Finance and Stochastics

, 14:49 | Cite as

Nonparametric estimation for a stochastic volatility model

  • F. Comte
  • V. Genon-Catalot
  • Y. Rozenholc


Consider discrete-time observations (X δ )1≤n+1 of the process X satisfying \(dX_{t}=\sqrt{V_{t}}dB_{t}\) , with V a one-dimensional positive diffusion process independent of the Brownian motion B. For both the drift and the diffusion coefficient of the unobserved diffusion V, we propose nonparametric least square estimators, and provide bounds for their risk. Estimators are chosen among a collection of functions belonging to a finite-dimensional space whose dimension is selected by a data driven procedure. Implementation on simulated data illustrates how the method works.


Diffusion coefficient Drift Mean square estimator Model selection Nonparametric estimation Penalized contrast Stochastic volatility 

Mathematics Subject Classification (2000)

62G08 62M05 62P05 

JEL Classification

C14 C87 


  1. 1.
    Aït-Sahalia, Y., Kimmel, R.: Maximum likelihood estimation of stochastic volatility models. J. Financ. Econ. 83, 413–452 (2007) CrossRefGoogle Scholar
  2. 2.
    Baraud, Y., Comte, F., Viennet, G.: Model selection for (auto-)regression with dependent data. ESAIM Probab. Stat. 5, 33–49 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Barron, A.R., Birgé, L., Massart, P.: Risk bounds for model selection via penalization. Probab. Theory Relat. Fields 113, 301–413 (1999) zbMATHCrossRefGoogle Scholar
  4. 4.
    Comte, F., Genon-Catalot, V., Rozenholc, Y.: Penalized nonparametric mean square estimation of the coefficients of diffusion processes. Bernoulli 13, 514–543 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Comte, F., Genon-Catalot, V., Rozenholc, Y.: Nonparametric adaptive estimation for integrated diffusions. Stoch. Process. Appl. 119, 811–834 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Comte, F., Rozenholc, Y.: A new algorithm for fixed design regression and denoising. Ann. Inst. Stat. Math. 56, 449–473 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cox, J.C., Ingersoll, J.E. Jr., Ross, S.A.: A theory of the term structure of interest rates. Econometrica 53, 385–407 (1985) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Genon-Catalot, V., Jeantheau, T., Larédo, C.: Parameter estimation for discretely observed stochastic volatility models. Bernoulli 5, 855–872 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Genon-Catalot, V., Jeantheau, T., Larédo, C.: Stochastic volatility models as hidden Markov models and statistical applications. Bernoulli 6, 1051–1079 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gloter, A.: Estimation du coefficient de diffusion de la volatilité d’un modèle à volatilité stochastique [Estimation of the volatility diffusion coefficient for a stochastic volatility model]. C. R. Acad. Sci. Paris, Sér. I Math. 330(3), 243–248 (2000) zbMATHMathSciNetGoogle Scholar
  11. 11.
    Gloter, A.: Efficient estimation of drift parameters in stochastic volatility models. Finance Stoch. 11, 495–519 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Heston, S.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6, 327–343 (1993) CrossRefGoogle Scholar
  13. 13.
    Hoffmann, M.: Adaptive estimation in diffusion processes. Stoch. Process. Appl. 79, 135–163 (1999) zbMATHCrossRefGoogle Scholar
  14. 14.
    Hoffmann, M.: Rate of convergence for parametric estimation in a stochastic volatility model. Stoch. Process. Appl. 97, 147–170 (2002) zbMATHCrossRefGoogle Scholar
  15. 15.
    Renò, R.: Nonparametric estimation of stochastic volatility models. Econ. Lett. 90, 390–395 (2006) CrossRefGoogle Scholar
  16. 16.
    Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes, and Martingales, vol. 2. Cambridge University Press, Cambridge (2000). Reprint of the second (1994) edition Google Scholar
  17. 17.
    Zhang, L., Mykland, P.A., Aït-Sahalia, Y.: A tale of two time scales: determining integrated volatility with noisy high-frequency data. J. Am. Stat. Assoc. 100, 1394–1411 (2005) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.MAP5 UMR 8145Université Paris DescartesParis cedex 06France

Personalised recommendations