• Bernoulli
  • Volume 5, Number 5 (1999), 855-872.

Parameter estimation for discretely observed stochastic volatility models

Valentine Genon-Catalot, Thierry Jeantheau, and Catherine Laredo

Full-text: Open access


This paper deals with parameter estimation for stochastic volatility models. We consider a two-dimensional diffusion process (Yt,Vt). Only (Yt) is observed at n discrete times with a regular sampling interval. The unobserved coordinate (Vt) rules the diffusion coefficient (volatility) of (Yt) and is an ergodic diffusion depending on unknown parameters. We build estimators of the parameters present in the stationary distribution of (Vt), based on appropriate functions of the observations. Consistency is proved under the asymptotic framework that the sampling interval tends to 0, while the number of observations and the length of the observation time tend to infinity. Asymptotic normality is obtained under an additional condition on the rate of convergence of the sampling interval. Examples of models from finance are treated, and numerical simulation results are given.

Article information

Bernoulli, Volume 5, Number 5 (1999), 855-872.

First available in Project Euclid: 12 February 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

diffusion processes discrete time observations mathematical finance parametric inference stochastic volatility


Genon-Catalot, Valentine; Jeantheau, Thierry; Laredo, Catherine. Parameter estimation for discretely observed stochastic volatility models. Bernoulli 5 (1999), no. 5, 855--872.

Export citation


  • [1] Barndorff-Nielsen, O.E. (1978) Hyperbolic distribution and distribution on hyperbolae. Scand. J. Statist., 5, 151-157.
  • [2] Barndorff-Nielsen, O.E. and Sørensen, M. (1994) A review of some aspects of asymptotic likelihood theory for stochastic processes. Int. Statist. Rev., 62, 133-165.
  • [3] Bibby, B. and Sørensen, M. (1995) Martingale estimation functions for discretely observed diffusion processes. Bernoulli, 1, 17-39.
  • [4] Black, F. and Scholes, M. (1973) The pricing of options and corporate liabilities. J. Political Economy, 81, 637-659.
  • [5] Blattberg, R. and Gonedes, N. (1974) A comparison of the stable and Student distributions as statistical models for stock prices. J. Business, 47, 244-280.
  • [6] Chesney, M. and Scott, L. (1989) Pricing European currency options: A comparison of the modified Black-Scholes model and a random variance model. J. Financial Quant. Anal., 24, 267-289.
  • [7] Cox, J.C., Ingersoll, J.E. and Ross, S.A. (1985) A theory of term structure of interest rates. Econometrica, 53, 385-407.
  • [8] Dacunha-Castelle, D. and Duflo, M. (1983) Probabilités et Statistiques, Vol. 2, Problèmes à Temps Mobile. Paris: Masson.
  • [9] Dacunha-Castelle, D. and Florens-Zmirou, D. (1986) Estimation of the coefficient of a diffusion from discrete observations. Stochastics, 19, 263-284.
  • [10] Donahl, G. (1987) On estimating the diffusion coefficient. J. Appl. Probab., 24, 105-114.
  • [11] Genon-Catalot, V. and Jacod, J. (1993) On the estimation of the diffusion coefficient for multidimensional diffusion processes. Ann. Inst. Henri Poincaré Probab. Statist., 29, 119-151.
  • [12] Genon-Catalot, V., Jeantheau, T. and Larédo, C. (1998) Limit theorems for discretely observed stochastic volatility models. Bernoulli, 4, 283-303.
  • [13] Ghysels, E., Harvey, A. and Renault, E. (1996) Stochastic volatility. Handbook Statist., 14, 119-192.
  • [14] Heston S.L. (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Stud., 6, 327-343.
  • [15] Hull, J. and White, A. (1987) The pricing of options on assets with stochastic volatilities. J. Finance, 42, 281-300.
  • [16] Kessler, M. (1996) Simple and explicit estimating functions for a discretely observed diffusion process. Prépublication du Laboratoire de Probabilités de l'Université de Paris VI.
  • [17] Kessler, M. (1997) Estimation of an ergodic diffusion from discrete observations. Scand. J. Statist., 24(2), 211-229.
  • [18] Kutoyants, Yu.A. (1984) Parameter Estimation for Stochastic Processes. Berlin: Heldermann.
  • [19] Larédo, C. (1990) A sufficient condition for asymptotic sufficiency of incomplete observations of a diffusion process. Ann. Statist., 18, 1158-1171.
  • [20] Luke, Y.L. (1969) The Special Functions and Their Approximations. Math. Sci. Engng, 53. New York: Academic Press.
  • [21] Madan, D.B. and Seneta, E. (1990) The variance gamma model for share market returns. J. Business, 63, 511-524.
  • [22] Nelson, D.B. (1990) ARCH models as diffusion approximations. J. Econometrics, 45, 7-38.
  • [23] Rogers, L.C.G. and Williams, D. (1987) Diffusions, Markov Processes and Martingales. Vol. 2, Ito Calculus. Wiley Ser. Probab. Math. Statist. New York: Wiley.