Annals of Statistics

A Fourier transform method for nonparametric estimation of multivariate volatility

Paul Malliavin and Maria Elvira Mancino

Full-text: Open access


We provide a nonparametric method for the computation of instantaneous multivariate volatility for continuous semi-martingales, which is based on Fourier analysis. The co-volatility is reconstructed as a stochastic function of time by establishing a connection between the Fourier transform of the prices process and the Fourier transform of the co-volatility process. A nonparametric estimator is derived given a discrete unevenly spaced and asynchronously sampled observations of the asset price processes. The asymptotic properties of the random estimator are studied: namely, consistency in probability uniformly in time and convergence in law to a mixture of Gaussian distributions.

Article information

Ann. Statist., Volume 37, Number 4 (2009), 1983-2010.

First available in Project Euclid: 18 June 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62F12: Asymptotic properties of estimators 42A38: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 62P20: Applications to economics [See also 91Bxx]

Continuous semi-martingale instantaneous co-volatility nonparametric estimation Fourier transform high frequency data


Malliavin, Paul; Mancino, Maria Elvira. A Fourier transform method for nonparametric estimation of multivariate volatility. Ann. Statist. 37 (2009), no. 4, 1983--2010. doi:10.1214/08-AOS633.

Export citation


  • [1] Aït-Sahalia, Y. (1996). Nonparametric pricing of interest rate derivative securities. Econometrica 64 527–560.
  • [2] Aït-Sahalia, Y. and Mykland, P. A. (2003). The effects of random and discrete sampling when estimating continuous-time diffusions. Econometrica 71 483–549.
  • [3] Aït-Sahalia, Y., Mykland, P. A. and Zhang, L. (2005). How often to sample a continuous time process in the presence of market microstructure noise. Rev. Financ. Stud. 18 351–416.
  • [4] Andersen, T., Bollerslev, T. and Diebold, F. (2002). Parametric and nonparametric volatility measurement. In Handbook of Financial Econometrics. (L. P. Hansen and Y. Ait-Sahalia, eds.) North-Holland, Amsterdam. Forthcoming. Available at
  • [5] Andersen, T., Bollerslev, T., Diebold, F. and Labys, P. (2001). The distribution of exchange rate volatility. J. Amer. Statist. Assoc. 96 42–55.
  • [6] Andersen, T., Bollerslev, T. and Meddahi, N. (2008). Realized volatility forecasting and market microstructure noise. Working Paper. Available at
  • [7] Bandi, F. M. and Phillips, P. C. B. (2003). Fully nonparametric estimation of scalar diffusion models. Econometrica 71 241–283.
  • [8] Barndorff-Nielsen, O. E. and Shephard, N. (2002). Econometric analysis of realised volatility and its use in estimating stochastic volatility models. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 253–280.
  • [9] Barndorff-Nielsen, O. E., Graversen, S. E., Jacod, J. and Shephard, N. (2006). Limit theorems for bipower variation in financial econometrics. Econometric Theory 22 677–719.
  • [10] Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A. and Shephard, N. (2008). Designing realised kernels to measure ex-post variation of equity prices in the presence of noise. Econometrica 76 1481–1536.
  • [11] Barucci, E., Malliavin, P., Mancino, M. E., Renò, R. and Thalmaier, A. (2003). The price-volatility feedback rate: An implementable mathematical indicator of market stability. Math. Finance 13 17–35.
  • [12] Barucci, E., Mancino, M. E. and Magno, D. (2008). Fourier volatility forecasting with high frequency data and microstructure noise. Working paper. Available at
  • [13] Barucci, E. and Renò, R. (2002). On measuring volatility and the GARCH forecasting performance. J. International Financial Markets, Institutions and Money 12 183–200.
  • [14] Comte, F. and Renault, E. (1998). Long memory in continuous time stochastic volatility models. Math. Finance 8 291–323.
  • [15] Corsi, F., Zumbach, G., Muller, U. and Dacorogna, M. (2001). Consistent high-precision volatility from high-frequency data. Economic Notes 30 183–204.
  • [16] Epps, T. (1979). Comovements in stock prices in the very short run. J. Amer. Statist. Assoc. 74 291–298.
  • [17] Florens-Zmirou, D. (1993). On estimating the diffusion coefficient from discrete observations. J. Appl. Probab. 30 790–804.
  • [18] Foster, D. P. and Nelson, D. B. (1996). Continuous record asymptotics for rolling sample variance estimators. Econometrica 64 139–174.
  • [19] Genon-Catalot, V., Laredo, C. and Picard, D. (1992). Nonparametric estimation of the diffusion coefficient by wavelet methods. Scand. J. Statist. 19 317–335.
  • [20] Hayashi, T. and Yoshida, N. (2005). On covariance estimation of nonsynchronously observed diffusion processes. Bernoulli 11 359–379.
  • [21] Hansen, P. R. and Lunde, A. (2005). A forecast comparison of volatility models: Does anything beat a GARCH (1, 1)? J. Appl. Econometrics 20 873–889.
  • [22] Hansen, P. R. and Lunde, A. (2006). Realized variance and market microstructure noise (with discussions). J. Bus. Econom. Statist. 24 127–218.
  • [23] Hoshikawa, T., Kanatani, T., Nagai, K. and Nashiyama, Y. (2008). Nonparametric estimation methods of integrated multivariate volatilities. Econometric Rev. 27 112–138.
  • [24] Iori, G. and Mattiussi, V. (2007). Currency futures volatility during the 1997 East Asian crisis: An application of Fourier analysis. In Debt, Risk and Liquidity in Futures Markets (B. A. Goss, ed.). Routledge, London. Available at
  • [25] Jacod, J. (1994). Limit of random measures associated with the increments of a Brownian semimartingale. Technical report, Univ. Paris VI.
  • [26] Jacod, J. (2000). Nonparametric kernel estimation of the coefficient of a diffusion. Scand. J. Statist. 27 83–96.
  • [27] Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267–307.
  • [28] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
  • [29] Kanatani, T. (2004). Integrated volatility measuring from unevenly sampled observations. Economics Bullettin 3 1–8.
  • [30] Malliavin, P. (1995). Integration and Probability. Graduate Texts in Mathematics 157. Springer, New York.
  • [31] Malliavin, P. (1997). Stochastic Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 313. Springer, Berlin.
  • [32] Malliavin, P. and Mancino, M. E. (2002). Fourier series method for measurement of multivariate volatilities. Finance Stoch. 6 49–61.
  • [33] Malliavin, P. and Mancino, M. E. (2002). Instantaneous liquidity rate, its econometric measurement by volatility feedback. C. R. Acad. Sci. Paris 334 505–508.
  • [34] Malliavin, P., Mancino, M. E. and Recchioni, M. C. (2007). A nonparametric calibration of the HJM geometry: An application of Itô calculus to financial statistics. Japan. J. Math. 2 55–77.
  • [35] Malliavin, P. and Thalmaier, A. (2005). Stochastic Calculus of Variations in Mathematical Finance. Springer, Berlin.
  • [36] Mancino, M. E. and Sanfelici, S. (2008). Robustness of the Fourier estimator of integrated volatility in presence of microstructure noise. Computational Statistics and Data Analysis 52 2966–2989.
  • [37] Mykland, P. and Zhang, L. (2006). Anova for diffusions. Ann. Statist. 34 1931–1963.
  • [38] Nielsen, M. O. and Frederiksen, P. H. (2008). Finite sample accuracy and choice of sampling frequency in integrated volatility estimation. J. Empirical Finance 15 265–286.
  • [39] Voev, V. and Lunde, A. (2007). Integrated covariance estimation using high-frequency data in the presence of noise. J. Financial Econometrics 5 68–104.
  • [40] Zamansky, M. (1949). Classes de saturation de certains procédés d’approximation des séries de Fourier des fonctions continues et applications à quelques problèmes d’approximation. Ann. Sci. École. Norm. Sup. (3) 66 19–93.
  • [41] Zhang, L. (2006). Efficient estimation of stochastic volatility using noisyobservations: A multi-scale approach. Bernoulli 12 1019–1043.
  • [42] Zhang, L. (2008). Estimating covariation: Epps effect, microstructure noise. Available at
  • [43] Zhang, L., Mykland, P. and Aït-Sahalia, Y. (2005). A tale of two time scales: Determining integrated volatility with noisy high frequency data. J. Amer. Statist. Assoc. 100 1394–1411.
  • [44] Zhou, B. (1996). High frequency data and volatility in foreign-exchange rates. J. Business and Economic Statistics 14 45–52.