleton, K. (2000): Transform Analysis and Asset Pricing for Affine Jump-Diffusions. Econometrica 68, 1342–1376.CrossRefMathSciNetGoogle Scholar
  42. Duffie, D. and Singleton, K. (1993): Simulated moments estimation of Markov models of asset prices. Econometrica 61, 929–952.zbMATHCrossRefMathSciNetGoogle Scholar
  43. Engle, R.F. (1982): Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation. Econometrica 50, 987–1008.zbMATHCrossRefMathSciNetGoogle Scholar
  44. Engle, R.F. (1995): ARCH: Selected Readings. Oxford University Press.Google Scholar
  45. Engle, R.F. (2000): The Econometrics of Ultra-High Frequency Data. Econometrica 68, 1–22.zbMATHCrossRefGoogle Scholar
  46. Engle, R. F., Ghysels, E. and Sohn, B. (2006): On the Economic Sources of Volatility. Discussion Paper NYU and UNC. Google Scholar
  47. Engle, R. F. and Lee, G. G. J. (1999): A Permanent and Transitory Component Model of Stock Retrun Volatility. In: Engle, R. F. and White, H. (Eds.): Cointegration, Causality, and Forecasting: A Festschrift in Honour of Clive W. J. Granger, 475–497. Oxford University Press.Google Scholar
  48. Engle, R.F., Lilien, D.M and Robins, R.P. (1987): Estimating Time-Varying Risk Premia in the Term Structure: the ARCH-M Model. Econometrica 55, 391–407.CrossRefGoogle Scholar
  49. Forsberg, L. and Ghysels, E. (2007): Why Do Absolute Returns Predict Volatility So Well. Journal of Financial Econometrics 5, 31–67.CrossRefGoogle Scholar
  50. Francq, C. and Zakoïan, J.M. (2000): Estimating weak GARCH representations. Econometric Theory 16, 692–728.zbMATHCrossRefMathSciNetGoogle Scholar
  51. Francq, C. and Zakoïan, J.M. (2004): Maximum Likelihood Estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10, 605–637.zbMATHCrossRefMathSciNetGoogle Scholar
  52. Francq, C. and Zakoïan, J.M. (2006): Linear-Representation Based Estimation of Stochastic Volatility Models. Scandinavian Journal of Statistics 33, 785–806.zbMATHCrossRefGoogle Scholar
  53. Franses, P.H., Van der Leij, M. and Paap, R. (2008): A Simple Test for GARCH against a Stochastic Volatility Model. Journal of Financial Econometrics 6, 291–306CrossRefGoogle Scholar
  54. French, K.F, Schwert, G.W. and Stambaugh, R.F. (1987): Expected Stock Returns and Volatility. Journal of Financial Economics 19, 3–29.CrossRefGoogle Scholar
  55. Gagliardini, P., Gouriéroux, C. and Renault, E. (2007): Efficient Derivative Pricing by Extended Method of Moments. Working Paper UNC. Google Scholar
  56. Gallant, A.R., Hsieh, D. and Tauchen, G. (1997): Estimation of Stochastic volatility Models with Diagnostics. Journal of Econometrics 81, 159–192.zbMATHCrossRefGoogle Scholar
  57. Gallant, A.R. and Tauchen, G. (1996): Which Moments to Match. Econometric Theory 12, 657–681.CrossRefMathSciNetGoogle Scholar
  58. Garcia, R., Lewis, M.A., Pastorello, S. and Renault, E. (2007): Estimation of Objective and Risk Neutral Distributions based on Moments of Integrated Volatility. Journal of Econometrics, forthcoming.Google Scholar
  59. Ghysels, E., Harvey, A. and Renault, E. (1996): Stochastic Volatility. In: Maddala, G.S. and Rao, C. R. (Eds.): Handbook of Statistics 14. Elsevier Science B.V.Google Scholar
  60. Ghysels, E., and Jasiak J. (1998): GARCH for irregularly spaced financial data: The ACD-GARCH model. Studies in Nonlinear Dynamics and Econometrics 2, 133–149.CrossRefGoogle Scholar
  61. Ghysels, E., Mykland, P. and Renault, E. (2007): In-Sample Asymptotics and Across-Sample Efficiency Gains for High Frequency Data Statistics. Working Paper.Google Scholar
  62. Ghysels, E., Santa–Clara, P. and Valkanov, R. (2005): There is a Risk–Return Tradeoff After All. Journal of Financial Economics 76, 509–548.CrossRefGoogle Scholar
  63. Ghysels, E., Santa–Clara, P. and Valkanov, R. (2006): Predicting Volatility: Getting the Most out of Return Data Sampled at Different Frequencies. Journal of Econometrics 131, 59–95.CrossRefMathSciNetGoogle Scholar
  64. Ghysels, E. and Sinko, A. (2006): Comments on Hansen and Lunde. Journal of Business and Economic Statistics 24, 192–194.CrossRefGoogle Scholar
  65. Ghysels, E., Sinko, A. and Valkanov, R. (2006): MIDAS Regressions: Further Results and New Directions. Econometric Reviews 26, 53–90.CrossRefMathSciNetGoogle Scholar
  66. Gourieroux, C. and Jasiak, J. (2006): Autoregressive Gamma Processes. Journal of Forecasting 25, 129–152.CrossRefMathSciNetGoogle Scholar
  67. Gouriéroux, C. and Monfort, A. (1992): Testing, encompassing, and simulating dynamic econometric models. Econometric Theory 11, 195–228.CrossRefGoogle Scholar
  68. Gouriéroux, C. and Monfort, A. (1996): Simulation–based econometric methods. Core Lectures, Oxford University Press.Google Scholar
  69. Gouriéroux C., Monfort, A. and Renault, E. (1993): Indirect Inference. Journal of Applied Econometrics 8, S85–S118.CrossRefGoogle Scholar
  70. Gouriéroux, C., Monfort, A. and Trognon, A. (1984): Pseudo–maximum likelihood methods theory. Econometrica 52, 681–700.zbMATHCrossRefMathSciNetGoogle Scholar
  71. Gouriéroux, C., Renault, E. and Touzi, N. (2000): Calibration by simulation for small sample bias correction. In: Mariano, R., Schuerman, T. and Weeks, M.J. (Eds.): Simulation–Based Inference in Econometrics. Cambridge University Press.Google Scholar
  72. Granger, C.W.J. and Newbold, P. (1976): Forecasting transformed series. Journal of the Royal Statistical Society Series B 38, 189–203.zbMATHMathSciNetGoogle Scholar
  73. Hall A.R. (2005): Generalized Method of Moments Oxford University Press, Oxford.zbMATHGoogle Scholar
  74. Hansen, L.P., Heaton, J.C. and Ogaki, M. (1988): Efficiency Bounds Implied by Multiperiod Conditional Moment Restrictions. Journal of the American Statistical Association 83, 863–871.zbMATHCrossRefMathSciNetGoogle Scholar
  75. Hansen, L.P., Heaton, J. and Yaron, A. (1996): Finite–Sample Properties of Some Alternative GMM Estimators. Journal of Business and Economic Statistics 14, 262–280.CrossRefGoogle Scholar
  76. 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
  77. Ingram, B.F. and Lee, B.S. (1991): Estimation by simulation of time series models. Journal of Econometrics 47, 197–207.zbMATHCrossRefMathSciNetGoogle Scholar
  78. Jacod J. (1994): Limit of random measures associated with the increments of a Brownian semimartingale. Preprint number 120, Laboratoire de probabilites Université Pierre et Marie Curie, Paris.Google Scholar
  79. Jacod J. (1997): On continuous conditional Gaussian martingales and stable convergence in law. In: Seminaire Probability, Lecture Notes in Mathematics 1655, 232–246. Springer Verlag.CrossRefMathSciNetGoogle Scholar
  80. Jacod J. and Shirayev, A.N. (1987): Limit Theorems for Stochastic Processes. Springer Verlag.Google Scholar
  81. Jacquier E., Polson, N.G. and Rossi, P.E. (1994): Bayesian analysis of stochastic volatility models (with discussion): Journal of Business and Economic Statistics 12, 371–417.CrossRefGoogle Scholar
  82. Ji, C., Renault, E. and Yoon, J. (2008): An Approximation Scheme for Option Pricing with Stochastic Volatility and Jumps. Working Paper. Google Scholar
  83. Jiang, G. and Tian, Y. (2005): Model–free implied volatility and its information content. Review of Financial Studies 18, 1305–1342.CrossRefGoogle Scholar
  84. Kim, S., Shephard, N. and Chib, S. (1998): Stochastic volatility: likelihood inference and comparison with ARCH models. Review of Economic Studies 45, 361–393.CrossRefGoogle Scholar
  85. Kobayashi, M. and Shi, X. (2005): Testing for EGARCH against Stochastic Volatility Models. Journal of Time Series Analysis 26, 135–150.zbMATHCrossRefMathSciNetGoogle Scholar
  86. Meddahi, N. (2001): An eigenfunction approach for volatility modeling. Working Paper 29–2001, CRDE, Université de Montréal.Google Scholar
  87. Meddahi, N. (2002): Moments of continuous time stochastic volatility models. Working Paper, Université de Montréal. Google Scholar
  88. Meddahi, N. and Renault, E. (2004): Temporal Aggregation of Volatility Models. Journal of Econometrics 119, 355–379.CrossRefMathSciNetGoogle Scholar
  89. Meddahi, N., Renault, E. and Werker, B. (2006): GARCH and Irregularly Spaced Data. Economic Letters 90, 200–204.CrossRefMathSciNetGoogle Scholar
  90. Melino, A. and Turnbull, S.M. (1990): Pricing foreign currency options with stochastic volatility. Journal of Econometrics 45, 239–265.CrossRefGoogle Scholar
  91. Monfort, A. (1996): A reappraisal of misspecified econometric models. Econometric Theory 12, 597–619.CrossRefMathSciNetGoogle Scholar
  92. Nelson, D.B. (1991): Conditional heteroskedasticity in asset pricing: a new approach. Econometrica 59, 347–370.zbMATHCrossRefMathSciNetGoogle Scholar
  93. Newey, W.K. and Smith, R.J. (2004): Higher Order Properties of GMM and Generalized Empirical Likelihood Estimators. Econometrica, 72, 219–255.zbMATHCrossRefMathSciNetGoogle Scholar
  94. Pastorello, S., Renault, E. and Touzi, N. (2000): Statistical inference for random–variance option pricing. Journal of Business and Economic Statistics 18, 358–367.CrossRefGoogle Scholar
  95. Renault, E. and Werker, B. (2008): Causality Effects in Return Volatility Measures with Random Times. Working Paper.Google Scholar
  96. Rosenberg, B. (1972): The behaviour of random variables with nonstationary variance and the distribution of security prices. WP11 GSBA, University of California, Berkeley.Google Scholar
  97. Ruiz, E. (1994): Quasi-Maximum Likelihood Estimation of Stochastic Volatility Models. Journal of Econometrics 63, 284–306.CrossRefGoogle Scholar
  98. Schwert, G.W. (1989): Why Does Stock Market Volatility Change Over Time? Journal of Finance 44, 1207–1239.CrossRefGoogle Scholar
  99. Shephard, N. (2005): Stochastic Volatility: Selected Readings. Oxford University Press, Oxford.zbMATHGoogle Scholar
  100. Smith, A.A. (1993): Estimating nonlinear time series models using simulated autoregressions. Journal of Applied Econometrics 8, S63–S84.CrossRefGoogle Scholar
  101. Taylor, S. J. (1986): Modelling Financial Time Series. Wiley, Chichester.zbMATHGoogle Scholar
  102. White, H. (1982): Maximum likelihood estimation of mis–specified models. Econometrica 50, 1–25.zbMATHCrossRefMathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.University of North CarolinaDepartment of EconomicsGardner HallNC

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